2807:
3673:
337:, allowing us to find the probability of a cause given its effect. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the individual is typical of the population as a whole. Based on Bayes law both the prevalence of a disease in a given population and the error rate of an infectious disease test have to be taken into account to evaluate the meaning of a positive test result correctly and avoid the
11574:
one that appears brighter than normal on a scan. This is not a foolproof test, as an echogenic bowel can be present in a perfectly healthy fetus. Parental genetic testing is very influential in this case, where a phenotypic facet can be overly influential in probability calculation. In the case of a fetus with an echogenic bowel, with a mother who has been tested and is known to be a CF carrier, the posterior probability that the fetus actually has the disease is very high (0.64). However, once the father has tested negative for CF, the posterior probability drops significantly (to 0.16).
5023:
7607:
11590:
275:
2493:
4656:
3343:
4723:
4552:
1092:
6886:
4418:
2802:{\displaystyle {\begin{aligned}P({\text{User}}\vert {\text{Positive}})&={\frac {P({\text{Positive}}\vert {\text{User}})P({\text{User}})}{P({\text{Positive}})}}\\&={\frac {P({\text{Positive}}\vert {\text{User}})P({\text{User}})}{P({\text{Positive}}\vert {\text{User}})P({\text{User}})+P({\text{Positive}}\vert {\text{Non-user}})P({\text{Non-user}})}}\\&={\frac {0.90\times 0.05}{0.90\times 0.05+0.20\times 0.95}}={\frac {0.045}{0.045+0.19}}\approx 19\%\end{aligned}}}
3668:{\displaystyle {\begin{aligned}P({\text{Cancer}}|{\text{Symptoms}})&={\frac {P({\text{Symptoms}}|{\text{Cancer}})P({\text{Cancer}})}{P({\text{Symptoms}})}}\\&={\frac {P({\text{Symptoms}}|{\text{Cancer}})P({\text{Cancer}})}{P({\text{Symptoms}}|{\text{Cancer}})P({\text{Cancer}})+P({\text{Symptoms}}|{\text{Non-Cancer}})P({\text{Non-Cancer}})}}\\&={\frac {1\times 0.00001}{1\times 0.00001+(10/99999)\times 0.99999}}={\frac {1}{11}}\approx 9.1\%\end{aligned}}}
5018:{\displaystyle {\begin{aligned}P({\text{Rare}}\vert {\text{Pattern}})&={\frac {P({\text{Pattern}}\vert {\text{Rare}})P({\text{Rare}})}{P({\text{Pattern}})}}\\&={\frac {P({\text{Pattern}}\vert {\text{Rare}})P({\text{Rare}})}{P({\text{Pattern}}\vert {\text{Rare}})P({\text{Rare}})+P({\text{Pattern}}\vert {\text{Common}})P({\text{Common}})}}\\&={\frac {0.98\times 0.001}{0.98\times 0.001+0.05\times 0.999}}\\&\approx 1.9\%\end{aligned}}}
4441:, probability measures a "degree of belief". Bayes' theorem links the degree of belief in a proposition before and after accounting for evidence. For example, suppose it is believed with 50% certainty that a coin is twice as likely to land heads than tails. If the coin is flipped a number of times and the outcomes observed, that degree of belief will probably rise or fall, but might even remain the same, depending on the results. For proposition
12557:
8756:
test is performed in serial testing, and that also turns out to be positive, then the posterior odds of actually having the disease becomes 10:1, which means a posterior probability of about 90.91%. The negative Bayes factor can be calculated to be 91%/(100%-90%)=9.1, so if the second test turns out to be negative, then the posterior odds of actually having the disease is 1:9.1, which means a posterior probability of about 9.9%.
43:
2941:
11383:) if she is a carrier, about 1 if she is a non-carrier (this is the Conditional Probability). The Joint Probability reconciles these two predictions by multiplying them together. The last line (the Posterior Probability) is calculated by dividing the Joint Probability for each hypothesis by the sum of both joint probabilities.
2925:. In this case, it says that the probability that someone tests positive is the probability that a user tests positive, times the probability of being a user, plus the probability that a non-user tests positive, times the probability of being a non-user. This is true because the classifications user and non-user form a
11275:, in order to predict whether an individual will develop a disease or pass one on to their children. Genetic testing and prediction is a common practice among couples who plan to have children but are concerned that they may both be carriers for a disease, especially within communities with low genetic variance.
10779:
10056:
3773:
A factory produces items using three machines—A, B, and C—which account for 20%, 30%, and 50% of its output respectively. Of the items produced by machine A, 5% are defective; similarly, 3% of machine B's items and 1% of machine C's are defective. If a randomly selected item is defective, what is the
11573:
Bayesian analysis can be done using phenotypic information associated with a genetic condition, and when combined with genetic testing this analysis becomes much more complicated. Cystic fibrosis, for example, can be identified in a fetus through an ultrasound looking for an echogenic bowel, meaning
11577:
Risk factor calculation is a powerful tool in genetic counseling and reproductive planning, but it cannot be treated as the only important factor to consider. As above, incomplete testing can yield falsely high probability of carrier status, and testing can be financially inaccessible or unfeasible
11270:
In genetics, Bayes' rule can be used to estimate the probability of an individual having a specific genotype. Many people seek to approximate their chances of being affected by a genetic disease or their likelihood of being a carrier for a recessive gene of interest. A Bayesian analysis can be done
8759:
The example above can also be understood with more solid numbers: Assume the patient taking the test is from a group of 1000 people, where 91 of them actually have the disease (prevalence of 9.1%). If all these 1000 people take the medical test, 82 of those with the disease will get a true positive
3259:
Even if 100% of patients with pancreatic cancer have a certain symptom, when someone has the same symptom, it does not mean that this person has a 100% chance of getting pancreatic cancer. Assuming the incidence rate of pancreatic cancer is 1/100000, while 10/99999 healthy individuals have the same
3010:
can be seen by showing that even if sensitivity is raised to 100% and specificity remains at 80%, the probability of someone testing positive really being a cannabis user only rises from 19% to 21%, but if the sensitivity is held at 90% and the specificity is increased to 95%, the probability rises
2353:
in his famous book from 1933. Kolmogorov underlines the importance of conditional probability by writing "I wish to call attention to ... and especially the theory of conditional probabilities and conditional expectations ..." in the
Preface. The Bayes theorem determines the posterior distribution
488:
By modern standards, we should refer to the Bayes–Price rule. Price discovered Bayes's work, recognized its importance, corrected it, contributed to the article, and found a use for it. The modern convention of employing Bayes's name alone is unfair but so entrenched that anything else makes little
2919:
8755:
of this disease is 9.09%, and if we take that as the prior probability, then the prior odds is about 1:10. So after receiving a positive test result, the posterior odds of actually having the disease becomes 1:1, which means that the posterior probability of having the disease is 50%. If a second
11342:
Example of a
Bayesian analysis table for a female individual's risk for a disease based on the knowledge that the disease is present in her siblings but not in her parents or any of her four children. Based solely on the status of the subject's siblings and parents, she is equally likely to be a
11669:
Laplace presented a refinement of Bayes' theorem in: Laplace (read: 1783 / published: 1785) "Mémoire sur les approximations des formules qui sont fonctions de très grands nombres", "Mémoires de l'Académie royale des
Sciences de Paris", 423–467. Reprinted in: Laplace, "Oeuvres complètes" (Paris,
11554:
After carrying out the same analysis on the patient's male partner (with a negative test result), the chances of their child being affected is equal to the product of the parents' respective posterior probabilities for being carriers times the chances that two carriers will produce an affected
8764:
of 9.9%), 827 of those without the disease will get a true negative result (specificity of 91.0%), and 82 of those without the disease will get a false positive result (false positive rate of 9.0%). Before taking any test, the patient's odds for having the disease is 91:909. After receiving a
3777:
Once again, the answer can be reached without using the formula by applying the conditions to a hypothetical number of cases. For example, if the factory produces 1,000 items, 200 will be produced by
Machine A, 300 by Machine B, and 500 by Machine C. Machine A will produce 5% × 200 = 10
4385:
Given that the item is defective, the probability that it was made by machine C is 5/24. Although machine C produces half of the total output, it produces a much smaller fraction of the defective items. Hence the knowledge that the item selected was defective enables us to replace the prior
11391:
Parental genetic testing can detect around 90% of known disease alleles in parents that can lead to carrier or affected status in their child. Cystic fibrosis is a heritable disease caused by an autosomal recessive mutation on the CFTR gene, located on the q arm of chromosome 7.
11399:
Because the patient is unaffected, she is either homozygous for the wild-type allele, or heterozygous. To establish prior probabilities, a
Punnett square is used, based on the knowledge that neither parent was affected by the disease but both could have been carriers:
11491:
Next, the patient undergoes genetic testing and tests negative for cystic fibrosis. This test has a 90% detection rate, so the conditional probabilities of a negative test are 1/10 and 1. Finally, the joint and posterior probabilities are calculated as before.
5285:, ..., are mutually exclusive and exhaustive, i.e., one of them is certain to occur but no two can occur together, we can determine the proportionality constant by using the fact that their probabilities must add up to one. For instance, for a given event
3260:
symptoms worldwide, the probability of having pancreatic cancer given the symptoms is only 9.1%, and the other 90.9% could be "false positives" (that is, falsely said to have cancer; "positive" is a confusing term when, as here, the test gives bad news).
8863:
6705:
4380:
5464:
8749:
9866:
5731:
2936:
In other words, even if someone tests positive, the probability that they are a cannabis user is only 19%—this is because in this group, only 5% of people are users, and most positives are false positives coming from the remaining 95%.
9145:
4213:
10577:
9886:
10969:
8017:
11216:
7895:
7749:
1550:
2815:
6875:
10327:
8254:
1932:
1360:
1217:
6001:
4046:
7596:
7443:
398:), also a statistician and philosopher. Bayes used conditional probability to provide an algorithm (his Proposition 9) that uses evidence to calculate limits on an unknown parameter. His work was published in 1763 as
7265:
7097:
5587:
4728:
4710:(Pattern | Rare) = 98%. Only 5% of members of the common subspecies have the pattern. The rare subspecies is 0.1% of the total population. How likely is the beetle having the pattern to be rare: what is
11657:
Laplace announced his independent discovery of Bayes' theorem in: Laplace (1774) "Mémoire sur la probabilité des causes par les événements", "Mémoires de l'Académie royale des
Sciences de MI (Savants Ă©trangers)",
11083:
10159:
11395:
Bayesian analysis of a female patient with a family history of cystic fibrosis (CF), who has tested negative for CF, demonstrating how this method was used to determine her risk of having a child born with CF:
3901:
1795:
1685:
3348:
2498:
2354:
from the prior distribution. Uniqueness requires continuity assumptions. Bayes' theorem can be generalized to include improper prior distributions such as the uniform distribution on the real line. Modern
9286:
3778:
defective items, Machine B 3% × 300 = 9, and
Machine C 1% × 500 = 5, for a total of 24. Thus, the likelihood that a randomly selected defective item was produced by machine C is 5/24 (~20.83%).
8132:
587:
429:
and chose one of the two solutions offered by Bayes. In 1765, Price was elected a Fellow of the Royal
Society in recognition of his work on the legacy of Bayes. On 27 April a letter sent to his friend
9619:
8547:
11468:
Given that the patient is unaffected, there are only three possibilities. Within these three, there are two scenarios in which the patient carries the mutant allele. Thus the prior probabilities are
2981:
2485:
2171:
6541:
11681:. English translation: Pierre Simon, Marquis de Laplace with F. W. Truscott and F. L. Emory, trans., "A Philosophical Essay on Probabilities" (New York, New York: John Wiley & Sons, 1902),
5155:
5263:
8944:
8430:
6443:
2337:
972:
10365:
9207:
8985:
8771:
6547:
4690:
are the events rare, common, pattern and no pattern. Percentages in parentheses are calculated. Three independent values are given, so it is possible to calculate the inverse tree.
10088:
4238:
10565:
11260:
5307:
6150:
10484:
4688:
400:
11343:
carrier as to be a non-carrier (this likelihood is denoted by the Prior
Hypothesis). However, the probability that the subject's four sons would all be unaffected is 1/16 (
9546:
8650:
9721:
8333:
10400:
9505:
9477:
9449:
9020:
6242:
1433:
832:
707:
10853:
10452:
9411:
1398:
1255:
669:
6089:
2203:
1972:
8297:
9376:
9353:
6293:
6188:
10206:
2045:
6047:
5598:
1032:
1003:
9028:
2249:
2018:
12109:
The Theory That Would Not Die: How Bayes' Rule
Cracked the Enigma Code, Hunted Down Russian Submarines & Emerged Triumphant from Two Centuries of Controversy
10774:{\displaystyle P(\omega _{A{\tilde {|}}B}^{S})={\frac {P(\omega _{B\vert A}^{S})a(A)}{P(\omega _{B\vert A}^{S})a(A)+P(\omega _{B\vert \lnot A}^{S})a(\lnot A)}}.}
10051:{\displaystyle (\omega _{A{\tilde {|}}B}^{S},\omega _{A{\tilde {|}}\lnot B}^{S})=(\omega _{B\vert A}^{S},\omega _{B\vert \lnot A}^{S}){\widetilde {\phi }}a_{A},}
4061:
10815:
10524:
10504:
10420:
10234:
10179:
9713:
9693:
9663:
9643:
9330:
9310:
8630:
8610:
8590:
8570:
8470:
8450:
7288:
2223:
2065:
1992:
1072:
1052:
916:
896:
872:
852:
795:
775:
751:
731:
630:
610:
11801:"An Essay towards solving a Problem in the Doctrine of Chance. By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter to John Canton, A.M.F.R.S."
10861:
7906:
480:, a blind English mathematician, some time before Bayes; that interpretation, however, has been disputed. Martyn Hooper and Sharon McGrayne have argued that
11094:
7769:
2914:{\displaystyle P({\text{Positive}})=P({\text{Positive}}\vert {\text{User}})P({\text{User}})+P({\text{Positive}}\vert {\text{Non-user}})P({\text{Non-user}})}
7635:
1441:
6724:
10239:
8140:
1806:
11662:: 621–656. Reprinted in: Laplace, "Oeuvres complètes" (Paris, France: Gauthier-Villars et fils, 1841), vol. 8, pp. 27–65. Available on-line at:
1263:
1117:
12569:
5850:
3927:
2440:(PPV) of a test is the proportion of persons who are actually positive out of all those testing positive, and can be calculated from a sample as:
7462:
7307:
11619:
7132:
6962:
419:
on 23 December 1763. Price edited Bayes's major work "An Essay Towards Solving a Problem in the Doctrine of Chances" (1763), which appeared in
5475:
305:
12541:"CFTR Gene – Genetics Home Reference". U.S. National Library of Medicine, National Institutes of Health, ghr.nlm.nih.gov/gene/CFTR#location.
10988:
10093:
7614:
A continuous event space is often conceptualized in terms of the numerator terms. It is then useful to eliminate the denominator using the
4421:
A geometric visualisation of Bayes' theorem using astronauts who may be suspicious (with eyebrows) and may be assassins (carrying daggers)
3806:
96:
12578:
1691:
1581:
2983:
visually by comparison of shaded areas. Note how small the pink area of true positives is compared to the blue area of false positives.
451:
from a prior probability, given evidence. He reproduced and extended Bayes's results in 1774, apparently unaware of Bayes's work. The
415:
Over two years, Richard Price significantly edited the unpublished manuscript, before sending it to a friend who read it aloud at the
408:(in modern terminology). On Bayes's death his family transferred his papers to a friend, the minister, philosopher, and mathematician
4563:
5168:
is fixed in the discussion, and we wish to consider the impact of its having been observed on our belief in various possible events
2487:
mean "the probability that someone is a cannabis user given that they test positive," which is what is meant by PPV. We can write:
434:
was read out at the Royal Society, and later published, where Price applies this work to population and computing 'life-annuities'.
12386:
IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2016), Baden-Baden, September 2016
9215:
443:
178:
8036:
2256:
2418:(TNR) = 0.80. Therefore, the test correctly identifies 80% of non-use for non-users, but also generates 20% false positives, or
504:
9562:
8478:
12622:
12603:
12414:
12310:
12183:
11719:
2947:
2451:
2070:
425:, and contains Bayes' theorem. Price wrote an introduction to the paper which provides some of the philosophical basis of
8761:
6453:
2407:(TPR) = 0.90. Therefore, it leads to 90% true positive results (correct identification of drug use) for cannabis users.
5065:
12269:
12122:
12018:
11966:
11926:
11851:
11776:
11742:
9668:
The corresponding formula in terms of probability calculus is Bayes' theorem, which in its expanded form involving the
8868:
which is consistent with the fact that there are 82 true positives and 82 false positives in the group of 1000 people.
12326:
5190:
4556:
298:
261:
8884:
8338:
6378:
188:
4639:
The role of Bayes' theorem is best visualized with tree diagrams. The two diagrams partition the same outcomes by
12720:
11630:
214:
91:
12532:"Types of CFTR Mutations". Cystic Fibrosis Foundation, www.cff.org/What-is-CF/Genetics/Types-of-CFTR-Mutations/.
11732:
11897:
4566:, probability measures a "proportion of outcomes". For example, suppose an experiment is performed many times.
152:
8858:{\displaystyle {\frac {91}{909}}\times {\frac {90.1\%}{9.0\%}}={\frac {91\times 90.1\%}{909\times 9.0\%}}=1:1}
6700:{\displaystyle \Rightarrow P(A_{i}|B)={\frac {P(B|A_{i})P(A_{i})}{\sum \limits _{j}P(B|A_{j})P(A_{j})}}\cdot }
12715:
5181:
4375:{\displaystyle P(X_{C}|Y)={\frac {P(Y|X_{C})P(X_{C})}{P(Y)}}={\frac {0.01\cdot 0.50}{0.024}}={\frac {5}{24}}}
921:
10332:
9153:
12710:
8949:
7103:
4647:
in opposite orders, to obtain the inverse probabilities. Bayes' theorem links the different partitionings.
291:
183:
121:
10064:
5459:{\displaystyle P(A|B)=c\cdot P(A)\cdot P(B|A){\text{ and }}P(\neg A|B)=c\cdot P(\neg A)\cdot P(B|\neg A).}
11677:
See also: Laplace, "Essai philosophique sur les probabilités" (Paris, France: Mme. Ve. Courcier , 1814),
9479:, where now beyond assigning true or false, we assign probability values to statements. The assertion of
8644:
8640:
6901:. In practice, these instances might be parametrized by writing the specified probability densities as a
4426:
3800:
denote the event that a randomly chosen item is defective. Then, we are given the following information:
10529:
11954:
11224:
6897:
with known probability distributions. There exists an instance of Bayes' theorem for each point in the
3906:
If the item was made by the first machine, then the probability that it is defective is 0.05; that is,
2448:
If sensitivity, specificity, and prevalence are known, PPV can be calculated using Bayes theorem. Let
633:
421:
173:
142:
12404:
8744:{\displaystyle \Lambda _{+}=P({\text{True Positive}})/P({\text{False Positive}})=90\%/(100\%-91\%)=10}
6102:
12143:
10457:
9861:{\displaystyle P(A\vert B)=P(B\vert A){\frac {a(A)}{P(B\vert A)\,a(A)+P(B\vert \neg A)\,a(\neg A)}}.}
4666:
2998:
The 1,000 people thus yields 235 positive tests, of which only 45 are genuine drug users, about 19%.
2437:
2346:
355:
235:
116:
5172:. In such a situation the denominator of the last expression, the probability of the given evidence
4539:
For more on the application of Bayes' theorem under the Bayesian interpretation of probability, see
3263:
Based on incidence rate, the following table presents the corresponding numbers per 100,000 people.
10980:
9510:
7615:
6370:
2922:
2355:
256:
168:
8302:
3337:
Which can then be used to calculate the probability of having cancer when you have the symptoms:
11682:
11671:
10370:
9482:
9454:
9419:
9414:
9413:, the two implications being certain are equivalent statements. In the probability formulas, the
8990:
8432:, and uses a similar abbreviation for the Bayes factor and for the conditional odds. The odds on
6245:
6207:
6191:
2930:
2376:
2372:
2342:
1403:
1108:
802:
710:
677:
334:
10820:
10571:, meaning that Bayes' theorem can be expressed in terms of projected probabilities of opinions:
10425:
9381:
8030:. The odds between two events is simply the ratio of the probabilities of the two events. Thus
1368:
1225:
639:
11958:
9553:
6902:
6062:
2176:
1945:
147:
11887:
11766:
12435:
12114:
11914:
11841:
11678:
11608:
11603:
8760:
result (sensitivity of 90.1%), 9 of those with the disease will get a false negative result (
8269:
7106:. To remain useful, Bayes' theorem can be formulated in terms of the relevant densities (see
6296:
5726:{\displaystyle c={\frac {1}{P(B|A)\cdot P(A)+P(B|\neg A)\cdot P(\neg A)}}={\frac {1}{P(B)}}.}
2388:
754:
448:
405:
363:
351:
50:
9548:. Relating the directions of implication, Bayes' theorem represents a generalization of the
9358:
9335:
9140:{\displaystyle P(\neg B\vert \neg A)=1-\left(1-P(A\vert B)\right){\frac {P(B)}{P(\neg A)}},}
6261:
6156:
2358:
methods have boosted the importance of Bayes' theorem including cases with improper priors.
466:
basis, writing in a 1973 book that Bayes' theorem "is to the theory of probability what the
12492:
Ogino, Shuji; Wilson, Robert B; Gold, Bert; Hawley, Pamela; Grody, Wayne W (October 2004).
12444:
12302:
10184:
6898:
4438:
4208:{\displaystyle P(Y)=\sum _{i}P(Y|X_{i})P(X_{i})=(0.05)(0.2)+(0.03)(0.3)+(0.01)(0.5)=0.024.}
2023:
1079:
452:
438:
230:
111:
81:
11670:
France: Gauthier-Villars et fils, 1844), vol. 10, pp. 295–338. Available on-line at:
6023:
1008:
979:
8:
11663:
11635:
11613:
11568:
6932:
with known probability distributions. In principle, Bayes' theorem applies to the events
3007:
2419:
2411:
2400:
2384:
2228:
1997:
1572:
875:
477:
467:
426:
362:) to obtain the probability of the model configuration given the observations (i.e., the
359:
62:
54:
34:
2929:, namely the set of people who take the drug test. This combined with the definition of
12510:
12493:
12466:
12439:
12240:
12107:
12084:
12034:
Edwards, A. W. F. (1986). "Is the Reference in Hartley (1749) to Bayesian Inference?".
11947:
11595:
10964:{\displaystyle P(A\vert B\cap C)={\frac {P(B\vert A\cap C)\,P(A\vert C)}{P(B\vert C)}}}
10800:
10509:
10489:
10405:
10219:
10164:
9698:
9678:
9648:
9628:
9315:
9295:
8615:
8595:
8575:
8555:
8455:
8435:
8263:, or in other words, the posterior is proportional to the prior times the likelihood.
8012:{\displaystyle \Lambda (A_{1}:A_{2}\vert B)={\frac {P(B\vert A_{1})}{P(B\vert A_{2})}}}
7606:
7273:
6317:
5161:
4540:
2926:
2415:
2404:
2380:
2208:
2050:
1977:
1057:
1037:
901:
881:
857:
837:
780:
760:
736:
716:
615:
595:
347:
279:
204:
106:
76:
12381:
12173:
11211:{\displaystyle P(A\cap B\cap C)=P(B\cap A\cap C)=P(B\vert A\cap C)\,P(A\vert C)\,P(C)}
9876:
Bayes' theorem represents a special case of deriving inverted conditional opinions in
7890:{\displaystyle O(A_{1}:A_{2}\vert B)=O(A_{1}:A_{2})\cdot \Lambda (A_{1}:A_{2}\vert B)}
4717:
From the extended form of Bayes' theorem (since any beetle is either rare or common),
12637:
12618:
12599:
12594:
Bolstad, William M.; Curran, James M. (2017). "Logic, Probability, and Uncertainty".
12515:
12471:
12410:
12306:
12275:
12265:
12244:
12232:
12179:
12118:
12088:
12079:
12062:
12014:
11962:
11922:
11893:
11847:
11772:
11738:
11715:
11589:
10209:
9669:
7744:{\displaystyle f_{Y}(y)=\int _{-\infty }^{\infty }f_{Y|X=\xi }(y)f_{X}(\xi )\,d\xi .}
6050:
5742:
2350:
1075:
431:
339:
274:
209:
58:
12440:"The evolving landscape of expanded carrier screening: challenges and opportunities"
10797:
A version of Bayes' theorem for 3 events results from the addition of a third event
10784:
Hence, the subjective Bayes' theorem represents a generalization of Bayes' theorem.
7610:
A way to conceptualize event spaces generated by continuous random variables X and Y
4655:
1545:{\displaystyle P(A\vert B)={\frac {P(B\vert A)P(A)}{P(B)}},{\text{ if }}P(B)\neq 0.}
12662:
12505:
12461:
12453:
12294:
12222:
12212:
12074:
12047:
12043:
11995:
11991:
11811:
9877:
8027:
6922:
463:
381:
101:
6889:
Bayes' theorem applied to an event space generated by continuous random variables
6870:{\displaystyle P(A|B)={\frac {P(B|A)P(A)}{P(B|A)P(A)+P(B|\neg A)P(\neg A)}}\cdot }
11942:
11272:
10322:{\displaystyle (\omega _{A{\tilde {|}}B}^{S},\omega _{A{\tilde {|}}\lnot B}^{S})}
8249:{\displaystyle O(A_{1}:A_{2}\vert B)={\frac {P(A_{1}\vert B)}{P(A_{2}\vert B)}},}
6715:
1561:
473:
459:
404:. Bayes studied how to compute a distribution for the probability parameter of a
137:
20:
10567:. The application of Bayes' theorem to projected probabilities of opinions is a
4429:
ascribed to the terms. The two predominant interpretations are described below.
1927:{\displaystyle f_{X\vert Y=y}(x)={\frac {f_{Y\vert X=x}(y)f_{X}(x)}{f_{Y}(y)}}.}
12494:"Bayesian analysis for cystic fibrosis risks in prenatal and carrier screening"
11221:
The desired result is obtained by identifying both expressions and solving for
9549:
1355:{\displaystyle P(B\vert A)={\frac {P(A\cap B)}{P(A)}},{\text{ if }}P(A)\neq 0.}
1212:{\displaystyle P(A\vert B)={\frac {P(A\cap B)}{P(B)}},{\text{ if }}P(B)\neq 0,}
12683:
12634:
Measuring Uncertainty : An Elementary Introduction to Bayesian Statistics
12457:
12356:
12279:
5844:
Another form of Bayes' theorem for two competing statements or hypotheses is:
4551:
2399:
Suppose, a particular test for whether someone has been using cannabis is 90%
12704:
12582:. Vol. 22 (11th ed.). Cambridge University Press. pp. 314–315.
12573:
12563:
12396:
12236:
5996:{\displaystyle P(A|B)={\frac {P(B|A)P(A)}{P(B|A)P(A)+P(B|\neg A)P(\neg A)}}.}
1091:
481:
416:
409:
12667:
12650:
4041:{\displaystyle P(Y|X_{A})=0.05,\quad P(Y|X_{B})=0.03,\quad P(Y|X_{C})=0.01.}
12519:
12475:
12400:
11816:
11800:
10568:
8260:
8023:
6918:
6885:
6332:
6328:
5268:
In words, the posterior is proportional to the prior times the likelihood.
476:
used a Bayesian argument to conclude that Bayes' theorem was discovered by
375:
330:
251:
12334:
8259:
Thus, the rule says that the posterior odds are the prior odds times the
7591:{\displaystyle f_{X|Y{=}y}(x)={\frac {f_{Y|X{=}x}(y)f_{X}(x)}{f_{Y}(y)}}.}
7438:{\displaystyle P(X{=}x|Y{=}y)={\frac {f_{Y|X{=}x}(y)P(X{=}x)}{f_{Y}(y)}}.}
5297:
are exclusive and exhaustive. Denoting the constant of proportionality by
12227:
11624:
7260:{\displaystyle f_{X|Y{=}y}(x)={\frac {P(Y{=}y|X{=}x)f_{X}(x)}{P(Y{=}y)}}}
7092:{\displaystyle P(X{=}x|Y{=}y)={\frac {P(Y{=}y|X{=}x)P(X{=}x)}{P(Y{=}y)}}}
4706:. A full 98% of the members of the rare subspecies have the pattern, so
4417:
2430:
12641:
12688:
12217:
12200:
12140:
Kendall's Advanced Theory of Statistics: Volume I – Distribution Theory
8752:
5582:{\displaystyle 1=c\cdot (P(B|A)\cdot P(A)+P(B|\neg A)\cdot P(\neg A)),}
4699:
4695:
2994:
50 of them are users and 45 of them give true positive (0.90 × 50)
2991:
950 are non-users and 190 of them give false positive (0.20 × 950)
2426:
2375:
provide a solution method for a number of popular puzzles, such as the
447:, used conditional probability to formulate the relation of an updated
12259:
11569:
Genetic testing done in parallel with other risk factor identification
10090:
denotes the operator for inverting conditional opinions. The argument
4225:
has occurred, and we want to calculate the conditional probability of
11919:
The History of Statistics: The Measurement of Uncertainty Before 1900
11078:{\displaystyle P(A\cap B\cap C)=P(A\vert B\cap C)\,P(B\vert C)\,P(C)}
10526:. Every subjective opinion has a corresponding projected probability
10213:
10154:{\displaystyle (\omega _{B\vert A}^{S},\omega _{B\vert \lnot A}^{S})}
9673:
19:"Bayes rule" redirects here. For the concept in decision theory, see
11308:
Conditional Probability that all four offspring will be unaffected
7102:
However, terms become 0 at points where either variable has finite
3896:{\displaystyle P(X_{A})=0.2,\quad P(X_{B})=0.3,\quad P(X_{C})=0.5.}
2433:
that a random person who tests positive is really a cannabis user?
1571:, Bayes' theorem may be analogously derived from the definition of
498:
Bayes' theorem is stated mathematically as the following equation:
12562:
This article incorporates text from a publication now in the
10236:. The pair of derivative inverted conditional opinions is denoted
8871:
1790:{\displaystyle f_{Y\vert X=x}(y)={\frac {f_{X,Y}(x,y)}{f_{X}(x)}}}
1680:{\displaystyle f_{X\vert Y=y}(x)={\frac {f_{X,Y}(x,y)}{f_{Y}(y)}}}
12693:
5180:. Bayes' theorem then shows that the posterior probabilities are
1074:
respectively without any given conditions; they are known as the
12199:
Taraldsen, Gunnar; Tufto, Jarle; Lindqvist, Bo H. (2021-07-24).
10161:
denotes a pair of binomial conditional opinions given by source
6447:
Or (using the multiplication rule for conditional probability),
10422:
can assign any subjective opinion to the conditional statement
4703:
11982:
Stigler, Stephen M. (1983). "Who Discovered Bayes' Theorem?".
11712:
Subjective Logic; A formalism for Reasoning Under Uncertainty.
10506:
with degrees of epistemic uncertainty, as expressed by source
9507:
is captured by certainty of the conditional, the assertion of
8765:
positive result, the patient's odds for having the disease is
2940:
9281:{\displaystyle P(A\vert B)=1\implies P(\neg B\vert \neg A)=1}
3788:
denote the event that a randomly chosen item was made by the
401:
An Essay Towards Solving a Problem in the Doctrine of Chances
42:
11278:
8127:{\displaystyle O(A_{1}:A_{2})={\frac {P(A_{1})}{P(A_{2})}},}
4698:
spots what might, due to the pattern on its back, be a rare
834:
is also a conditional probability: the probability of event
12420:
11798:
7760:
582:{\displaystyle P(A\vert B)={\frac {P(B\vert A)P(A)}{P(B)}}}
387:
11654:
Laplace refined Bayes's theorem over a period of decades:
8472:. Bayes' rule can then be written in the abbreviated form
3781:
This problem can also be solved using Bayes' theorem: Let
1257:
is the probability of both A and B being true. Similarly,
12433:
11804:
Philosophical Transactions of the Royal Society of London
10402:, i.e. in addition to assigning a probability the source
9614:{\displaystyle (B\implies A)\iff (\neg A\implies \neg B)}
8542:{\displaystyle O(A\vert B)=O(A)\cdot \Lambda (A\vert B),}
12434:
Kraft, Stephanie A; Duenas, Devan; Wilfond, Benjamin S;
12357:"Bayes Theorem - Formula, Statement, Proof | Bayes Rule"
9625:
In this relation between implications, the positions of
4486:, is the degree of belief after incorporating news that
12198:
11311:(1/2) ⋅ (1/2) ⋅ (1/2) ⋅ (1/2) = 1/16
8634:
posterior odds equals prior odds times likelihood ratio
2976:{\displaystyle P({\text{User}}\vert {\text{Positive}})}
2480:{\displaystyle P({\text{User}}\vert {\text{Positive}})}
12395:
12009:
de Vaux, Richard; Velleman, Paul; Bock, David (2016).
12002:
462:
put Bayes's algorithm and Laplace's formulation on an
12491:
12100:
12098:
11227:
11097:
10991:
10864:
10823:
10803:
10580:
10532:
10512:
10492:
10460:
10428:
10408:
10373:
10335:
10242:
10222:
10187:
10167:
10096:
10067:
9889:
9724:
9701:
9681:
9651:
9631:
9565:
9513:
9485:
9457:
9422:
9384:
9361:
9338:
9318:
9298:
9218:
9156:
9031:
8993:
8952:
8887:
8774:
8653:
8618:
8598:
8578:
8558:
8481:
8458:
8438:
8341:
8305:
8272:
8143:
8039:
7909:
7772:
7638:
7465:
7310:
7276:
7135:
6965:
6727:
6550:
6456:
6381:
6264:
6210:
6159:
6105:
6065:
6026:
5853:
5601:
5478:
5310:
5193:
5068:
4726:
4669:
4241:
4064:
3930:
3809:
3346:
3168:
100% sensitive, 80% specific, PPV=50/240 ≈ 21%
2950:
2818:
2496:
2454:
2259:
2231:
2211:
2179:
2166:{\displaystyle P_{X,Y}(dx,dy)=P_{Y}^{x}(dy)P_{X}(dx)}
2073:
2053:
2026:
2000:
1980:
1948:
1809:
1694:
1584:
1444:
1406:
1371:
1266:
1228:
1120:
1107:
Bayes' theorem may be derived from the definition of
1060:
1040:
1011:
982:
924:
904:
884:
860:
840:
805:
783:
763:
739:
719:
680:
642:
618:
598:
507:
390:
12632:
Schmitt, Samuel A. (1969). "Accumulating Evidence".
11724:
11585:
3088:
90% sensitive, 80% specific, PPV=45/235 ≈ 19%
384:
12487:
12485:
12008:
6536:{\displaystyle P(B)={\sum _{j}P(B|A_{j})P(A_{j})},}
3248:90% sensitive, 95% specific, PPV=45/92 ≈ 49%
12106:
12095:
11946:
11254:
11210:
11077:
10963:
10847:
10809:
10773:
10559:
10518:
10498:
10478:
10446:
10414:
10394:
10359:
10321:
10228:
10200:
10173:
10153:
10082:
10050:
9860:
9707:
9687:
9657:
9637:
9613:
9540:
9499:
9471:
9443:
9405:
9370:
9347:
9324:
9304:
9280:
9201:
9139:
9014:
8979:
8946:twice, one may use Bayes' theorem to also express
8938:
8857:
8743:
8624:
8604:
8584:
8564:
8541:
8464:
8444:
8424:
8327:
8291:
8248:
8126:
8011:
7889:
7743:
7590:
7437:
7282:
7259:
7091:
6869:
6699:
6535:
6437:
6287:
6236:
6182:
6144:
6091:is the corresponding initial degree of belief in
6083:
6041:
5995:
5725:
5581:
5458:
5257:
5150:{\displaystyle P(A|B)={\frac {P(B|A)P(A)}{P(B)}}.}
5149:
5017:
4682:
4397:) = 1/2 by the smaller posterior probability
4374:
4207:
4040:
3895:
3667:
2975:
2913:
2801:
2479:
2331:
2243:
2217:
2197:
2165:
2059:
2039:
2012:
1986:
1966:
1926:
1789:
1679:
1555:
1544:
1427:
1392:
1354:
1249:
1211:
1066:
1046:
1026:
997:
966:
910:
890:
866:
846:
826:
789:
769:
745:
725:
701:
663:
624:
604:
581:
346:One of the many applications of Bayes' theorem is
12598:(3rd ed.). New York: Wiley. pp. 59–82.
4425:The interpretation of Bayes' rule depends on the
2429:, meaning 5% of people use cannabis, what is the
12702:
12482:
11737:. Wales: University of Wales Press. p. 44.
5184:to the numerator, so the last equation becomes:
4555:Illustration of frequentist interpretation with
455:of probability was developed mainly by Laplace.
354:, where it is used to invert the probability of
12382:Generalising Bayes' Theorem in Subjective Logic
8872:Correspondence to other mathematical frameworks
5258:{\displaystyle P(A|B)\propto P(A)\cdot P(B|A).}
4051:To answer the original question, we first find
1400:and substituting into the above expression for
12651:"Laplace's 1774 Memoir on Inverse Probability"
12258:Robert, Christian P.; Casella, George (2004).
11620:Why Most Published Research Findings Are False
8939:{\displaystyle P(\neg B\vert A)=1-P(B\vert A)}
8425:{\displaystyle O(A)=O(A:\neg A)=P(A)/(1-P(A))}
4659:Tree diagram illustrating the beetle example.
4602:) is the proportion of outcomes with property
4574:) is the proportion of outcomes with property
4218:Hence, 2.4% of the total output is defective.
12593:
12257:
11921:. Harvard University Press. pp. 99–138.
11846:. Cambridge University Press. p. xxiii.
11386:
10792:
6438:{\displaystyle P(B)=\sum _{j}P(B\cap A_{j}),}
4546:
3001:
299:
16:Mathematical rule for inverting probabilities
11935:
11237:
11186:
11161:
11053:
11028:
10949:
10929:
10904:
10874:
10855:on which all probabilities are conditioned:
10731:
10686:
10642:
10435:
10383:
10344:
10132:
10108:
10004:
9980:
9824:
9790:
9755:
9734:
9523:
9432:
9260:
9228:
9085:
9044:
9003:
8965:
8927:
8900:
8527:
8491:
8231:
8204:
8173:
7990:
7963:
7939:
7878:
7802:
4919:
4880:
4842:
4774:
4742:
4439:Bayesian (or epistemological) interpretation
4055:(Y). That can be done in the following way:
2962:
2886:
2847:
2689:
2650:
2612:
2544:
2512:
2466:
2332:{\displaystyle P_{X}^{y}(A)=E(1_{A}(X)|Y=y)}
1855:
1818:
1703:
1593:
1478:
1454:
1416:
1276:
1130:
955:
934:
815:
690:
541:
517:
12636:. Reading: Addison-Wesley. pp. 61–99.
12013:(4th ed.). Pearson. pp. 380–381.
11799:Bayes, Thomas & Price, Richard (1763).
11520:Conditional Probability of a negative test
7754:
2366:
874:is true. It can also be interpreted as the
374:Bayes' theorem is named after the Reverend
12409:. Massachusetts: MIT Press. p. 1208.
12156:
11889:Classical Probability in the Enlightenment
10367:generalizes the probabilistic conditional
9601:
9597:
9587:
9583:
9576:
9572:
9493:
9489:
9465:
9461:
9247:
9243:
9209:. From this we can read off the inference
8647:of 91%, then the positive Bayes factor is
8452:is by definition the odds for and against
4432:
3774:probability it was produced by machine C?
333:) gives a mathematical rule for inverting
306:
292:
12666:
12509:
12465:
12226:
12216:
12201:"Improper priors and improper posteriors"
12137:
12078:
11815:
11279:Using pedigree to calculate probabilities
11195:
11176:
11062:
11043:
10919:
9836:
9799:
7731:
5469:Adding these two formulas we deduce that
12567:
12389:
12159:Foundations of the Theory of Probability
12104:
12063:"Richard Price, Bayes' theorem, and God"
11941:
11873:
11824:
11792:
11706:
11704:
10486:is the belief in the truth of statement
7605:
6884:
6194:or likelihood, the degree of belief in
4654:
4550:
4416:
2939:
1090:
179:Integrated nested Laplace approximations
12648:
12631:
12033:
11981:
11912:
11674:. Bayes' theorem is stated on page 301.
11666:. Bayes' theorem appears on p. 29.
11504:Hypothesis 2: Patient is not a carrier
11292:Hypothesis 2: Patient is not a carrier
6248:or likelihood, the degree of belief in
6006:For an epistemological interpretation:
2341:Existence and uniqueness of the needed
967:{\displaystyle P(B\vert A)=L(A\vert B)}
493:
358:given a model configuration (i.e., the
12703:
12060:
11885:
10360:{\displaystyle \omega _{A\vert B}^{S}}
9202:{\displaystyle P(\neg A)=1-P(A)\neq 0}
8876:
5835: P(¬B) = 1−P(B)
5160:In many applications, for instance in
3678:
11892:. Princeton Univ Press. p. 268.
11867:
11839:
11764:
11730:
11701:
8980:{\displaystyle P(\neg B\vert \neg A)}
8639:For example, if a medical test has a
5813:= P(¬A|¬B)⋅P(¬B)
4464:, is the initial degree of belief in
2444:PPV = True positive / Tested positive
12615:Bayesian Statistics: An Introduction
12171:
10083:{\displaystyle {\widetilde {\phi }}}
9451:generalizes the logical implication
8552:or, in words, the posterior odds on
5176:, is fixed; what we want to vary is
12649:Stigler, Stephen M. (August 1986).
12612:
12596:Introduction to Bayesian Statistics
12292:
12175:Probability based on Radon measures
11501:Hypothesis 1: Patient is a carrier
11289:Hypothesis 1: Patient is a carrier
9871:
6880:
6636:
5810:P(¬B|¬A)⋅P(¬A)
5736:
1974:be the conditional distribution of
1034:are the probabilities of observing
444:Théorie analytique des probabilités
13:
12587:
12511:10.1097/01.GIM.0000139511.83336.8F
12205:Scandinavian Journal of Statistics
11265:
10787:
10756:
10734:
10560:{\displaystyle P(\omega _{A}^{S})}
10303:
10135:
10007:
9950:
9843:
9827:
9602:
9591:
9362:
9339:
9263:
9254:
9163:
9122:
9047:
9038:
8968:
8959:
8894:
8837:
8823:
8802:
8794:
8729:
8720:
8706:
8655:
8518:
8369:
8319:
7910:
7849:
7674:
7669:
7601:
6852:
6837:
6225:
6112:
6072:
6053:, the initial degree of belief in
5978:
5963:
5684:
5666:
5564:
5546:
5444:
5418:
5386:
5008:
4625:) is the proportion of those with
4586:) is the proportion with property
4412:
3658:
2792:
14:
12732:
12676:
11255:{\displaystyle P(A\vert B\cap C)}
2067:. The joint distribution is then
484:'s contribution was substantial:
12568:Mitchell, John Malcolm (1911). "
12555:
12080:10.1111/j.1740-9713.2013.00638.x
11771:. SUNY Press. pp. 243–244.
11588:
10454:. A binomial subjective opinion
7113:
6361:). It is then useful to compute
6311:
6145:{\displaystyle P(\neg A)=1-P(A)}
2933:results in the above statement.
380:
273:
189:Approximate Bayesian computation
41:
12548:
12535:
12526:
12427:
12373:
12349:
12319:
12286:
12261:Monte Carlo Statistical Methods
12251:
12192:
12165:
12150:
12131:
12054:
12027:
11975:
11906:
11631:Regular conditional probability
10479:{\displaystyle \omega _{A}^{S}}
8592:times the likelihood ratio for
7107:
5787:= P(A|¬B)⋅P(¬B)
4683:{\displaystyle {\overline {P}}}
4004:
3967:
3867:
3838:
2921:is a direct application of the
2394:
2173:. The conditional distribution
1937:
1556:For continuous random variables
753:is true. It is also called the
441:in 1774, and later in his 1812
215:Maximum a posteriori estimation
12406:Probabilistic Graphical Models
12327:"Bayes' Theorem: Introduction"
12048:10.1080/00031305.1986.10475370
11996:10.1080/00031305.1983.10483122
11879:
11833:
11758:
11648:
11578:when a parent is not present.
11461:(affected by cystic fibrosis)
11249:
11231:
11205:
11199:
11192:
11180:
11173:
11155:
11146:
11128:
11119:
11101:
11072:
11066:
11059:
11047:
11040:
11022:
11013:
10995:
10955:
10943:
10935:
10923:
10916:
10898:
10886:
10868:
10833:
10827:
10762:
10753:
10747:
10720:
10711:
10705:
10699:
10675:
10667:
10661:
10655:
10631:
10619:
10603:
10599:
10584:
10554:
10536:
10441:
10429:
10389:
10377:
10316:
10297:
10293:
10262:
10258:
10243:
10148:
10097:
10020:
9969:
9963:
9944:
9940:
9909:
9905:
9890:
9849:
9840:
9833:
9818:
9809:
9803:
9796:
9784:
9776:
9770:
9761:
9749:
9740:
9728:
9608:
9598:
9588:
9584:
9580:
9573:
9566:
9529:
9517:
9490:
9462:
9438:
9426:
9394:
9388:
9269:
9251:
9244:
9234:
9222:
9190:
9184:
9169:
9160:
9128:
9119:
9111:
9105:
9091:
9079:
9053:
9035:
9009:
8997:
8974:
8956:
8933:
8921:
8906:
8891:
8732:
8714:
8697:
8689:
8678:
8670:
8533:
8521:
8512:
8506:
8497:
8485:
8419:
8416:
8410:
8398:
8390:
8384:
8375:
8360:
8351:
8345:
8237:
8218:
8210:
8191:
8179:
8147:
8115:
8102:
8094:
8081:
8069:
8043:
8003:
7984:
7976:
7957:
7945:
7913:
7884:
7852:
7843:
7817:
7808:
7776:
7728:
7722:
7709:
7703:
7688:
7655:
7649:
7579:
7573:
7558:
7552:
7539:
7533:
7516:
7498:
7492:
7475:
7426:
7420:
7405:
7391:
7385:
7379:
7362:
7344:
7329:
7314:
7251:
7237:
7229:
7223:
7210:
7195:
7180:
7168:
7162:
7145:
7083:
7069:
7061:
7047:
7041:
7026:
7011:
6999:
6984:
6969:
6858:
6849:
6843:
6833:
6826:
6817:
6811:
6805:
6798:
6791:
6783:
6777:
6771:
6764:
6757:
6745:
6738:
6731:
6688:
6675:
6669:
6655:
6648:
6630:
6617:
6611:
6597:
6590:
6578:
6571:
6557:
6551:
6526:
6513:
6507:
6493:
6486:
6466:
6460:
6429:
6410:
6391:
6385:
6282:
6275:
6268:
6231:
6221:
6214:
6177:
6170:
6163:
6139:
6133:
6118:
6109:
6078:
6069:
6036:
6030:
5984:
5975:
5969:
5959:
5952:
5943:
5937:
5931:
5924:
5917:
5909:
5903:
5897:
5890:
5883:
5871:
5864:
5857:
5714:
5708:
5690:
5681:
5672:
5662:
5655:
5646:
5640:
5631:
5624:
5617:
5573:
5570:
5561:
5552:
5542:
5535:
5526:
5520:
5511:
5504:
5497:
5491:
5450:
5440:
5433:
5424:
5415:
5400:
5393:
5383:
5372:
5365:
5358:
5349:
5343:
5328:
5321:
5314:
5249:
5242:
5235:
5226:
5220:
5211:
5204:
5197:
5138:
5132:
5124:
5118:
5112:
5105:
5098:
5086:
5079:
5072:
5038:
4941:
4933:
4927:
4911:
4902:
4894:
4888:
4872:
4864:
4856:
4850:
4834:
4812:
4804:
4796:
4788:
4782:
4766:
4750:
4734:
4332:
4326:
4318:
4305:
4299:
4285:
4278:
4266:
4259:
4245:
4196:
4190:
4187:
4181:
4175:
4169:
4166:
4160:
4154:
4148:
4145:
4139:
4133:
4120:
4114:
4100:
4093:
4074:
4068:
4029:
4015:
4008:
3992:
3978:
3971:
3955:
3941:
3934:
3884:
3871:
3855:
3842:
3826:
3813:
3627:
3613:
3571:
3563:
3557:
3548:
3539:
3530:
3522:
3516:
3507:
3498:
3490:
3482:
3476:
3467:
3458:
3436:
3428:
3420:
3412:
3406:
3397:
3388:
3372:
3363:
3354:
3254:
2970:
2954:
2944:Using a frequency box to show
2908:
2900:
2894:
2878:
2869:
2861:
2855:
2839:
2830:
2822:
2711:
2703:
2697:
2681:
2672:
2664:
2658:
2642:
2634:
2626:
2620:
2604:
2582:
2574:
2566:
2558:
2552:
2536:
2520:
2504:
2474:
2458:
2326:
2313:
2309:
2303:
2290:
2281:
2275:
2160:
2151:
2138:
2129:
2108:
2090:
1915:
1909:
1894:
1888:
1875:
1869:
1838:
1832:
1781:
1775:
1760:
1748:
1723:
1717:
1671:
1665:
1650:
1638:
1613:
1607:
1533:
1527:
1510:
1504:
1496:
1490:
1484:
1472:
1460:
1448:
1422:
1410:
1387:
1375:
1343:
1337:
1320:
1314:
1306:
1294:
1282:
1270:
1244:
1232:
1197:
1191:
1174:
1168:
1160:
1148:
1136:
1124:
1021:
1015:
992:
986:
961:
949:
940:
928:
821:
809:
696:
684:
652:
646:
573:
567:
559:
553:
547:
535:
523:
511:
1:
12161:. Chelsea Publishing Company.
11768:David Hartley on Human Nature
11694:
11336:(1/2) / (1/32 + 1/2) = 16/17
11333:(1/32) / (1/32 + 1/2) = 1/17
10974:
9541:{\displaystyle P(A\vert B)=1}
7629:), this becomes an integral:
5803:P(B|¬A)⋅P(¬A)
4427:interpretation of probability
2987:If 1,000 people were tested:
2422:(FPR) = 0.20, for non-users.
1102:
12692:. April 5, 2017 – via
12138:Stuart, A.; Ord, K. (1994),
11913:Stigler, Stephen M. (1986).
11452:Heterozygous (a CF carrier)
11438:type allele (a non-carrier)
11420:type allele (a non-carrier)
11322:(1/2) ⋅ (1/16) = 1/32
8328:{\displaystyle A_{2}=\neg A}
4714:(Rare | Pattern)?
4675:
122:Principle of maximum entropy
7:
11581:
11271:based on family history or
10395:{\displaystyle P(A\vert B)}
9500:{\displaystyle B\implies A}
9472:{\displaystyle B\implies A}
9444:{\displaystyle P(A\vert B)}
9015:{\displaystyle P(A\vert B)}
6237:{\displaystyle P(B|\neg A)}
6013:and evidence or background
5293:itself and its complement ¬
3921:) = 0.05. Overall, we have
2361:
1428:{\displaystyle P(A\vert B)}
827:{\displaystyle P(B\vert A)}
713:: the probability of event
702:{\displaystyle P(A\vert B)}
350:, a particular approach to
92:Bernstein–von Mises theorem
10:
12737:
12157:Kolmogorov, A.N. (1933) .
11955:Cambridge University Press
11387:Using genetic test results
10848:{\displaystyle P(C)>0,}
10793:Bayes theorem for 3 events
10447:{\displaystyle (A\vert B)}
10329:. The conditional opinion
9406:{\displaystyle P(B)\neq 0}
9332:, we infer that certainly
6710:In the special case where
6303:after taking into account
4650:
4564:frequentist interpretation
4547:Frequentist interpretation
3002:Sensitivity or specificity
2371:Bayes' rule and computing
1393:{\displaystyle P(A\cap B)}
1250:{\displaystyle P(A\cap B)}
664:{\displaystyle P(B)\neq 0}
422:Philosophical Transactions
369:
18:
12458:10.1038/s41436-018-0273-4
12036:The American Statistician
11984:The American Statistician
11886:Daston, Lorraine (1988).
11843:Price: Political Writings
8572:equals the prior odds on
8266:In the special case that
6084:{\displaystyle P(\neg A)}
5824:
5806:= P(¬A|B)⋅P(B)
5765:
5033:
3751:
3700:
3317:
3282:
3247:
3227:
3190:
3167:
3147:
3110:
3087:
3067:
3030:
2438:Positive predictive value
2373:conditional probabilities
2349:. This was formulated by
2198:{\displaystyle P_{X}^{y}}
1967:{\displaystyle P_{Y}^{x}}
335:conditional probabilities
117:Principle of indifference
12105:McGrayne, S. B. (2011).
11641:
11436:Homozygous for the wild-
11418:Homozygous for the wild-
9552:law, which in classical
7755:Bayes' rule in odds form
7616:law of total probability
6371:law of total probability
5028:
2923:Law of Total Probability
2367:Recreational mathematics
2356:Markov chain Monte Carlo
2345:is a consequence of the
1086:
437:Independently of Bayes,
169:Markov chain Monte Carlo
12617:(4th ed.). Wiley.
12579:Encyclopædia Britannica
12061:Hooper, Martyn (2013).
11840:Price, Richard (1991).
11830:Holland, pp. 46–7.
11765:Allen, Richard (1999).
11088:And, on the other hand
9415:conditional probability
9292:In words: If certainly
9022:and without negations:
8292:{\displaystyle A_{1}=A}
7290:is a density function.
6252:given that proposition
6246:conditional probability
6198:given that proposition
6192:conditional probability
5784:P(¬B|A)⋅P(A)
4609:outcomes with property
4527:represents the support
4433:Bayesian interpretation
2931:conditional probability
2377:Three Prisoners problem
2343:conditional expectation
2047:be the distribution of
1435:yields Bayes' theorem:
1109:conditional probability
711:conditional probability
458:About 200 years later,
453:Bayesian interpretation
174:Laplace's approximation
161:Posterior approximation
12721:Theorems in statistics
12613:Lee, Peter M. (2012).
12293:Lee, Peter M. (2012).
12011:Stats, Data and Models
11817:10.1098/rstl.1763.0053
11542:Posterior Probability
11330:Posterior Probability
11325:(1/2) ⋅ 1 = 1/2
11256:
11212:
11079:
10965:
10849:
10811:
10775:
10561:
10520:
10500:
10480:
10448:
10416:
10396:
10361:
10323:
10230:
10202:
10175:
10155:
10084:
10052:
9862:
9709:
9689:
9659:
9639:
9615:
9542:
9501:
9473:
9445:
9407:
9372:
9371:{\displaystyle \neg B}
9349:
9348:{\displaystyle \neg A}
9326:
9306:
9282:
9203:
9141:
9016:
8981:
8940:
8859:
8745:
8626:
8606:
8586:
8566:
8543:
8466:
8446:
8426:
8329:
8293:
8250:
8128:
8013:
7891:
7745:
7611:
7592:
7439:
7284:
7261:
7093:
6914:
6871:
6701:
6537:
6439:
6289:
6288:{\displaystyle P(A|B)}
6238:
6184:
6183:{\displaystyle P(B|A)}
6146:
6085:
6043:
5997:
5750: Background
5727:
5583:
5460:
5259:
5151:
5019:
4691:
4684:
4559:
4422:
4376:
4209:
4042:
3897:
3669:
2984:
2977:
2915:
2803:
2481:
2333:
2251:is then determined by
2245:
2219:
2199:
2167:
2061:
2041:
2014:
1988:
1968:
1928:
1791:
1681:
1546:
1429:
1394:
1356:
1251:
1213:
1099:
1068:
1048:
1028:
999:
968:
912:
892:
868:
848:
828:
791:
771:
747:
727:
703:
665:
626:
606:
583:
491:
280:Mathematics portal
223:Evidence approximation
12668:10.1214/ss/1177013620
12438:(24 September 2018).
12115:Yale University Press
11915:"Inverse Probability"
11609:Inductive probability
11604:Bayesian epistemology
11257:
11213:
11080:
10966:
10850:
10812:
10776:
10562:
10521:
10501:
10481:
10449:
10417:
10397:
10362:
10324:
10231:
10203:
10201:{\displaystyle a_{A}}
10176:
10156:
10085:
10053:
9863:
9710:
9690:
9660:
9640:
9616:
9556:can be expressed as:
9543:
9502:
9474:
9446:
9408:
9373:
9350:
9327:
9307:
9283:
9204:
9142:
9017:
8982:
8941:
8860:
8746:
8627:
8607:
8587:
8567:
8544:
8467:
8447:
8427:
8330:
8294:
8251:
8129:
8014:
7892:
7746:
7609:
7593:
7440:
7285:
7262:
7094:
6888:
6872:
6702:
6538:
6440:
6335:is given in terms of
6299:, the probability of
6297:posterior probability
6290:
6239:
6185:
6147:
6086:
6044:
5998:
5728:
5584:
5461:
5260:
5152:
5020:
4685:
4658:
4554:
4420:
4377:
4232:. By Bayes' theorem,
4210:
4043:
3898:
3670:
2978:
2943:
2916:
2804:
2482:
2410:The test is also 80%
2389:Two Envelopes problem
2347:Radon–Nikodym theorem
2334:
2246:
2220:
2200:
2168:
2062:
2042:
2040:{\displaystyle P_{X}}
2015:
1989:
1969:
1929:
1792:
1682:
1547:
1430:
1395:
1357:
1252:
1214:
1094:
1069:
1049:
1029:
1000:
969:
913:
893:
869:
854:occurring given that
849:
829:
792:
772:
755:posterior probability
748:
733:occurring given that
728:
704:
666:
627:
607:
584:
486:
449:posterior probability
406:binomial distribution
364:posterior probability
352:statistical inference
184:Variational inference
12716:Probability theorems
12498:Genetics in Medicine
12445:Genetics in Medicine
12379:Audun Jøsang, 2016,
11949:Scientific Inference
11731:Frame, Paul (2015).
11710:Audun Jøsang, 2016,
11225:
11095:
10989:
10862:
10821:
10801:
10578:
10530:
10510:
10490:
10458:
10426:
10406:
10371:
10333:
10240:
10220:
10185:
10165:
10094:
10065:
9887:
9722:
9699:
9679:
9649:
9629:
9563:
9511:
9483:
9455:
9420:
9382:
9359:
9336:
9316:
9296:
9216:
9154:
9029:
8991:
8950:
8885:
8772:
8651:
8616:
8596:
8576:
8556:
8479:
8456:
8436:
8339:
8303:
8270:
8141:
8037:
7907:
7770:
7636:
7463:
7308:
7274:
7133:
6963:
6725:
6548:
6454:
6379:
6262:
6208:
6157:
6103:
6063:
6042:{\displaystyle P(A)}
6024:
5851:
5599:
5476:
5308:
5191:
5066:
4724:
4667:
4239:
4062:
3928:
3807:
3796: = A,B,C). Let
3344:
2948:
2816:
2494:
2452:
2257:
2229:
2209:
2177:
2071:
2051:
2024:
1998:
1978:
1946:
1807:
1692:
1582:
1442:
1404:
1369:
1264:
1226:
1118:
1080:marginal probability
1058:
1038:
1027:{\displaystyle P(B)}
1009:
998:{\displaystyle P(A)}
980:
922:
902:
882:
858:
838:
803:
781:
761:
737:
717:
678:
640:
616:
596:
505:
494:Statement of theorem
439:Pierre-Simon Laplace
262:Posterior predictive
231:Evidence lower bound
112:Likelihood principle
82:Bayesian probability
12711:Bayesian statistics
12684:"The Bayesian Trap"
12655:Statistical Science
12436:Goddard, Katrina AB
12299:Bayesian Statistics
12178:. New York: Wiley.
11636:Bayesian persuasion
11614:Quantum Bayesianism
10746:
10698:
10654:
10618:
10553:
10475:
10356:
10315:
10277:
10181:, and the argument
10147:
10120:
10019:
9992:
9962:
9924:
9715:, is expressed as:
9554:propositional logic
8877:Propositional logic
8762:false negative rate
7678:
7104:probability density
6921:Ω generated by two
5780:= P(A|B)⋅P(B)
5745:
3679:Defective item rate
2420:false positive rate
2274:
2244:{\displaystyle Y=y}
2194:
2128:
2013:{\displaystyle X=x}
1963:
1573:conditional density
1560:For two continuous
478:Nicholas Saunderson
468:Pythagorean theorem
460:Sir Harold Jeffreys
427:Bayesian statistics
360:likelihood function
35:Bayesian statistics
29:Part of a series on
12331:Trinity University
12218:10.1111/sjos.12550
12172:Tjur, Tue (1980).
11623:, a 2005 essay in
11596:Mathematics portal
11531:Joint Probability
11509:Prior Probability
11319:Joint Probability
11297:Prior Probability
11252:
11208:
11075:
10961:
10845:
10807:
10771:
10723:
10678:
10634:
10587:
10557:
10539:
10516:
10496:
10476:
10461:
10444:
10412:
10392:
10357:
10336:
10319:
10281:
10246:
10226:
10198:
10171:
10151:
10124:
10100:
10080:
10048:
9996:
9972:
9928:
9893:
9858:
9705:
9685:
9655:
9635:
9611:
9538:
9497:
9469:
9441:
9403:
9368:
9345:
9322:
9302:
9278:
9199:
9137:
9012:
8977:
8936:
8855:
8741:
8622:
8612:given information
8602:
8582:
8562:
8539:
8462:
8442:
8422:
8325:
8289:
8246:
8124:
8009:
7887:
7759:Bayes' theorem in
7741:
7661:
7612:
7588:
7435:
7280:
7257:
7122:is continuous and
7089:
6915:
6867:
6697:
6644:
6533:
6482:
6435:
6406:
6285:
6234:
6180:
6142:
6081:
6039:
5993:
5741:
5723:
5579:
5456:
5255:
5162:Bayesian inference
5147:
5015:
5013:
4692:
4680:
4560:
4541:Bayesian inference
4423:
4372:
4221:We are given that
4205:
4089:
4038:
3893:
3665:
3663:
3006:The importance of
2985:
2973:
2927:partition of a set
2911:
2799:
2797:
2477:
2416:true negative rate
2405:true positive rate
2381:Monty Hall problem
2329:
2260:
2241:
2215:
2195:
2180:
2163:
2114:
2057:
2037:
2010:
1984:
1964:
1949:
1924:
1787:
1677:
1542:
1425:
1390:
1352:
1247:
1209:
1100:
1064:
1044:
1024:
995:
964:
908:
888:
864:
844:
824:
787:
767:
743:
723:
699:
661:
622:
602:
579:
348:Bayesian inference
205:Bayesian estimator
153:Hierarchical model
77:Bayesian inference
12624:978-1-118-33257-3
12605:978-1-118-09156-2
12416:978-0-262-01319-2
12337:on 21 August 2004
12312:978-1-1183-3257-3
12185:978-0-471-27824-5
11734:Liberty's Apostle
11720:978-3-319-42337-1
11627:by John Ioannidis
11552:
11551:
11466:
11465:
11340:
11339:
10959:
10810:{\displaystyle C}
10766:
10606:
10519:{\displaystyle S}
10499:{\displaystyle A}
10415:{\displaystyle S}
10300:
10265:
10229:{\displaystyle A}
10210:prior probability
10174:{\displaystyle S}
10077:
10032:
9947:
9912:
9853:
9708:{\displaystyle A}
9688:{\displaystyle a}
9670:prior probability
9658:{\displaystyle B}
9638:{\displaystyle A}
9325:{\displaystyle A}
9305:{\displaystyle B}
9132:
8841:
8806:
8783:
8695:
8676:
8625:{\displaystyle B}
8605:{\displaystyle A}
8585:{\displaystyle A}
8565:{\displaystyle A}
8465:{\displaystyle A}
8445:{\displaystyle A}
8241:
8119:
8007:
7583:
7430:
7283:{\displaystyle f}
7255:
7087:
6862:
6692:
6635:
6473:
6397:
6099:is false, where
6051:prior probability
5988:
5842:
5841:
5832: P(B)
5817:P(¬A) =
5777:P(B|A)⋅P(A)
5743:Contingency table
5718:
5694:
5378:
5142:
5059:) ≠0,
4993:
4945:
4939:
4925:
4917:
4900:
4886:
4878:
4862:
4848:
4840:
4816:
4810:
4794:
4780:
4772:
4748:
4740:
4678:
4636:(the posterior).
4578:(the prior) and
4370:
4357:
4336:
4080:
3771:
3770:
3650:
3637:
3575:
3569:
3555:
3545:
3528:
3514:
3504:
3488:
3474:
3464:
3440:
3434:
3418:
3404:
3394:
3370:
3360:
3335:
3334:
3252:
3251:
3172:
3171:
3092:
3091:
2968:
2960:
2906:
2892:
2884:
2867:
2853:
2845:
2828:
2784:
2763:
2715:
2709:
2695:
2687:
2670:
2656:
2648:
2632:
2618:
2610:
2586:
2580:
2564:
2550:
2542:
2518:
2510:
2472:
2464:
2385:Two Child problem
2218:{\displaystyle X}
2060:{\displaystyle X}
1987:{\displaystyle Y}
1919:
1785:
1675:
1522:
1514:
1332:
1324:
1186:
1178:
1076:prior probability
1067:{\displaystyle B}
1047:{\displaystyle A}
911:{\displaystyle B}
891:{\displaystyle A}
867:{\displaystyle A}
847:{\displaystyle B}
790:{\displaystyle B}
770:{\displaystyle A}
746:{\displaystyle B}
726:{\displaystyle A}
625:{\displaystyle B}
605:{\displaystyle A}
577:
470:is to geometry".
432:Benjamin Franklin
340:base-rate fallacy
316:
315:
210:Credible interval
143:Linear regression
12728:
12697:
12672:
12670:
12645:
12628:
12609:
12583:
12561:
12559:
12558:
12542:
12539:
12533:
12530:
12524:
12523:
12513:
12489:
12480:
12479:
12469:
12431:
12425:
12424:
12419:. Archived from
12393:
12387:
12377:
12371:
12370:
12368:
12367:
12353:
12347:
12346:
12344:
12342:
12333:. Archived from
12323:
12317:
12316:
12290:
12284:
12283:
12255:
12249:
12248:
12230:
12220:
12196:
12190:
12189:
12169:
12163:
12162:
12154:
12148:
12147:
12135:
12129:
12128:
12112:
12102:
12093:
12092:
12082:
12058:
12052:
12051:
12031:
12025:
12024:
12006:
12000:
11999:
11979:
11973:
11972:
11953:(3rd ed.).
11952:
11943:Jeffreys, Harold
11939:
11933:
11932:
11910:
11904:
11903:
11883:
11877:
11871:
11865:
11864:
11862:
11860:
11837:
11831:
11828:
11822:
11821:
11819:
11796:
11790:
11789:
11787:
11785:
11762:
11756:
11755:
11753:
11751:
11728:
11722:
11714:Springer, Cham,
11708:
11688:
11652:
11598:
11593:
11592:
11564:
11563:
11559:
11495:
11494:
11487:
11486:
11482:
11477:
11476:
11472:
11403:
11402:
11382:
11381:
11377:
11372:
11371:
11367:
11362:
11361:
11357:
11352:
11351:
11347:
11283:
11282:
11261:
11259:
11258:
11253:
11217:
11215:
11214:
11209:
11084:
11082:
11081:
11076:
10970:
10968:
10967:
10962:
10960:
10958:
10938:
10893:
10854:
10852:
10851:
10846:
10816:
10814:
10813:
10808:
10780:
10778:
10777:
10772:
10767:
10765:
10745:
10740:
10697:
10692:
10670:
10653:
10648:
10626:
10617:
10612:
10608:
10607:
10602:
10597:
10566:
10564:
10563:
10558:
10552:
10547:
10525:
10523:
10522:
10517:
10505:
10503:
10502:
10497:
10485:
10483:
10482:
10477:
10474:
10469:
10453:
10451:
10450:
10445:
10421:
10419:
10418:
10413:
10401:
10399:
10398:
10393:
10366:
10364:
10363:
10358:
10355:
10350:
10328:
10326:
10325:
10320:
10314:
10309:
10302:
10301:
10296:
10291:
10276:
10271:
10267:
10266:
10261:
10256:
10235:
10233:
10232:
10227:
10207:
10205:
10204:
10199:
10197:
10196:
10180:
10178:
10177:
10172:
10160:
10158:
10157:
10152:
10146:
10141:
10119:
10114:
10089:
10087:
10086:
10081:
10079:
10078:
10070:
10057:
10055:
10054:
10049:
10044:
10043:
10034:
10033:
10025:
10018:
10013:
9991:
9986:
9961:
9956:
9949:
9948:
9943:
9938:
9923:
9918:
9914:
9913:
9908:
9903:
9878:subjective logic
9872:Subjective logic
9867:
9865:
9864:
9859:
9854:
9852:
9779:
9765:
9714:
9712:
9711:
9706:
9694:
9692:
9691:
9686:
9664:
9662:
9661:
9656:
9644:
9642:
9641:
9636:
9620:
9618:
9617:
9612:
9547:
9545:
9544:
9539:
9506:
9504:
9503:
9498:
9478:
9476:
9475:
9470:
9450:
9448:
9447:
9442:
9412:
9410:
9409:
9404:
9377:
9375:
9374:
9369:
9354:
9352:
9351:
9346:
9331:
9329:
9328:
9323:
9311:
9309:
9308:
9303:
9287:
9285:
9284:
9279:
9208:
9206:
9205:
9200:
9146:
9144:
9143:
9138:
9133:
9131:
9114:
9100:
9098:
9094:
9021:
9019:
9018:
9013:
8986:
8984:
8983:
8978:
8945:
8943:
8942:
8937:
8864:
8862:
8861:
8856:
8842:
8840:
8826:
8812:
8807:
8805:
8797:
8789:
8784:
8776:
8750:
8748:
8747:
8742:
8713:
8696:
8693:
8685:
8677:
8674:
8663:
8662:
8631:
8629:
8628:
8623:
8611:
8609:
8608:
8603:
8591:
8589:
8588:
8583:
8571:
8569:
8568:
8563:
8548:
8546:
8545:
8540:
8471:
8469:
8468:
8463:
8451:
8449:
8448:
8443:
8431:
8429:
8428:
8423:
8397:
8334:
8332:
8331:
8326:
8315:
8314:
8298:
8296:
8295:
8290:
8282:
8281:
8255:
8253:
8252:
8247:
8242:
8240:
8230:
8229:
8213:
8203:
8202:
8186:
8172:
8171:
8159:
8158:
8133:
8131:
8130:
8125:
8120:
8118:
8114:
8113:
8097:
8093:
8092:
8076:
8068:
8067:
8055:
8054:
8028:likelihood ratio
8018:
8016:
8015:
8010:
8008:
8006:
8002:
8001:
7979:
7975:
7974:
7952:
7938:
7937:
7925:
7924:
7896:
7894:
7893:
7888:
7877:
7876:
7864:
7863:
7842:
7841:
7829:
7828:
7801:
7800:
7788:
7787:
7750:
7748:
7747:
7742:
7721:
7720:
7702:
7701:
7691:
7677:
7672:
7648:
7647:
7597:
7595:
7594:
7589:
7584:
7582:
7572:
7571:
7561:
7551:
7550:
7532:
7531:
7527:
7519:
7505:
7491:
7490:
7486:
7478:
7456:are continuous,
7444:
7442:
7441:
7436:
7431:
7429:
7419:
7418:
7408:
7401:
7378:
7377:
7373:
7365:
7351:
7340:
7332:
7324:
7297:is discrete and
7289:
7287:
7286:
7281:
7266:
7264:
7263:
7258:
7256:
7254:
7247:
7232:
7222:
7221:
7206:
7198:
7190:
7175:
7161:
7160:
7156:
7148:
7098:
7096:
7095:
7090:
7088:
7086:
7079:
7064:
7057:
7037:
7029:
7021:
7006:
6995:
6987:
6979:
6923:random variables
6881:Random variables
6876:
6874:
6873:
6868:
6863:
6861:
6836:
6801:
6786:
6767:
6752:
6741:
6706:
6704:
6703:
6698:
6693:
6691:
6687:
6686:
6668:
6667:
6658:
6643:
6633:
6629:
6628:
6610:
6609:
6600:
6585:
6574:
6569:
6568:
6542:
6540:
6539:
6534:
6529:
6525:
6524:
6506:
6505:
6496:
6481:
6444:
6442:
6441:
6436:
6428:
6427:
6405:
6316:Often, for some
6294:
6292:
6291:
6286:
6278:
6243:
6241:
6240:
6235:
6224:
6189:
6187:
6186:
6181:
6173:
6151:
6149:
6148:
6143:
6090:
6088:
6087:
6082:
6048:
6046:
6045:
6040:
6009:For proposition
6002:
6000:
5999:
5994:
5989:
5987:
5962:
5927:
5912:
5893:
5878:
5867:
5746:
5740:
5737:Alternative form
5732:
5730:
5729:
5724:
5719:
5717:
5700:
5695:
5693:
5665:
5627:
5609:
5588:
5586:
5585:
5580:
5545:
5507:
5465:
5463:
5462:
5457:
5443:
5396:
5379:
5376:
5368:
5324:
5264:
5262:
5261:
5256:
5245:
5207:
5156:
5154:
5153:
5148:
5143:
5141:
5127:
5108:
5093:
5082:
5051:, provided that
5024:
5022:
5021:
5016:
5014:
4998:
4994:
4992:
4969:
4958:
4950:
4946:
4944:
4940:
4937:
4926:
4923:
4918:
4915:
4901:
4898:
4887:
4884:
4879:
4876:
4867:
4863:
4860:
4849:
4846:
4841:
4838:
4829:
4821:
4817:
4815:
4811:
4808:
4799:
4795:
4792:
4781:
4778:
4773:
4770:
4761:
4749:
4746:
4741:
4738:
4689:
4687:
4686:
4681:
4679:
4671:
4632:those with
4526:
4524:
4523:
4513:
4510:
4381:
4379:
4378:
4373:
4371:
4363:
4358:
4353:
4342:
4337:
4335:
4321:
4317:
4316:
4298:
4297:
4288:
4273:
4262:
4257:
4256:
4214:
4212:
4211:
4206:
4132:
4131:
4113:
4112:
4103:
4088:
4047:
4045:
4044:
4039:
4028:
4027:
4018:
3991:
3990:
3981:
3954:
3953:
3944:
3902:
3900:
3899:
3894:
3883:
3882:
3854:
3853:
3825:
3824:
3683:
3682:
3674:
3672:
3671:
3666:
3664:
3651:
3643:
3638:
3636:
3623:
3599:
3588:
3580:
3576:
3574:
3570:
3567:
3556:
3553:
3551:
3546:
3543:
3529:
3526:
3515:
3512:
3510:
3505:
3502:
3493:
3489:
3486:
3475:
3472:
3470:
3465:
3462:
3453:
3445:
3441:
3439:
3435:
3432:
3423:
3419:
3416:
3405:
3402:
3400:
3395:
3392:
3383:
3371:
3368:
3366:
3361:
3358:
3266:
3265:
3174:
3173:
3094:
3093:
3014:
3013:
2982:
2980:
2979:
2974:
2969:
2966:
2961:
2958:
2920:
2918:
2917:
2912:
2907:
2904:
2893:
2890:
2885:
2882:
2868:
2865:
2854:
2851:
2846:
2843:
2829:
2826:
2812:The denominator
2808:
2806:
2805:
2800:
2798:
2785:
2783:
2769:
2764:
2762:
2739:
2728:
2720:
2716:
2714:
2710:
2707:
2696:
2693:
2688:
2685:
2671:
2668:
2657:
2654:
2649:
2646:
2637:
2633:
2630:
2619:
2616:
2611:
2608:
2599:
2591:
2587:
2585:
2581:
2578:
2569:
2565:
2562:
2551:
2548:
2543:
2540:
2531:
2519:
2516:
2511:
2508:
2486:
2484:
2483:
2478:
2473:
2470:
2465:
2462:
2338:
2336:
2335:
2330:
2316:
2302:
2301:
2273:
2268:
2250:
2248:
2247:
2242:
2224:
2222:
2221:
2216:
2204:
2202:
2201:
2196:
2193:
2188:
2172:
2170:
2169:
2164:
2150:
2149:
2127:
2122:
2089:
2088:
2066:
2064:
2063:
2058:
2046:
2044:
2043:
2038:
2036:
2035:
2019:
2017:
2016:
2011:
1993:
1991:
1990:
1985:
1973:
1971:
1970:
1965:
1962:
1957:
1933:
1931:
1930:
1925:
1920:
1918:
1908:
1907:
1897:
1887:
1886:
1868:
1867:
1845:
1831:
1830:
1796:
1794:
1793:
1788:
1786:
1784:
1774:
1773:
1763:
1747:
1746:
1730:
1716:
1715:
1686:
1684:
1683:
1678:
1676:
1674:
1664:
1663:
1653:
1637:
1636:
1620:
1606:
1605:
1562:random variables
1551:
1549:
1548:
1543:
1523:
1520:
1515:
1513:
1499:
1467:
1434:
1432:
1431:
1426:
1399:
1397:
1396:
1391:
1361:
1359:
1358:
1353:
1333:
1330:
1325:
1323:
1309:
1289:
1256:
1254:
1253:
1248:
1218:
1216:
1215:
1210:
1187:
1184:
1179:
1177:
1163:
1143:
1098:
1095:Visual proof of
1073:
1071:
1070:
1065:
1053:
1051:
1050:
1045:
1033:
1031:
1030:
1025:
1004:
1002:
1001:
996:
973:
971:
970:
965:
917:
915:
914:
909:
897:
895:
894:
889:
873:
871:
870:
865:
853:
851:
850:
845:
833:
831:
830:
825:
796:
794:
793:
788:
776:
774:
773:
768:
752:
750:
749:
744:
732:
730:
729:
724:
708:
706:
705:
700:
670:
668:
667:
662:
631:
629:
628:
623:
611:
609:
608:
603:
588:
586:
585:
580:
578:
576:
562:
530:
397:
396:
393:
392:
389:
386:
308:
301:
294:
278:
277:
244:Model evaluation
45:
26:
25:
12736:
12735:
12731:
12730:
12729:
12727:
12726:
12725:
12701:
12700:
12682:
12679:
12625:
12606:
12590:
12588:Further reading
12556:
12554:
12551:
12546:
12545:
12540:
12536:
12531:
12527:
12490:
12483:
12432:
12428:
12417:
12394:
12390:
12378:
12374:
12365:
12363:
12355:
12354:
12350:
12340:
12338:
12325:
12324:
12320:
12313:
12291:
12287:
12272:
12256:
12252:
12197:
12193:
12186:
12170:
12166:
12155:
12151:
12136:
12132:
12125:
12103:
12096:
12059:
12055:
12032:
12028:
12021:
12007:
12003:
11980:
11976:
11969:
11940:
11936:
11929:
11911:
11907:
11900:
11884:
11880:
11872:
11868:
11858:
11856:
11854:
11838:
11834:
11829:
11825:
11797:
11793:
11783:
11781:
11779:
11763:
11759:
11749:
11747:
11745:
11729:
11725:
11709:
11702:
11697:
11692:
11691:
11653:
11649:
11644:
11594:
11587:
11584:
11571:
11561:
11557:
11556:
11484:
11480:
11479:
11474:
11470:
11469:
11437:
11428:(a CF carrier)
11427:
11419:
11413:
11411:
11410:
11408:
11389:
11379:
11375:
11374:
11369:
11365:
11364:
11359:
11355:
11354:
11349:
11345:
11344:
11281:
11273:genetic testing
11268:
11266:Use in genetics
11226:
11223:
11222:
11096:
11093:
11092:
10990:
10987:
10986:
10977:
10939:
10894:
10892:
10863:
10860:
10859:
10822:
10819:
10818:
10802:
10799:
10798:
10795:
10790:
10788:Generalizations
10741:
10727:
10693:
10682:
10671:
10649:
10638:
10627:
10625:
10613:
10598:
10596:
10595:
10591:
10579:
10576:
10575:
10548:
10543:
10531:
10528:
10527:
10511:
10508:
10507:
10491:
10488:
10487:
10470:
10465:
10459:
10456:
10455:
10427:
10424:
10423:
10407:
10404:
10403:
10372:
10369:
10368:
10351:
10340:
10334:
10331:
10330:
10310:
10292:
10290:
10289:
10285:
10272:
10257:
10255:
10254:
10250:
10241:
10238:
10237:
10221:
10218:
10217:
10192:
10188:
10186:
10183:
10182:
10166:
10163:
10162:
10142:
10128:
10115:
10104:
10095:
10092:
10091:
10069:
10068:
10066:
10063:
10062:
10039:
10035:
10024:
10023:
10014:
10000:
9987:
9976:
9957:
9939:
9937:
9936:
9932:
9919:
9904:
9902:
9901:
9897:
9888:
9885:
9884:
9874:
9780:
9766:
9764:
9723:
9720:
9719:
9700:
9697:
9696:
9680:
9677:
9676:
9650:
9647:
9646:
9630:
9627:
9626:
9564:
9561:
9560:
9512:
9509:
9508:
9484:
9481:
9480:
9456:
9453:
9452:
9421:
9418:
9417:
9383:
9380:
9379:
9360:
9357:
9356:
9337:
9334:
9333:
9317:
9314:
9313:
9297:
9294:
9293:
9217:
9214:
9213:
9155:
9152:
9151:
9115:
9101:
9099:
9069:
9065:
9030:
9027:
9026:
8992:
8989:
8988:
8951:
8948:
8947:
8886:
8883:
8882:
8879:
8874:
8827:
8813:
8811:
8798:
8790:
8788:
8775:
8773:
8770:
8769:
8709:
8692:
8681:
8673:
8658:
8654:
8652:
8649:
8648:
8617:
8614:
8613:
8597:
8594:
8593:
8577:
8574:
8573:
8557:
8554:
8553:
8480:
8477:
8476:
8457:
8454:
8453:
8437:
8434:
8433:
8393:
8340:
8337:
8336:
8310:
8306:
8304:
8301:
8300:
8277:
8273:
8271:
8268:
8267:
8225:
8221:
8214:
8198:
8194:
8187:
8185:
8167:
8163:
8154:
8150:
8142:
8139:
8138:
8109:
8105:
8098:
8088:
8084:
8077:
8075:
8063:
8059:
8050:
8046:
8038:
8035:
8034:
7997:
7993:
7980:
7970:
7966:
7953:
7951:
7933:
7929:
7920:
7916:
7908:
7905:
7904:
7872:
7868:
7859:
7855:
7837:
7833:
7824:
7820:
7796:
7792:
7783:
7779:
7771:
7768:
7767:
7757:
7716:
7712:
7687:
7683:
7679:
7673:
7665:
7643:
7639:
7637:
7634:
7633:
7623:
7604:
7567:
7563:
7562:
7546:
7542:
7523:
7515:
7511:
7507:
7506:
7504:
7482:
7474:
7470:
7466:
7464:
7461:
7460:
7414:
7410:
7409:
7397:
7369:
7361:
7357:
7353:
7352:
7350:
7336:
7328:
7320:
7309:
7306:
7305:
7301:is continuous,
7275:
7272:
7271:
7243:
7233:
7217:
7213:
7202:
7194:
7186:
7176:
7174:
7152:
7144:
7140:
7136:
7134:
7131:
7130:
7116:
7075:
7065:
7053:
7033:
7025:
7017:
7007:
7005:
6991:
6983:
6975:
6964:
6961:
6960:
6883:
6832:
6797:
6787:
6763:
6753:
6751:
6737:
6726:
6723:
6722:
6716:binary variable
6682:
6678:
6663:
6659:
6654:
6639:
6634:
6624:
6620:
6605:
6601:
6596:
6586:
6584:
6570:
6564:
6560:
6549:
6546:
6545:
6520:
6516:
6501:
6497:
6492:
6477:
6472:
6455:
6452:
6451:
6423:
6419:
6401:
6380:
6377:
6376:
6359:
6344:
6325:
6314:
6274:
6263:
6260:
6259:
6220:
6209:
6206:
6205:
6169:
6158:
6155:
6154:
6104:
6101:
6100:
6064:
6061:
6060:
6025:
6022:
6021:
5958:
5923:
5913:
5889:
5879:
5877:
5863:
5852:
5849:
5848:
5818:
5812:
5805:
5798:
5786:
5779:
5762:
5755:
5753:
5751:
5739:
5704:
5699:
5661:
5623:
5613:
5608:
5600:
5597:
5596:
5541:
5503:
5477:
5474:
5473:
5439:
5392:
5377: and
5375:
5364:
5320:
5309:
5306:
5305:
5284:
5277:
5241:
5203:
5192:
5189:
5188:
5128:
5104:
5094:
5092:
5078:
5067:
5064:
5063:
5041:
5036:
5031:
5012:
5011:
4996:
4995:
4970:
4959:
4957:
4948:
4947:
4936:
4922:
4914:
4897:
4883:
4875:
4868:
4859:
4845:
4837:
4830:
4828:
4819:
4818:
4807:
4800:
4791:
4777:
4769:
4762:
4760:
4753:
4745:
4737:
4727:
4725:
4722:
4721:
4670:
4668:
4665:
4664:
4653:
4621: |
4598: |
4549:
4514:
4511:
4505: |
4497:
4496:
4494:
4478: |
4435:
4415:
4413:Interpretations
4409:) = 5/24.
4405: |
4404:
4396:
4362:
4343:
4341:
4322:
4312:
4308:
4293:
4289:
4284:
4274:
4272:
4258:
4252:
4248:
4240:
4237:
4236:
4231:
4127:
4123:
4108:
4104:
4099:
4084:
4063:
4060:
4059:
4023:
4019:
4014:
3986:
3982:
3977:
3949:
3945:
3940:
3929:
3926:
3925:
3920:
3914: |
3878:
3874:
3849:
3845:
3820:
3816:
3808:
3805:
3804:
3786:
3692:
3690:
3688:
3681:
3662:
3661:
3642:
3619:
3600:
3589:
3587:
3578:
3577:
3566:
3552:
3547:
3542:
3525:
3511:
3506:
3501:
3494:
3485:
3471:
3466:
3461:
3454:
3452:
3443:
3442:
3431:
3424:
3415:
3401:
3396:
3391:
3384:
3382:
3375:
3367:
3362:
3357:
3347:
3345:
3342:
3341:
3274:
3271:
3257:
3182:
3179:
3102:
3099:
3022:
3019:
3004:
2965:
2957:
2949:
2946:
2945:
2903:
2889:
2881:
2864:
2850:
2842:
2825:
2817:
2814:
2813:
2796:
2795:
2773:
2768:
2740:
2729:
2727:
2718:
2717:
2706:
2692:
2684:
2667:
2653:
2645:
2638:
2629:
2615:
2607:
2600:
2598:
2589:
2588:
2577:
2570:
2561:
2547:
2539:
2532:
2530:
2523:
2515:
2507:
2497:
2495:
2492:
2491:
2469:
2461:
2453:
2450:
2449:
2397:
2369:
2364:
2312:
2297:
2293:
2269:
2264:
2258:
2255:
2254:
2230:
2227:
2226:
2210:
2207:
2206:
2189:
2184:
2178:
2175:
2174:
2145:
2141:
2123:
2118:
2078:
2074:
2072:
2069:
2068:
2052:
2049:
2048:
2031:
2027:
2025:
2022:
2021:
1999:
1996:
1995:
1979:
1976:
1975:
1958:
1953:
1947:
1944:
1943:
1940:
1903:
1899:
1898:
1882:
1878:
1851:
1847:
1846:
1844:
1814:
1810:
1808:
1805:
1804:
1769:
1765:
1764:
1736:
1732:
1731:
1729:
1699:
1695:
1693:
1690:
1689:
1659:
1655:
1654:
1626:
1622:
1621:
1619:
1589:
1585:
1583:
1580:
1579:
1558:
1519:
1500:
1468:
1466:
1443:
1440:
1439:
1405:
1402:
1401:
1370:
1367:
1366:
1329:
1310:
1290:
1288:
1265:
1262:
1261:
1227:
1224:
1223:
1183:
1164:
1144:
1142:
1119:
1116:
1115:
1105:
1096:
1089:
1059:
1056:
1055:
1039:
1036:
1035:
1010:
1007:
1006:
981:
978:
977:
923:
920:
919:
903:
900:
899:
883:
880:
879:
859:
856:
855:
839:
836:
835:
804:
801:
800:
782:
779:
778:
762:
759:
758:
738:
735:
734:
718:
715:
714:
679:
676:
675:
641:
638:
637:
617:
614:
613:
597:
594:
593:
590:
563:
531:
529:
506:
503:
502:
496:
474:Stephen Stigler
383:
379:
372:
321:(alternatively
312:
272:
257:Model averaging
236:Nested sampling
148:Empirical Bayes
138:Conjugate prior
107:Cromwell's rule
24:
21:Bayes estimator
17:
12:
11:
5:
12734:
12724:
12723:
12718:
12713:
12699:
12698:
12678:
12677:External links
12675:
12674:
12673:
12661:(3): 359–363.
12646:
12629:
12623:
12610:
12604:
12589:
12586:
12585:
12584:
12574:Chisholm, Hugh
12570:Price, Richard
12550:
12547:
12544:
12543:
12534:
12525:
12504:(5): 439–449.
12481:
12452:(4): 790–797.
12426:
12423:on 2014-04-27.
12415:
12388:
12372:
12348:
12318:
12311:
12285:
12271:978-1475741452
12270:
12250:
12211:(3): 969–991.
12191:
12184:
12164:
12149:
12130:
12124:978-0300188226
12123:
12094:
12053:
12042:(2): 109–110.
12026:
12020:978-0321986498
12019:
12001:
11990:(4): 290–296.
11974:
11968:978-0521180788
11967:
11934:
11928:978-0674403413
11927:
11905:
11898:
11878:
11876:, p. 314.
11866:
11853:978-0521409698
11852:
11832:
11823:
11791:
11778:978-0791494516
11777:
11757:
11744:978-1783162161
11743:
11723:
11699:
11698:
11696:
11693:
11690:
11689:
11687:
11686:
11675:
11667:
11646:
11645:
11643:
11640:
11639:
11638:
11633:
11628:
11616:
11611:
11606:
11600:
11599:
11583:
11580:
11570:
11567:
11550:
11549:
11546:
11543:
11539:
11538:
11535:
11532:
11528:
11527:
11524:
11521:
11517:
11516:
11513:
11510:
11506:
11505:
11502:
11499:
11464:
11463:
11457:
11454:
11447:
11446:
11443:
11440:
11431:
11430:
11422:
11414:
11409:
11406:
11388:
11385:
11338:
11337:
11334:
11331:
11327:
11326:
11323:
11320:
11316:
11315:
11312:
11309:
11305:
11304:
11301:
11298:
11294:
11293:
11290:
11287:
11280:
11277:
11267:
11264:
11251:
11248:
11245:
11242:
11239:
11236:
11233:
11230:
11219:
11218:
11207:
11204:
11201:
11198:
11194:
11191:
11188:
11185:
11182:
11179:
11175:
11172:
11169:
11166:
11163:
11160:
11157:
11154:
11151:
11148:
11145:
11142:
11139:
11136:
11133:
11130:
11127:
11124:
11121:
11118:
11115:
11112:
11109:
11106:
11103:
11100:
11086:
11085:
11074:
11071:
11068:
11065:
11061:
11058:
11055:
11052:
11049:
11046:
11042:
11039:
11036:
11033:
11030:
11027:
11024:
11021:
11018:
11015:
11012:
11009:
11006:
11003:
11000:
10997:
10994:
10976:
10973:
10972:
10971:
10957:
10954:
10951:
10948:
10945:
10942:
10937:
10934:
10931:
10928:
10925:
10922:
10918:
10915:
10912:
10909:
10906:
10903:
10900:
10897:
10891:
10888:
10885:
10882:
10879:
10876:
10873:
10870:
10867:
10844:
10841:
10838:
10835:
10832:
10829:
10826:
10806:
10794:
10791:
10789:
10786:
10782:
10781:
10770:
10764:
10761:
10758:
10755:
10752:
10749:
10744:
10739:
10736:
10733:
10730:
10726:
10722:
10719:
10716:
10713:
10710:
10707:
10704:
10701:
10696:
10691:
10688:
10685:
10681:
10677:
10674:
10669:
10666:
10663:
10660:
10657:
10652:
10647:
10644:
10641:
10637:
10633:
10630:
10624:
10621:
10616:
10611:
10605:
10601:
10594:
10590:
10586:
10583:
10556:
10551:
10546:
10542:
10538:
10535:
10515:
10495:
10473:
10468:
10464:
10443:
10440:
10437:
10434:
10431:
10411:
10391:
10388:
10385:
10382:
10379:
10376:
10354:
10349:
10346:
10343:
10339:
10318:
10313:
10308:
10305:
10299:
10295:
10288:
10284:
10280:
10275:
10270:
10264:
10260:
10253:
10249:
10245:
10225:
10195:
10191:
10170:
10150:
10145:
10140:
10137:
10134:
10131:
10127:
10123:
10118:
10113:
10110:
10107:
10103:
10099:
10076:
10073:
10059:
10058:
10047:
10042:
10038:
10031:
10028:
10022:
10017:
10012:
10009:
10006:
10003:
9999:
9995:
9990:
9985:
9982:
9979:
9975:
9971:
9968:
9965:
9960:
9955:
9952:
9946:
9942:
9935:
9931:
9927:
9922:
9917:
9911:
9907:
9900:
9896:
9892:
9880:expressed as:
9873:
9870:
9869:
9868:
9857:
9851:
9848:
9845:
9842:
9839:
9835:
9832:
9829:
9826:
9823:
9820:
9817:
9814:
9811:
9808:
9805:
9802:
9798:
9795:
9792:
9789:
9786:
9783:
9778:
9775:
9772:
9769:
9763:
9760:
9757:
9754:
9751:
9748:
9745:
9742:
9739:
9736:
9733:
9730:
9727:
9704:
9684:
9654:
9634:
9623:
9622:
9610:
9607:
9604:
9600:
9596:
9593:
9590:
9586:
9582:
9579:
9575:
9571:
9568:
9550:contraposition
9537:
9534:
9531:
9528:
9525:
9522:
9519:
9516:
9496:
9492:
9488:
9468:
9464:
9460:
9440:
9437:
9434:
9431:
9428:
9425:
9402:
9399:
9396:
9393:
9390:
9387:
9367:
9364:
9344:
9341:
9321:
9301:
9290:
9289:
9277:
9274:
9271:
9268:
9265:
9262:
9259:
9256:
9253:
9250:
9246:
9242:
9239:
9236:
9233:
9230:
9227:
9224:
9221:
9198:
9195:
9192:
9189:
9186:
9183:
9180:
9177:
9174:
9171:
9168:
9165:
9162:
9159:
9148:
9147:
9136:
9130:
9127:
9124:
9121:
9118:
9113:
9110:
9107:
9104:
9097:
9093:
9090:
9087:
9084:
9081:
9078:
9075:
9072:
9068:
9064:
9061:
9058:
9055:
9052:
9049:
9046:
9043:
9040:
9037:
9034:
9011:
9008:
9005:
9002:
8999:
8996:
8976:
8973:
8970:
8967:
8964:
8961:
8958:
8955:
8935:
8932:
8929:
8926:
8923:
8920:
8917:
8914:
8911:
8908:
8905:
8902:
8899:
8896:
8893:
8890:
8878:
8875:
8873:
8870:
8866:
8865:
8854:
8851:
8848:
8845:
8839:
8836:
8833:
8830:
8825:
8822:
8819:
8816:
8810:
8804:
8801:
8796:
8793:
8787:
8782:
8779:
8751:. Now, if the
8740:
8737:
8734:
8731:
8728:
8725:
8722:
8719:
8716:
8712:
8708:
8705:
8702:
8699:
8694:False Positive
8691:
8688:
8684:
8680:
8672:
8669:
8666:
8661:
8657:
8621:
8601:
8581:
8561:
8550:
8549:
8538:
8535:
8532:
8529:
8526:
8523:
8520:
8517:
8514:
8511:
8508:
8505:
8502:
8499:
8496:
8493:
8490:
8487:
8484:
8461:
8441:
8421:
8418:
8415:
8412:
8409:
8406:
8403:
8400:
8396:
8392:
8389:
8386:
8383:
8380:
8377:
8374:
8371:
8368:
8365:
8362:
8359:
8356:
8353:
8350:
8347:
8344:
8324:
8321:
8318:
8313:
8309:
8288:
8285:
8280:
8276:
8257:
8256:
8245:
8239:
8236:
8233:
8228:
8224:
8220:
8217:
8212:
8209:
8206:
8201:
8197:
8193:
8190:
8184:
8181:
8178:
8175:
8170:
8166:
8162:
8157:
8153:
8149:
8146:
8135:
8134:
8123:
8117:
8112:
8108:
8104:
8101:
8096:
8091:
8087:
8083:
8080:
8074:
8071:
8066:
8062:
8058:
8053:
8049:
8045:
8042:
8022:is called the
8020:
8019:
8005:
8000:
7996:
7992:
7989:
7986:
7983:
7978:
7973:
7969:
7965:
7962:
7959:
7956:
7950:
7947:
7944:
7941:
7936:
7932:
7928:
7923:
7919:
7915:
7912:
7898:
7897:
7886:
7883:
7880:
7875:
7871:
7867:
7862:
7858:
7854:
7851:
7848:
7845:
7840:
7836:
7832:
7827:
7823:
7819:
7816:
7813:
7810:
7807:
7804:
7799:
7795:
7791:
7786:
7782:
7778:
7775:
7756:
7753:
7752:
7751:
7740:
7737:
7734:
7730:
7727:
7724:
7719:
7715:
7711:
7708:
7705:
7700:
7697:
7694:
7690:
7686:
7682:
7676:
7671:
7668:
7664:
7660:
7657:
7654:
7651:
7646:
7642:
7621:
7603:
7600:
7599:
7598:
7587:
7581:
7578:
7575:
7570:
7566:
7560:
7557:
7554:
7549:
7545:
7541:
7538:
7535:
7530:
7526:
7522:
7518:
7514:
7510:
7503:
7500:
7497:
7494:
7489:
7485:
7481:
7477:
7473:
7469:
7446:
7445:
7434:
7428:
7425:
7422:
7417:
7413:
7407:
7404:
7400:
7396:
7393:
7390:
7387:
7384:
7381:
7376:
7372:
7368:
7364:
7360:
7356:
7349:
7346:
7343:
7339:
7335:
7331:
7327:
7323:
7319:
7316:
7313:
7279:
7268:
7267:
7253:
7250:
7246:
7242:
7239:
7236:
7231:
7228:
7225:
7220:
7216:
7212:
7209:
7205:
7201:
7197:
7193:
7189:
7185:
7182:
7179:
7173:
7170:
7167:
7164:
7159:
7155:
7151:
7147:
7143:
7139:
7115:
7112:
7100:
7099:
7085:
7082:
7078:
7074:
7071:
7068:
7063:
7060:
7056:
7052:
7049:
7046:
7043:
7040:
7036:
7032:
7028:
7024:
7020:
7016:
7013:
7010:
7004:
7001:
6998:
6994:
6990:
6986:
6982:
6978:
6974:
6971:
6968:
6948: = {
6936: = {
6882:
6879:
6878:
6877:
6866:
6860:
6857:
6854:
6851:
6848:
6845:
6842:
6839:
6835:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6810:
6807:
6804:
6800:
6796:
6793:
6790:
6785:
6782:
6779:
6776:
6773:
6770:
6766:
6762:
6759:
6756:
6750:
6747:
6744:
6740:
6736:
6733:
6730:
6708:
6707:
6696:
6690:
6685:
6681:
6677:
6674:
6671:
6666:
6662:
6657:
6653:
6650:
6647:
6642:
6638:
6632:
6627:
6623:
6619:
6616:
6613:
6608:
6604:
6599:
6595:
6592:
6589:
6583:
6580:
6577:
6573:
6567:
6563:
6559:
6556:
6553:
6543:
6532:
6528:
6523:
6519:
6515:
6512:
6509:
6504:
6500:
6495:
6491:
6488:
6485:
6480:
6476:
6471:
6468:
6465:
6462:
6459:
6434:
6431:
6426:
6422:
6418:
6415:
6412:
6409:
6404:
6400:
6396:
6393:
6390:
6387:
6384:
6357:
6342:
6323:
6313:
6310:
6309:
6308:
6284:
6281:
6277:
6273:
6270:
6267:
6257:
6233:
6230:
6227:
6223:
6219:
6216:
6213:
6203:
6179:
6176:
6172:
6168:
6165:
6162:
6152:
6141:
6138:
6135:
6132:
6129:
6126:
6123:
6120:
6117:
6114:
6111:
6108:
6080:
6077:
6074:
6071:
6068:
6058:
6038:
6035:
6032:
6029:
6004:
6003:
5992:
5986:
5983:
5980:
5977:
5974:
5971:
5968:
5965:
5961:
5957:
5954:
5951:
5948:
5945:
5942:
5939:
5936:
5933:
5930:
5926:
5922:
5919:
5916:
5911:
5908:
5905:
5902:
5899:
5896:
5892:
5888:
5885:
5882:
5876:
5873:
5870:
5866:
5862:
5859:
5856:
5840:
5839:
5836:
5833:
5830:
5826:
5825:
5822:
5821:
5814:
5807:
5800:
5794:
5793:
5788:
5781:
5774:
5770:
5769:
5766:
5764:
5759:
5756:
5752:
5749:
5738:
5735:
5734:
5733:
5722:
5716:
5713:
5710:
5707:
5703:
5698:
5692:
5689:
5686:
5683:
5680:
5677:
5674:
5671:
5668:
5664:
5660:
5657:
5654:
5651:
5648:
5645:
5642:
5639:
5636:
5633:
5630:
5626:
5622:
5619:
5616:
5612:
5607:
5604:
5590:
5589:
5578:
5575:
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5544:
5540:
5537:
5534:
5531:
5528:
5525:
5522:
5519:
5516:
5513:
5510:
5506:
5502:
5499:
5496:
5493:
5490:
5487:
5484:
5481:
5467:
5466:
5455:
5452:
5449:
5446:
5442:
5438:
5435:
5432:
5429:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5395:
5391:
5388:
5385:
5382:
5374:
5371:
5367:
5363:
5360:
5357:
5354:
5351:
5348:
5345:
5342:
5339:
5336:
5333:
5330:
5327:
5323:
5319:
5316:
5313:
5282:
5275:
5266:
5265:
5254:
5251:
5248:
5244:
5240:
5237:
5234:
5231:
5228:
5225:
5222:
5219:
5216:
5213:
5210:
5206:
5202:
5199:
5196:
5158:
5157:
5146:
5140:
5137:
5134:
5131:
5126:
5123:
5120:
5117:
5114:
5111:
5107:
5103:
5100:
5097:
5091:
5088:
5085:
5081:
5077:
5074:
5071:
5040:
5037:
5035:
5032:
5030:
5027:
5026:
5025:
5010:
5007:
5004:
5001:
4999:
4997:
4991:
4988:
4985:
4982:
4979:
4976:
4973:
4968:
4965:
4962:
4956:
4953:
4951:
4949:
4943:
4935:
4932:
4929:
4921:
4913:
4910:
4907:
4904:
4896:
4893:
4890:
4882:
4874:
4871:
4866:
4858:
4855:
4852:
4844:
4836:
4833:
4827:
4824:
4822:
4820:
4814:
4806:
4803:
4798:
4790:
4787:
4784:
4776:
4768:
4765:
4759:
4756:
4754:
4752:
4744:
4736:
4733:
4730:
4729:
4677:
4674:
4652:
4649:
4548:
4545:
4537:
4536:
4491:
4469:
4434:
4431:
4414:
4411:
4402:
4394:
4383:
4382:
4369:
4366:
4361:
4356:
4352:
4349:
4346:
4340:
4334:
4331:
4328:
4325:
4320:
4315:
4311:
4307:
4304:
4301:
4296:
4292:
4287:
4283:
4280:
4277:
4271:
4268:
4265:
4261:
4255:
4251:
4247:
4244:
4229:
4216:
4215:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4135:
4130:
4126:
4122:
4119:
4116:
4111:
4107:
4102:
4098:
4095:
4092:
4087:
4083:
4079:
4076:
4073:
4070:
4067:
4049:
4048:
4037:
4034:
4031:
4026:
4022:
4017:
4013:
4010:
4007:
4003:
4000:
3997:
3994:
3989:
3985:
3980:
3976:
3973:
3970:
3966:
3963:
3960:
3957:
3952:
3948:
3943:
3939:
3936:
3933:
3918:
3904:
3903:
3892:
3889:
3886:
3881:
3877:
3873:
3870:
3866:
3863:
3860:
3857:
3852:
3848:
3844:
3841:
3837:
3834:
3831:
3828:
3823:
3819:
3815:
3812:
3784:
3769:
3768:
3765:
3762:
3757:
3753:
3752:
3749:
3748:
3745:
3742:
3737:
3733:
3732:
3729:
3726:
3723:
3719:
3718:
3715:
3712:
3709:
3705:
3704:
3701:
3699:
3696:
3693:
3689:
3686:
3680:
3677:
3676:
3675:
3660:
3657:
3654:
3649:
3646:
3641:
3635:
3632:
3629:
3626:
3622:
3618:
3615:
3612:
3609:
3606:
3603:
3598:
3595:
3592:
3586:
3583:
3581:
3579:
3573:
3565:
3562:
3559:
3550:
3541:
3538:
3535:
3532:
3524:
3521:
3518:
3509:
3500:
3497:
3492:
3484:
3481:
3478:
3469:
3460:
3457:
3451:
3448:
3446:
3444:
3438:
3430:
3427:
3422:
3414:
3411:
3408:
3399:
3390:
3387:
3381:
3378:
3376:
3374:
3365:
3356:
3353:
3350:
3349:
3333:
3332:
3329:
3326:
3323:
3319:
3318:
3315:
3314:
3311:
3308:
3305:
3301:
3300:
3297:
3294:
3291:
3287:
3286:
3283:
3281:
3278:
3275:
3272:
3269:
3256:
3253:
3250:
3249:
3245:
3244:
3241:
3238:
3233:
3229:
3228:
3225:
3224:
3221:
3218:
3215:
3211:
3210:
3207:
3204:
3199:
3195:
3194:
3191:
3189:
3186:
3183:
3180:
3177:
3170:
3169:
3165:
3164:
3161:
3158:
3153:
3149:
3148:
3145:
3144:
3141:
3138:
3135:
3131:
3130:
3127:
3124:
3119:
3115:
3114:
3111:
3109:
3106:
3103:
3100:
3097:
3090:
3089:
3085:
3084:
3081:
3078:
3073:
3069:
3068:
3065:
3064:
3061:
3058:
3055:
3051:
3050:
3047:
3044:
3039:
3035:
3034:
3031:
3029:
3026:
3023:
3020:
3017:
3003:
3000:
2996:
2995:
2992:
2972:
2964:
2956:
2953:
2910:
2902:
2899:
2896:
2888:
2880:
2877:
2874:
2871:
2863:
2860:
2857:
2849:
2841:
2838:
2835:
2832:
2824:
2821:
2810:
2809:
2794:
2791:
2788:
2782:
2779:
2776:
2772:
2767:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2738:
2735:
2732:
2726:
2723:
2721:
2719:
2713:
2705:
2702:
2699:
2691:
2683:
2680:
2677:
2674:
2666:
2663:
2660:
2652:
2644:
2641:
2636:
2628:
2625:
2622:
2614:
2606:
2603:
2597:
2594:
2592:
2590:
2584:
2576:
2573:
2568:
2560:
2557:
2554:
2546:
2538:
2535:
2529:
2526:
2524:
2522:
2514:
2506:
2503:
2500:
2499:
2476:
2468:
2460:
2457:
2446:
2445:
2425:Assuming 0.05
2403:, meaning the
2396:
2393:
2368:
2365:
2363:
2360:
2328:
2325:
2322:
2319:
2315:
2311:
2308:
2305:
2300:
2296:
2292:
2289:
2286:
2283:
2280:
2277:
2272:
2267:
2263:
2240:
2237:
2234:
2214:
2192:
2187:
2183:
2162:
2159:
2156:
2153:
2148:
2144:
2140:
2137:
2134:
2131:
2126:
2121:
2117:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2087:
2084:
2081:
2077:
2056:
2034:
2030:
2009:
2006:
2003:
1983:
1961:
1956:
1952:
1939:
1936:
1935:
1934:
1923:
1917:
1914:
1911:
1906:
1902:
1896:
1893:
1890:
1885:
1881:
1877:
1874:
1871:
1866:
1863:
1860:
1857:
1854:
1850:
1843:
1840:
1837:
1834:
1829:
1826:
1823:
1820:
1817:
1813:
1798:
1797:
1783:
1780:
1777:
1772:
1768:
1762:
1759:
1756:
1753:
1750:
1745:
1742:
1739:
1735:
1728:
1725:
1722:
1719:
1714:
1711:
1708:
1705:
1702:
1698:
1687:
1673:
1670:
1667:
1662:
1658:
1652:
1649:
1646:
1643:
1640:
1635:
1632:
1629:
1625:
1618:
1615:
1612:
1609:
1604:
1601:
1598:
1595:
1592:
1588:
1557:
1554:
1553:
1552:
1541:
1538:
1535:
1532:
1529:
1526:
1521: if
1518:
1512:
1509:
1506:
1503:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1424:
1421:
1418:
1415:
1412:
1409:
1389:
1386:
1383:
1380:
1377:
1374:
1363:
1362:
1351:
1348:
1345:
1342:
1339:
1336:
1331: if
1328:
1322:
1319:
1316:
1313:
1308:
1305:
1302:
1299:
1296:
1293:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1246:
1243:
1240:
1237:
1234:
1231:
1220:
1219:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1185: if
1182:
1176:
1173:
1170:
1167:
1162:
1159:
1156:
1153:
1150:
1147:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1104:
1101:
1097:Bayes' theorem
1088:
1085:
1084:
1083:
1063:
1043:
1023:
1020:
1017:
1014:
994:
991:
988:
985:
975:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
907:
898:given a fixed
887:
863:
843:
823:
820:
817:
814:
811:
808:
798:
786:
766:
742:
722:
698:
695:
692:
689:
686:
683:
660:
657:
654:
651:
648:
645:
621:
601:
575:
572:
569:
566:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
528:
525:
522:
519:
516:
513:
510:
500:
495:
492:
371:
368:
319:Bayes' theorem
314:
313:
311:
310:
303:
296:
288:
285:
284:
283:
282:
267:
266:
265:
264:
259:
254:
246:
245:
241:
240:
239:
238:
233:
225:
224:
220:
219:
218:
217:
212:
207:
199:
198:
194:
193:
192:
191:
186:
181:
176:
171:
163:
162:
158:
157:
156:
155:
150:
145:
140:
132:
131:
130:Model building
127:
126:
125:
124:
119:
114:
109:
104:
99:
94:
89:
87:Bayes' theorem
84:
79:
71:
70:
66:
65:
47:
46:
38:
37:
31:
30:
15:
9:
6:
4:
3:
2:
12733:
12722:
12719:
12717:
12714:
12712:
12709:
12708:
12706:
12695:
12691:
12690:
12685:
12681:
12680:
12669:
12664:
12660:
12656:
12652:
12647:
12643:
12639:
12635:
12630:
12626:
12620:
12616:
12611:
12607:
12601:
12597:
12592:
12591:
12581:
12580:
12575:
12571:
12565:
12564:public domain
12553:
12552:
12538:
12529:
12521:
12517:
12512:
12507:
12503:
12499:
12495:
12488:
12486:
12477:
12473:
12468:
12463:
12459:
12455:
12451:
12447:
12446:
12441:
12437:
12430:
12422:
12418:
12412:
12408:
12407:
12402:
12398:
12392:
12385:
12383:
12376:
12362:
12358:
12352:
12336:
12332:
12328:
12322:
12314:
12308:
12304:
12300:
12296:
12289:
12281:
12277:
12273:
12267:
12263:
12262:
12254:
12246:
12242:
12238:
12234:
12229:
12228:11250/2984409
12224:
12219:
12214:
12210:
12206:
12202:
12195:
12187:
12181:
12177:
12176:
12168:
12160:
12153:
12145:
12144:Edward Arnold
12141:
12134:
12126:
12120:
12116:
12111:
12110:
12101:
12099:
12090:
12086:
12081:
12076:
12072:
12068:
12064:
12057:
12049:
12045:
12041:
12037:
12030:
12022:
12016:
12012:
12005:
11997:
11993:
11989:
11985:
11978:
11970:
11964:
11960:
11956:
11951:
11950:
11944:
11938:
11930:
11924:
11920:
11916:
11909:
11901:
11895:
11891:
11890:
11882:
11875:
11874:Mitchell 1911
11870:
11855:
11849:
11845:
11844:
11836:
11827:
11818:
11813:
11809:
11805:
11802:
11795:
11780:
11774:
11770:
11769:
11761:
11746:
11740:
11736:
11735:
11727:
11721:
11717:
11713:
11707:
11705:
11700:
11684:
11680:
11676:
11673:
11668:
11665:
11661:
11656:
11655:
11651:
11647:
11637:
11634:
11632:
11629:
11626:
11622:
11621:
11617:
11615:
11612:
11610:
11607:
11605:
11602:
11601:
11597:
11591:
11586:
11579:
11575:
11566:
11547:
11544:
11541:
11540:
11536:
11533:
11530:
11529:
11525:
11522:
11519:
11518:
11514:
11511:
11508:
11507:
11503:
11500:
11497:
11496:
11493:
11489:
11462:
11458:
11455:
11453:
11449:
11448:
11444:
11441:
11439:
11433:
11432:
11429:
11423:
11421:
11415:
11405:
11404:
11401:
11397:
11393:
11384:
11335:
11332:
11329:
11328:
11324:
11321:
11318:
11317:
11313:
11310:
11307:
11306:
11302:
11299:
11296:
11295:
11291:
11288:
11285:
11284:
11276:
11274:
11263:
11246:
11243:
11240:
11234:
11228:
11202:
11196:
11189:
11183:
11177:
11170:
11167:
11164:
11158:
11152:
11149:
11143:
11140:
11137:
11134:
11131:
11125:
11122:
11116:
11113:
11110:
11107:
11104:
11098:
11091:
11090:
11089:
11069:
11063:
11056:
11050:
11044:
11037:
11034:
11031:
11025:
11019:
11016:
11010:
11007:
11004:
11001:
10998:
10992:
10985:
10984:
10983:
10982:
10952:
10946:
10940:
10932:
10926:
10920:
10913:
10910:
10907:
10901:
10895:
10889:
10883:
10880:
10877:
10871:
10865:
10858:
10857:
10856:
10842:
10839:
10836:
10830:
10824:
10804:
10785:
10768:
10759:
10750:
10742:
10737:
10728:
10724:
10717:
10714:
10708:
10702:
10694:
10689:
10683:
10679:
10672:
10664:
10658:
10650:
10645:
10639:
10635:
10628:
10622:
10614:
10609:
10592:
10588:
10581:
10574:
10573:
10572:
10570:
10549:
10544:
10540:
10533:
10513:
10493:
10471:
10466:
10462:
10438:
10432:
10409:
10386:
10380:
10374:
10352:
10347:
10341:
10337:
10311:
10306:
10286:
10282:
10278:
10273:
10268:
10251:
10247:
10223:
10215:
10211:
10193:
10189:
10168:
10143:
10138:
10129:
10125:
10121:
10116:
10111:
10105:
10101:
10074:
10071:
10045:
10040:
10036:
10029:
10026:
10015:
10010:
10001:
9997:
9993:
9988:
9983:
9977:
9973:
9966:
9958:
9953:
9933:
9929:
9925:
9920:
9915:
9898:
9894:
9883:
9882:
9881:
9879:
9855:
9846:
9837:
9830:
9821:
9815:
9812:
9806:
9800:
9793:
9787:
9781:
9773:
9767:
9758:
9752:
9746:
9743:
9737:
9731:
9725:
9718:
9717:
9716:
9702:
9682:
9675:
9671:
9666:
9665:get flipped.
9652:
9632:
9605:
9594:
9577:
9569:
9559:
9558:
9557:
9555:
9551:
9535:
9532:
9526:
9520:
9514:
9494:
9486:
9466:
9458:
9435:
9429:
9423:
9416:
9400:
9397:
9391:
9385:
9365:
9342:
9319:
9299:
9275:
9272:
9266:
9257:
9248:
9240:
9237:
9231:
9225:
9219:
9212:
9211:
9210:
9196:
9193:
9187:
9181:
9178:
9175:
9172:
9166:
9157:
9134:
9125:
9116:
9108:
9102:
9095:
9088:
9082:
9076:
9073:
9070:
9066:
9062:
9059:
9056:
9050:
9041:
9032:
9025:
9024:
9023:
9006:
9000:
8994:
8971:
8962:
8953:
8930:
8924:
8918:
8915:
8912:
8909:
8903:
8897:
8888:
8869:
8852:
8849:
8846:
8843:
8834:
8831:
8828:
8820:
8817:
8814:
8808:
8799:
8791:
8785:
8780:
8777:
8768:
8767:
8766:
8763:
8757:
8754:
8738:
8735:
8726:
8723:
8717:
8710:
8703:
8700:
8686:
8682:
8675:True Positive
8667:
8664:
8659:
8646:
8643:of 90% and a
8642:
8637:
8635:
8632:. In short,
8619:
8599:
8579:
8559:
8536:
8530:
8524:
8515:
8509:
8503:
8500:
8494:
8488:
8482:
8475:
8474:
8473:
8459:
8439:
8413:
8407:
8404:
8401:
8394:
8387:
8381:
8378:
8372:
8366:
8363:
8357:
8354:
8348:
8342:
8335:, one writes
8322:
8316:
8311:
8307:
8286:
8283:
8278:
8274:
8264:
8262:
8243:
8234:
8226:
8222:
8215:
8207:
8199:
8195:
8188:
8182:
8176:
8168:
8164:
8160:
8155:
8151:
8144:
8137:
8136:
8121:
8110:
8106:
8099:
8089:
8085:
8078:
8072:
8064:
8060:
8056:
8051:
8047:
8040:
8033:
8032:
8031:
8029:
8025:
7998:
7994:
7987:
7981:
7971:
7967:
7960:
7954:
7948:
7942:
7934:
7930:
7926:
7921:
7917:
7903:
7902:
7901:
7881:
7873:
7869:
7865:
7860:
7856:
7846:
7838:
7834:
7830:
7825:
7821:
7814:
7811:
7805:
7797:
7793:
7789:
7784:
7780:
7773:
7766:
7765:
7764:
7762:
7738:
7735:
7732:
7725:
7717:
7713:
7706:
7698:
7695:
7692:
7684:
7680:
7666:
7662:
7658:
7652:
7644:
7640:
7632:
7631:
7630:
7628:
7624:
7617:
7608:
7602:Extended form
7585:
7576:
7568:
7564:
7555:
7547:
7543:
7536:
7528:
7524:
7520:
7512:
7508:
7501:
7495:
7487:
7483:
7479:
7471:
7467:
7459:
7458:
7457:
7455:
7451:
7432:
7423:
7415:
7411:
7402:
7398:
7394:
7388:
7382:
7374:
7370:
7366:
7358:
7354:
7347:
7341:
7337:
7333:
7325:
7321:
7317:
7311:
7304:
7303:
7302:
7300:
7296:
7291:
7277:
7248:
7244:
7240:
7234:
7226:
7218:
7214:
7207:
7203:
7199:
7191:
7187:
7183:
7177:
7171:
7165:
7157:
7153:
7149:
7141:
7137:
7129:
7128:
7127:
7126:is discrete,
7125:
7121:
7111:
7109:
7105:
7080:
7076:
7072:
7066:
7058:
7054:
7050:
7044:
7038:
7034:
7030:
7022:
7018:
7014:
7008:
7002:
6996:
6992:
6988:
6980:
6976:
6972:
6966:
6959:
6958:
6957:
6955:
6952: =
6951:
6947:
6943:
6940: =
6939:
6935:
6931:
6927:
6924:
6920:
6912:
6908:
6904:
6900:
6896:
6892:
6887:
6864:
6855:
6846:
6840:
6829:
6823:
6820:
6814:
6808:
6802:
6794:
6788:
6780:
6774:
6768:
6760:
6754:
6748:
6742:
6734:
6728:
6721:
6720:
6719:
6717:
6713:
6694:
6683:
6679:
6672:
6664:
6660:
6651:
6645:
6640:
6625:
6621:
6614:
6606:
6602:
6593:
6587:
6581:
6575:
6565:
6561:
6554:
6544:
6530:
6521:
6517:
6510:
6502:
6498:
6489:
6483:
6478:
6474:
6469:
6463:
6457:
6450:
6449:
6448:
6445:
6432:
6424:
6420:
6416:
6413:
6407:
6402:
6398:
6394:
6388:
6382:
6374:
6372:
6368:
6364:
6360:
6354: |
6353:
6349:
6345:
6338:
6334:
6330:
6326:
6319:
6312:Extended form
6306:
6302:
6298:
6279:
6271:
6265:
6258:
6255:
6251:
6247:
6228:
6217:
6211:
6204:
6201:
6197:
6193:
6174:
6166:
6160:
6153:
6136:
6130:
6127:
6124:
6121:
6115:
6106:
6098:
6094:
6075:
6066:
6059:
6056:
6052:
6033:
6027:
6020:
6019:
6018:
6016:
6012:
6007:
5990:
5981:
5972:
5966:
5955:
5949:
5946:
5940:
5934:
5928:
5920:
5914:
5906:
5900:
5894:
5886:
5880:
5874:
5868:
5860:
5854:
5847:
5846:
5845:
5837:
5834:
5831:
5828:
5827:
5823:
5820:
5815:
5811:
5808:
5804:
5801:
5796:
5795:
5792:
5789:
5785:
5782:
5778:
5775:
5772:
5771:
5767:
5760:
5757:
5748:
5747:
5744:
5720:
5711:
5705:
5701:
5696:
5687:
5678:
5675:
5669:
5658:
5652:
5649:
5643:
5637:
5634:
5628:
5620:
5614:
5610:
5605:
5602:
5595:
5594:
5593:
5576:
5567:
5558:
5555:
5549:
5538:
5532:
5529:
5523:
5517:
5514:
5508:
5500:
5494:
5488:
5485:
5482:
5479:
5472:
5471:
5470:
5453:
5447:
5436:
5430:
5427:
5421:
5412:
5409:
5406:
5403:
5397:
5389:
5380:
5369:
5361:
5355:
5352:
5346:
5340:
5337:
5334:
5331:
5325:
5317:
5311:
5304:
5303:
5302:
5300:
5296:
5292:
5288:
5281:
5274:
5269:
5252:
5246:
5238:
5232:
5229:
5223:
5217:
5214:
5208:
5200:
5194:
5187:
5186:
5185:
5183:
5179:
5175:
5171:
5167:
5163:
5144:
5135:
5129:
5121:
5115:
5109:
5101:
5095:
5089:
5083:
5075:
5069:
5062:
5061:
5060:
5058:
5054:
5050:
5046:
5005:
5002:
5000:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4966:
4963:
4960:
4954:
4952:
4930:
4908:
4905:
4891:
4869:
4853:
4831:
4825:
4823:
4801:
4785:
4763:
4757:
4755:
4731:
4720:
4719:
4718:
4715:
4713:
4709:
4705:
4701:
4697:
4672:
4662:
4657:
4648:
4646:
4642:
4637:
4635:
4631:
4628:
4624:
4620:
4616:
4612:
4608:
4605:
4601:
4597:
4593:
4589:
4585:
4581:
4577:
4573:
4569:
4565:
4558:
4557:tree diagrams
4553:
4544:
4542:
4534:
4531:provides for
4530:
4521:
4517:
4508:
4504:
4500:
4493:the quotient
4492:
4489:
4485:
4481:
4477:
4473:
4470:
4467:
4463:
4459:
4455:
4452:
4451:
4450:
4448:
4445:and evidence
4444:
4440:
4430:
4428:
4419:
4410:
4408:
4400:
4393:
4389:
4367:
4364:
4359:
4354:
4350:
4347:
4344:
4338:
4329:
4323:
4313:
4309:
4302:
4294:
4290:
4281:
4275:
4269:
4263:
4253:
4249:
4242:
4235:
4234:
4233:
4228:
4224:
4219:
4202:
4199:
4193:
4184:
4178:
4172:
4163:
4157:
4151:
4142:
4136:
4128:
4124:
4117:
4109:
4105:
4096:
4090:
4085:
4081:
4077:
4071:
4065:
4058:
4057:
4056:
4054:
4035:
4032:
4024:
4020:
4011:
4005:
4001:
3998:
3995:
3987:
3983:
3974:
3968:
3964:
3961:
3958:
3950:
3946:
3937:
3931:
3924:
3923:
3922:
3917:
3913:
3909:
3890:
3887:
3879:
3875:
3868:
3864:
3861:
3858:
3850:
3846:
3839:
3835:
3832:
3829:
3821:
3817:
3810:
3803:
3802:
3801:
3799:
3795:
3792:machine (for
3791:
3787:
3779:
3775:
3766:
3763:
3761:
3758:
3755:
3754:
3750:
3746:
3743:
3741:
3738:
3735:
3734:
3730:
3727:
3724:
3721:
3720:
3716:
3713:
3710:
3707:
3706:
3702:
3697:
3694:
3685:
3684:
3655:
3652:
3647:
3644:
3639:
3633:
3630:
3624:
3620:
3616:
3610:
3607:
3604:
3601:
3596:
3593:
3590:
3584:
3582:
3560:
3536:
3533:
3519:
3495:
3479:
3455:
3449:
3447:
3425:
3409:
3385:
3379:
3377:
3351:
3340:
3339:
3338:
3330:
3327:
3324:
3321:
3320:
3316:
3312:
3309:
3306:
3303:
3302:
3298:
3295:
3292:
3289:
3288:
3284:
3279:
3276:
3268:
3267:
3264:
3261:
3246:
3242:
3239:
3237:
3234:
3231:
3230:
3226:
3222:
3219:
3216:
3213:
3212:
3208:
3205:
3203:
3200:
3197:
3196:
3192:
3187:
3184:
3176:
3175:
3166:
3162:
3159:
3157:
3154:
3151:
3150:
3146:
3142:
3139:
3136:
3133:
3132:
3128:
3125:
3123:
3120:
3117:
3116:
3112:
3107:
3104:
3096:
3095:
3086:
3082:
3079:
3077:
3074:
3071:
3070:
3066:
3062:
3059:
3056:
3053:
3052:
3048:
3045:
3043:
3040:
3037:
3036:
3032:
3027:
3024:
3016:
3015:
3012:
3009:
2999:
2993:
2990:
2989:
2988:
2951:
2942:
2938:
2934:
2932:
2928:
2924:
2897:
2875:
2872:
2858:
2836:
2833:
2819:
2789:
2786:
2780:
2777:
2774:
2770:
2765:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2736:
2733:
2730:
2724:
2722:
2700:
2678:
2675:
2661:
2639:
2623:
2601:
2595:
2593:
2571:
2555:
2533:
2527:
2525:
2501:
2490:
2489:
2488:
2455:
2443:
2442:
2441:
2439:
2434:
2432:
2428:
2423:
2421:
2417:
2413:
2408:
2406:
2402:
2392:
2390:
2386:
2382:
2378:
2374:
2359:
2357:
2352:
2348:
2344:
2339:
2323:
2320:
2317:
2306:
2298:
2294:
2287:
2284:
2278:
2270:
2265:
2261:
2252:
2238:
2235:
2232:
2212:
2190:
2185:
2181:
2157:
2154:
2146:
2142:
2135:
2132:
2124:
2119:
2115:
2111:
2105:
2102:
2099:
2096:
2093:
2085:
2082:
2079:
2075:
2054:
2032:
2028:
2007:
2004:
2001:
1981:
1959:
1954:
1950:
1921:
1912:
1904:
1900:
1891:
1883:
1879:
1872:
1864:
1861:
1858:
1852:
1848:
1841:
1835:
1827:
1824:
1821:
1815:
1811:
1803:
1802:
1801:
1778:
1770:
1766:
1757:
1754:
1751:
1743:
1740:
1737:
1733:
1726:
1720:
1712:
1709:
1706:
1700:
1696:
1688:
1668:
1660:
1656:
1647:
1644:
1641:
1633:
1630:
1627:
1623:
1616:
1610:
1602:
1599:
1596:
1590:
1586:
1578:
1577:
1576:
1574:
1570:
1566:
1563:
1539:
1536:
1530:
1524:
1516:
1507:
1501:
1493:
1487:
1481:
1475:
1469:
1463:
1457:
1451:
1445:
1438:
1437:
1436:
1419:
1413:
1407:
1384:
1381:
1378:
1372:
1349:
1346:
1340:
1334:
1326:
1317:
1311:
1303:
1300:
1297:
1291:
1285:
1279:
1273:
1267:
1260:
1259:
1258:
1241:
1238:
1235:
1229:
1206:
1203:
1200:
1194:
1188:
1180:
1171:
1165:
1157:
1154:
1151:
1145:
1139:
1133:
1127:
1121:
1114:
1113:
1112:
1110:
1093:
1081:
1077:
1061:
1041:
1018:
1012:
989:
983:
976:
958:
952:
946:
943:
937:
931:
925:
905:
885:
877:
861:
841:
818:
812:
806:
799:
784:
764:
756:
740:
720:
712:
693:
687:
681:
674:
673:
672:
658:
655:
649:
643:
635:
619:
599:
589:
570:
564:
556:
550:
544:
538:
532:
526:
520:
514:
508:
499:
490:
485:
483:
482:Richard Price
479:
475:
471:
469:
465:
461:
456:
454:
450:
446:
445:
440:
435:
433:
428:
424:
423:
418:
417:Royal Society
413:
411:
410:Richard Price
407:
403:
402:
395:
377:
367:
365:
361:
357:
353:
349:
344:
342:
341:
336:
332:
328:
324:
320:
309:
304:
302:
297:
295:
290:
289:
287:
286:
281:
276:
271:
270:
269:
268:
263:
260:
258:
255:
253:
250:
249:
248:
247:
243:
242:
237:
234:
232:
229:
228:
227:
226:
222:
221:
216:
213:
211:
208:
206:
203:
202:
201:
200:
196:
195:
190:
187:
185:
182:
180:
177:
175:
172:
170:
167:
166:
165:
164:
160:
159:
154:
151:
149:
146:
144:
141:
139:
136:
135:
134:
133:
129:
128:
123:
120:
118:
115:
113:
110:
108:
105:
103:
102:Cox's theorem
100:
98:
95:
93:
90:
88:
85:
83:
80:
78:
75:
74:
73:
72:
68:
67:
64:
60:
56:
52:
49:
48:
44:
40:
39:
36:
33:
32:
28:
27:
22:
12687:
12658:
12654:
12633:
12614:
12595:
12577:
12549:Bibliography
12537:
12528:
12501:
12497:
12449:
12443:
12429:
12421:the original
12405:
12401:Friedman, N.
12391:
12380:
12375:
12364:. Retrieved
12360:
12351:
12339:. Retrieved
12335:the original
12330:
12321:
12298:
12288:
12264:. Springer.
12260:
12253:
12208:
12204:
12194:
12174:
12167:
12158:
12152:
12139:
12133:
12108:
12073:(1): 36–39.
12070:
12067:Significance
12066:
12056:
12039:
12035:
12029:
12010:
12004:
11987:
11983:
11977:
11948:
11937:
11918:
11908:
11888:
11881:
11869:
11857:. Retrieved
11842:
11835:
11826:
11807:
11803:
11794:
11782:. Retrieved
11767:
11760:
11748:. Retrieved
11733:
11726:
11711:
11659:
11650:
11618:
11576:
11572:
11553:
11490:
11467:
11460:
11451:
11435:
11426:Heterozygous
11425:
11417:
11398:
11394:
11390:
11341:
11269:
11220:
11087:
10978:
10796:
10783:
10569:homomorphism
10208:denotes the
10060:
9875:
9667:
9624:
9291:
9149:
8987:in terms of
8880:
8867:
8758:
8638:
8633:
8551:
8265:
8261:Bayes factor
8258:
8024:Bayes factor
8021:
7899:
7758:
7626:
7619:
7613:
7453:
7449:
7447:
7298:
7294:
7292:
7269:
7123:
7119:
7117:
7101:
6953:
6949:
6945:
6941:
6937:
6933:
6929:
6925:
6919:sample space
6916:
6910:
6906:
6894:
6890:
6711:
6709:
6446:
6375:
6369:) using the
6366:
6362:
6355:
6351:
6347:
6340:
6336:
6329:sample space
6321:
6315:
6304:
6300:
6253:
6249:
6199:
6195:
6096:
6092:
6054:
6014:
6010:
6008:
6005:
5843:
5819:1−P(A)
5816:
5809:
5802:
5790:
5783:
5776:
5591:
5468:
5298:
5294:
5290:
5289:, the event
5286:
5279:
5272:
5270:
5267:
5182:proportional
5177:
5173:
5169:
5165:
5164:, the event
5159:
5056:
5052:
5048:
5044:
5042:
4716:
4711:
4707:
4696:entomologist
4693:
4660:
4644:
4640:
4638:
4633:
4629:
4626:
4622:
4618:
4614:
4610:
4606:
4603:
4599:
4595:
4591:
4587:
4583:
4579:
4575:
4571:
4567:
4561:
4538:
4532:
4528:
4519:
4515:
4506:
4502:
4498:
4487:
4483:
4479:
4475:
4471:
4465:
4461:
4457:
4453:
4446:
4442:
4436:
4424:
4406:
4398:
4391:
4387:
4386:probability
4384:
4226:
4222:
4220:
4217:
4052:
4050:
3915:
3911:
3907:
3905:
3797:
3793:
3789:
3782:
3780:
3776:
3772:
3759:
3739:
3336:
3262:
3258:
3235:
3201:
3155:
3121:
3075:
3041:
3005:
2997:
2986:
2935:
2811:
2447:
2435:
2424:
2409:
2398:
2395:Drug testing
2370:
2340:
2253:
1941:
1938:General case
1799:
1568:
1564:
1559:
1365:Solving for
1364:
1221:
1106:
591:
501:
497:
487:
472:
457:
442:
436:
420:
414:
399:
376:Thomas Bayes
373:
356:observations
345:
338:
331:Thomas Bayes
326:
322:
318:
317:
252:Bayes factor
86:
12295:"Chapter 1"
11810:: 370–418.
11750:23 February
11625:metascience
11555:offspring (
11498:Hypothesis
11286:Hypothesis
8645:specificity
8641:sensitivity
7270:where each
7114:Simple form
6917:Consider a
6333:event space
5754:Proposition
5043:For events
5039:Simple form
3255:Cancer rate
3008:specificity
2431:probability
1800:Therefore,
327:Bayes' rule
12705:Categories
12689:Veritasium
12397:Koller, D.
12366:2023-10-20
12280:1159112760
11957:. p.
11899:0691084971
11695:References
10981:chain rule
10979:Using the
10975:Derivation
10212:(aka. the
8753:prevalence
7108:Derivation
5271:If events
4700:subspecies
3695:Defective
3568:Non-Cancer
3554:Non-Cancer
2427:prevalence
2414:, meaning
2351:Kolmogorov
1103:For events
876:likelihood
323:Bayes' law
197:Estimators
69:Background
55:Likelihood
12245:237736986
12237:0303-6898
12089:153704746
11244:∩
11168:∩
11141:∩
11135:∩
11114:∩
11108:∩
11035:∩
11008:∩
11002:∩
10911:∩
10881:∩
10757:¬
10735:¬
10725:ω
10680:ω
10636:ω
10604:~
10589:ω
10541:ω
10463:ω
10338:ω
10304:¬
10298:~
10283:ω
10263:~
10248:ω
10214:base rate
10136:¬
10126:ω
10102:ω
10075:~
10072:ϕ
10030:~
10027:ϕ
10008:¬
9998:ω
9974:ω
9951:¬
9945:~
9930:ω
9910:~
9895:ω
9844:¬
9828:¬
9674:base rate
9603:¬
9599:⟹
9592:¬
9585:⟺
9574:⟹
9491:⟹
9463:⟹
9398:≠
9363:¬
9340:¬
9264:¬
9255:¬
9245:⟹
9194:≠
9179:−
9164:¬
9123:¬
9074:−
9063:−
9048:¬
9039:¬
8969:¬
8960:¬
8916:−
8895:¬
8838:%
8832:×
8824:%
8818:×
8803:%
8795:%
8786:×
8730:%
8724:−
8721:%
8707:%
8656:Λ
8519:Λ
8516:⋅
8405:−
8370:¬
8320:¬
7911:Λ
7850:Λ
7847:⋅
7761:odds form
7736:ξ
7726:ξ
7699:ξ
7675:∞
7670:∞
7667:−
7663:∫
6865:⋅
6853:¬
6838:¬
6695:⋅
6637:∑
6552:⇒
6475:∑
6417:∩
6399:∑
6327:} of the
6318:partition
6256:is false.
6226:¬
6128:−
6113:¬
6073:¬
5979:¬
5964:¬
5685:¬
5676:⋅
5667:¬
5635:⋅
5565:¬
5556:⋅
5547:¬
5515:⋅
5489:⋅
5445:¬
5428:⋅
5419:¬
5410:⋅
5387:¬
5353:⋅
5338:⋅
5230:⋅
5215:∝
5009:%
5003:≈
4987:×
4975:×
4964:×
4676:¯
4484:posterior
4348:⋅
4082:∑
3698:Flawless
3687:Condition
3659:%
3653:≈
3631:×
3605:×
3594:×
3214:Non-user
3188:Negative
3185:Positive
3134:Non-user
3108:Negative
3105:Positive
3054:Non-user
3028:Negative
3025:Positive
2793:%
2787:≈
2757:×
2745:×
2734:×
2401:sensitive
1537:≠
1382:∩
1347:≠
1301:∩
1239:∩
1201:≠
1155:∩
656:≠
464:axiomatic
97:Coherence
51:Posterior
12520:15371910
12476:30245516
12403:(2009).
12341:5 August
11945:(1973).
11582:See also
11314:About 1
9695:of only
9378:. Where
9355:implies
9312:implies
7448:If both
6903:function
6202:is true.
6095:, that
5799:(not A)
5301:we have
4490:is true.
3544:Symptoms
3503:Symptoms
3463:Symptoms
3433:Symptoms
3393:Symptoms
3369:Symptoms
3011:to 49%.
2967:Positive
2905:Non-user
2891:Non-user
2883:Positive
2844:Positive
2827:Positive
2708:Non-user
2694:Non-user
2686:Positive
2647:Positive
2609:Positive
2579:Positive
2541:Positive
2517:Positive
2471:Positive
2412:specific
2387:and the
2362:Examples
2020:and let
918:because
329:, after
63:Evidence
12694:YouTube
12576:(ed.).
12566::
12467:6752283
12361:Cuemath
11859:16 June
11784:16 June
11679:page 10
11672:Gallica
11664:Gallica
11560:⁄
11483:⁄
11473:⁄
11378:⁄
11373:⋅
11368:⁄
11363:⋅
11358:⁄
11353:⋅
11348:⁄
10817:, with
6295:is the
6244:is the
6190:is the
6049:is the
5797:¬A
5763:(not B)
5761:¬B
4916:Pattern
4877:Pattern
4839:Pattern
4809:Pattern
4771:Pattern
4747:Pattern
4661:R, C, P
4651:Example
4562:In the
4525:
4495:
4482:), the
4460:), the
4437:In the
3691:Machine
3634:0.99999
3608:0.00001
3597:0.00001
3331:100000
3270:Symptom
370:History
12640:
12621:
12602:
12572:". In
12560:
12518:
12474:
12464:
12413:
12309:
12278:
12268:
12243:
12235:
12182:
12146:, §8.7
12121:
12087:
12017:
11965:
11925:
11896:
11850:
11775:
11741:
11718:
11412:Father
11407:Mother
10061:where
9645:resp.
8881:Using
7900:where
7618:. For
6944:} and
6899:domain
6346:) and
6331:, the
5829:Total
5768:Total
5034:Events
4938:Common
4924:Common
4704:beetle
4630:out of
4613:, and
4607:out of
4203:0.024.
3756:Total
3703:Total
3527:Cancer
3513:Cancer
3487:Cancer
3473:Cancer
3417:Cancer
3403:Cancer
3359:Cancer
3328:99989
3322:Total
3313:99999
3310:99989
3285:Total
3273:Cancer
3232:Total
3193:Total
3181:Actual
3152:Total
3113:Total
3101:Actual
3072:Total
3033:Total
3021:Actual
2383:, the
2379:, the
2225:given
1994:given
1222:where
777:given
634:events
592:where
489:sense.
12303:Wiley
12241:S2CID
12085:S2CID
11683:p. 15
11642:Notes
11534:1/15
11523:1/10
10216:) of
9150:when
6714:is a
6093:not-A
5029:Forms
4990:0.999
4978:0.001
4967:0.001
4462:prior
4355:0.024
4036:0.01.
3767:1000
3625:99999
3243:1000
3198:User
3163:1000
3118:User
3083:1000
3038:User
2775:0.045
2771:0.045
1087:Proof
709:is a
59:Prior
12642:5013
12638:OCLC
12619:ISBN
12600:ISBN
12516:PMID
12472:PMID
12411:ISBN
12343:2014
12307:ISBN
12276:OCLC
12266:ISBN
12233:ISSN
12180:ISBN
12119:ISBN
12015:ISBN
11963:ISBN
11923:ISBN
11894:ISBN
11861:2013
11848:ISBN
11786:2013
11773:ISBN
11752:2021
11739:ISBN
11716:ISBN
11548:5/6
11545:1/6
11537:1/3
11515:1/3
11512:2/3
11478:and
11303:1/2
11300:1/2
10837:>
8821:90.1
8792:90.1
8299:and
7763:is:
7452:and
6928:and
6909:and
6893:and
5791:P(A)
5047:and
4984:0.05
4972:0.98
4961:0.98
4899:Rare
4885:Rare
4861:Rare
4847:Rare
4793:Rare
4779:Rare
4739:Rare
4663:and
4643:and
4351:0.50
4345:0.01
4185:0.01
4164:0.03
4143:0.05
3999:0.03
3962:0.05
3891:0.5.
3764:976
3747:500
3744:495
3731:300
3728:291
3717:200
3714:190
3290:Yes
3277:Yes
3240:908
3223:950
3220:903
3178:Test
3160:760
3143:950
3140:760
3137:190
3098:Test
3080:765
3063:950
3060:760
3057:190
3018:Test
2959:User
2866:User
2852:User
2781:0.19
2760:0.95
2754:0.20
2748:0.05
2742:0.90
2737:0.05
2731:0.90
2669:User
2655:User
2631:User
2617:User
2563:User
2549:User
2509:User
2463:User
2436:The
1942:Let
1567:and
1078:and
1054:and
1005:and
636:and
632:are
612:and
12663:doi
12506:doi
12462:PMC
12454:doi
12223:hdl
12213:doi
12075:doi
12044:doi
11992:doi
11812:doi
11565:).
11459:MM
11456:MW
11445:MW
11442:WW
8835:9.0
8829:909
8800:9.0
8781:909
8718:100
8026:or
7293:If
7118:If
7110:).
6956:}.
6905:of
5592:or
5006:1.9
4702:of
4694:An
4194:0.5
4173:0.3
4152:0.2
3862:0.3
3833:0.2
3711:10
3656:9.1
3325:11
3307:10
3304:No
3280:No
3217:47
3209:50
3156:240
3129:50
3076:235
3049:50
2205:of
878:of
757:of
366:).
325:or
12707::
12686:.
12657:.
12653:.
12514:.
12500:.
12496:.
12484:^
12470:.
12460:.
12450:21
12448:.
12442:.
12399:;
12359:.
12329:.
12305:.
12301:.
12297:.
12274:.
12239:.
12231:.
12221:.
12209:49
12207:.
12203:.
12142:,
12117:.
12113:.
12097:^
12083:.
12071:10
12069:.
12065:.
12040:40
12038:.
11988:37
11986:.
11961:.
11959:31
11917:.
11808:53
11806:.
11703:^
11526:1
11488:.
11450:M
11434:W
11424:M
11416:W
11262:.
8815:91
8778:91
8739:10
8727:91
8704:90
8636:.
6718::
6373::
6017:,
5838:1
5773:A
5278:,
4590:.
4543:.
4449:,
4401:(X
4368:24
3760:24
3736:C
3725:9
3722:B
3708:A
3648:11
3617:10
3299:1
3296:0
3293:1
3236:92
3206:5
3202:45
3126:0
3122:50
3046:5
3042:45
2790:19
2391:.
1575::
1540:0.
1350:0.
1111::
671:.
412:.
388:eÉŞ
343:.
61:Ă·
57:Ă—
53:=
12696:.
12671:.
12665::
12659:1
12644:.
12627:.
12608:.
12522:.
12508::
12502:6
12478:.
12456::
12384:.
12369:.
12345:.
12315:.
12282:.
12247:.
12225::
12215::
12188:.
12127:.
12091:.
12077::
12050:.
12046::
12023:.
11998:.
11994::
11971:.
11931:.
11902:.
11863:.
11820:.
11814::
11788:.
11754:.
11685:.
11660:4
11562:4
11558:1
11485:3
11481:1
11475:3
11471:2
11380:2
11376:1
11370:2
11366:1
11360:2
11356:1
11350:2
11346:1
11250:)
11247:C
11241:B
11238:|
11235:A
11232:(
11229:P
11206:)
11203:C
11200:(
11197:P
11193:)
11190:C
11187:|
11184:A
11181:(
11178:P
11174:)
11171:C
11165:A
11162:|
11159:B
11156:(
11153:P
11150:=
11147:)
11144:C
11138:A
11132:B
11129:(
11126:P
11123:=
11120:)
11117:C
11111:B
11105:A
11102:(
11099:P
11073:)
11070:C
11067:(
11064:P
11060:)
11057:C
11054:|
11051:B
11048:(
11045:P
11041:)
11038:C
11032:B
11029:|
11026:A
11023:(
11020:P
11017:=
11014:)
11011:C
11005:B
10999:A
10996:(
10993:P
10956:)
10953:C
10950:|
10947:B
10944:(
10941:P
10936:)
10933:C
10930:|
10927:A
10924:(
10921:P
10917:)
10914:C
10908:A
10905:|
10902:B
10899:(
10896:P
10890:=
10887:)
10884:C
10878:B
10875:|
10872:A
10869:(
10866:P
10843:,
10840:0
10834:)
10831:C
10828:(
10825:P
10805:C
10769:.
10763:)
10760:A
10754:(
10751:a
10748:)
10743:S
10738:A
10732:|
10729:B
10721:(
10718:P
10715:+
10712:)
10709:A
10706:(
10703:a
10700:)
10695:S
10690:A
10687:|
10684:B
10676:(
10673:P
10668:)
10665:A
10662:(
10659:a
10656:)
10651:S
10646:A
10643:|
10640:B
10632:(
10629:P
10623:=
10620:)
10615:S
10610:B
10600:|
10593:A
10585:(
10582:P
10555:)
10550:S
10545:A
10537:(
10534:P
10514:S
10494:A
10472:S
10467:A
10442:)
10439:B
10436:|
10433:A
10430:(
10410:S
10390:)
10387:B
10384:|
10381:A
10378:(
10375:P
10353:S
10348:B
10345:|
10342:A
10317:)
10312:S
10307:B
10294:|
10287:A
10279:,
10274:S
10269:B
10259:|
10252:A
10244:(
10224:A
10194:A
10190:a
10169:S
10149:)
10144:S
10139:A
10133:|
10130:B
10122:,
10117:S
10112:A
10109:|
10106:B
10098:(
10046:,
10041:A
10037:a
10021:)
10016:S
10011:A
10005:|
10002:B
9994:,
9989:S
9984:A
9981:|
9978:B
9970:(
9967:=
9964:)
9959:S
9954:B
9941:|
9934:A
9926:,
9921:S
9916:B
9906:|
9899:A
9891:(
9856:.
9850:)
9847:A
9841:(
9838:a
9834:)
9831:A
9825:|
9822:B
9819:(
9816:P
9813:+
9810:)
9807:A
9804:(
9801:a
9797:)
9794:A
9791:|
9788:B
9785:(
9782:P
9777:)
9774:A
9771:(
9768:a
9762:)
9759:A
9756:|
9753:B
9750:(
9747:P
9744:=
9741:)
9738:B
9735:|
9732:A
9729:(
9726:P
9703:A
9683:a
9672:/
9653:B
9633:A
9621:.
9609:)
9606:B
9595:A
9589:(
9581:)
9578:A
9570:B
9567:(
9536:1
9533:=
9530:)
9527:B
9524:|
9521:A
9518:(
9515:P
9495:A
9487:B
9467:A
9459:B
9439:)
9436:B
9433:|
9430:A
9427:(
9424:P
9401:0
9395:)
9392:B
9389:(
9386:P
9366:B
9343:A
9320:A
9300:B
9288:.
9276:1
9273:=
9270:)
9267:A
9261:|
9258:B
9252:(
9249:P
9241:1
9238:=
9235:)
9232:B
9229:|
9226:A
9223:(
9220:P
9197:0
9191:)
9188:A
9185:(
9182:P
9176:1
9173:=
9170:)
9167:A
9161:(
9158:P
9135:,
9129:)
9126:A
9120:(
9117:P
9112:)
9109:B
9106:(
9103:P
9096:)
9092:)
9089:B
9086:|
9083:A
9080:(
9077:P
9071:1
9067:(
9060:1
9057:=
9054:)
9051:A
9045:|
9042:B
9036:(
9033:P
9010:)
9007:B
9004:|
9001:A
8998:(
8995:P
8975:)
8972:A
8966:|
8963:B
8957:(
8954:P
8934:)
8931:A
8928:|
8925:B
8922:(
8919:P
8913:1
8910:=
8907:)
8904:A
8901:|
8898:B
8892:(
8889:P
8853:1
8850::
8847:1
8844:=
8809:=
8736:=
8733:)
8715:(
8711:/
8701:=
8698:)
8690:(
8687:P
8683:/
8679:)
8671:(
8668:P
8665:=
8660:+
8620:B
8600:A
8580:A
8560:A
8537:,
8534:)
8531:B
8528:|
8525:A
8522:(
8513:)
8510:A
8507:(
8504:O
8501:=
8498:)
8495:B
8492:|
8489:A
8486:(
8483:O
8460:A
8440:A
8420:)
8417:)
8414:A
8411:(
8408:P
8402:1
8399:(
8395:/
8391:)
8388:A
8385:(
8382:P
8379:=
8376:)
8373:A
8367::
8364:A
8361:(
8358:O
8355:=
8352:)
8349:A
8346:(
8343:O
8323:A
8317:=
8312:2
8308:A
8287:A
8284:=
8279:1
8275:A
8244:,
8238:)
8235:B
8232:|
8227:2
8223:A
8219:(
8216:P
8211:)
8208:B
8205:|
8200:1
8196:A
8192:(
8189:P
8183:=
8180:)
8177:B
8174:|
8169:2
8165:A
8161::
8156:1
8152:A
8148:(
8145:O
8122:,
8116:)
8111:2
8107:A
8103:(
8100:P
8095:)
8090:1
8086:A
8082:(
8079:P
8073:=
8070:)
8065:2
8061:A
8057::
8052:1
8048:A
8044:(
8041:O
8004:)
7999:2
7995:A
7991:|
7988:B
7985:(
7982:P
7977:)
7972:1
7968:A
7964:|
7961:B
7958:(
7955:P
7949:=
7946:)
7943:B
7940:|
7935:2
7931:A
7927::
7922:1
7918:A
7914:(
7885:)
7882:B
7879:|
7874:2
7870:A
7866::
7861:1
7857:A
7853:(
7844:)
7839:2
7835:A
7831::
7826:1
7822:A
7818:(
7815:O
7812:=
7809:)
7806:B
7803:|
7798:2
7794:A
7790::
7785:1
7781:A
7777:(
7774:O
7739:.
7733:d
7729:)
7723:(
7718:X
7714:f
7710:)
7707:y
7704:(
7696:=
7693:X
7689:|
7685:Y
7681:f
7659:=
7656:)
7653:y
7650:(
7645:Y
7641:f
7627:y
7625:(
7622:Y
7620:f
7586:.
7580:)
7577:y
7574:(
7569:Y
7565:f
7559:)
7556:x
7553:(
7548:X
7544:f
7540:)
7537:y
7534:(
7529:x
7525:=
7521:X
7517:|
7513:Y
7509:f
7502:=
7499:)
7496:x
7493:(
7488:y
7484:=
7480:Y
7476:|
7472:X
7468:f
7454:Y
7450:X
7433:.
7427:)
7424:y
7421:(
7416:Y
7412:f
7406:)
7403:x
7399:=
7395:X
7392:(
7389:P
7386:)
7383:y
7380:(
7375:x
7371:=
7367:X
7363:|
7359:Y
7355:f
7348:=
7345:)
7342:y
7338:=
7334:Y
7330:|
7326:x
7322:=
7318:X
7315:(
7312:P
7299:Y
7295:X
7278:f
7252:)
7249:y
7245:=
7241:Y
7238:(
7235:P
7230:)
7227:x
7224:(
7219:X
7215:f
7211:)
7208:x
7204:=
7200:X
7196:|
7192:y
7188:=
7184:Y
7181:(
7178:P
7172:=
7169:)
7166:x
7163:(
7158:y
7154:=
7150:Y
7146:|
7142:X
7138:f
7124:Y
7120:X
7084:)
7081:y
7077:=
7073:Y
7070:(
7067:P
7062:)
7059:x
7055:=
7051:X
7048:(
7045:P
7042:)
7039:x
7035:=
7031:X
7027:|
7023:y
7019:=
7015:Y
7012:(
7009:P
7003:=
7000:)
6997:y
6993:=
6989:Y
6985:|
6981:x
6977:=
6973:X
6970:(
6967:P
6954:y
6950:Y
6946:B
6942:x
6938:X
6934:A
6930:Y
6926:X
6913:.
6911:y
6907:x
6895:Y
6891:X
6859:)
6856:A
6850:(
6847:P
6844:)
6841:A
6834:|
6830:B
6827:(
6824:P
6821:+
6818:)
6815:A
6812:(
6809:P
6806:)
6803:A
6799:|
6795:B
6792:(
6789:P
6784:)
6781:A
6778:(
6775:P
6772:)
6769:A
6765:|
6761:B
6758:(
6755:P
6749:=
6746:)
6743:B
6739:|
6735:A
6732:(
6729:P
6712:A
6689:)
6684:j
6680:A
6676:(
6673:P
6670:)
6665:j
6661:A
6656:|
6652:B
6649:(
6646:P
6641:j
6631:)
6626:i
6622:A
6618:(
6615:P
6612:)
6607:i
6603:A
6598:|
6594:B
6591:(
6588:P
6582:=
6579:)
6576:B
6572:|
6566:i
6562:A
6558:(
6555:P
6531:,
6527:)
6522:j
6518:A
6514:(
6511:P
6508:)
6503:j
6499:A
6494:|
6490:B
6487:(
6484:P
6479:j
6470:=
6467:)
6464:B
6461:(
6458:P
6433:,
6430:)
6425:j
6421:A
6414:B
6411:(
6408:P
6403:j
6395:=
6392:)
6389:B
6386:(
6383:P
6367:B
6365:(
6363:P
6358:j
6356:A
6352:B
6350:(
6348:P
6343:j
6341:A
6339:(
6337:P
6324:j
6322:A
6320:{
6307:.
6305:B
6301:A
6283:)
6280:B
6276:|
6272:A
6269:(
6266:P
6254:A
6250:B
6232:)
6229:A
6222:|
6218:B
6215:(
6212:P
6200:A
6196:B
6178:)
6175:A
6171:|
6167:B
6164:(
6161:P
6140:)
6137:A
6134:(
6131:P
6125:1
6122:=
6119:)
6116:A
6110:(
6107:P
6097:A
6079:)
6076:A
6070:(
6067:P
6057:.
6055:A
6037:)
6034:A
6031:(
6028:P
6015:B
6011:A
5991:.
5985:)
5982:A
5976:(
5973:P
5970:)
5967:A
5960:|
5956:B
5953:(
5950:P
5947:+
5944:)
5941:A
5938:(
5935:P
5932:)
5929:A
5925:|
5921:B
5918:(
5915:P
5910:)
5907:A
5904:(
5901:P
5898:)
5895:A
5891:|
5887:B
5884:(
5881:P
5875:=
5872:)
5869:B
5865:|
5861:A
5858:(
5855:P
5758:B
5721:.
5715:)
5712:B
5709:(
5706:P
5702:1
5697:=
5691:)
5688:A
5682:(
5679:P
5673:)
5670:A
5663:|
5659:B
5656:(
5653:P
5650:+
5647:)
5644:A
5641:(
5638:P
5632:)
5629:A
5625:|
5621:B
5618:(
5615:P
5611:1
5606:=
5603:c
5577:,
5574:)
5571:)
5568:A
5562:(
5559:P
5553:)
5550:A
5543:|
5539:B
5536:(
5533:P
5530:+
5527:)
5524:A
5521:(
5518:P
5512:)
5509:A
5505:|
5501:B
5498:(
5495:P
5492:(
5486:c
5483:=
5480:1
5454:.
5451:)
5448:A
5441:|
5437:B
5434:(
5431:P
5425:)
5422:A
5416:(
5413:P
5407:c
5404:=
5401:)
5398:B
5394:|
5390:A
5384:(
5381:P
5373:)
5370:A
5366:|
5362:B
5359:(
5356:P
5350:)
5347:A
5344:(
5341:P
5335:c
5332:=
5329:)
5326:B
5322:|
5318:A
5315:(
5312:P
5299:c
5295:A
5291:A
5287:A
5283:2
5280:A
5276:1
5273:A
5253:.
5250:)
5247:A
5243:|
5239:B
5236:(
5233:P
5227:)
5224:A
5221:(
5218:P
5212:)
5209:B
5205:|
5201:A
5198:(
5195:P
5178:A
5174:B
5170:A
5166:B
5145:.
5139:)
5136:B
5133:(
5130:P
5125:)
5122:A
5119:(
5116:P
5113:)
5110:A
5106:|
5102:B
5099:(
5096:P
5090:=
5087:)
5084:B
5080:|
5076:A
5073:(
5070:P
5057:B
5055:(
5053:P
5049:B
5045:A
4981:+
4955:=
4942:)
4934:(
4931:P
4928:)
4920:|
4912:(
4909:P
4906:+
4903:)
4895:(
4892:P
4889:)
4881:|
4873:(
4870:P
4865:)
4857:(
4854:P
4851:)
4843:|
4835:(
4832:P
4826:=
4813:)
4805:(
4802:P
4797:)
4789:(
4786:P
4783:)
4775:|
4767:(
4764:P
4758:=
4751:)
4743:|
4735:(
4732:P
4712:P
4708:P
4673:P
4645:B
4641:A
4634:B
4627:A
4623:B
4619:A
4617:(
4615:P
4611:A
4604:B
4600:A
4596:B
4594:(
4592:P
4588:B
4584:B
4582:(
4580:P
4576:A
4572:A
4570:(
4568:P
4535:.
4533:A
4529:B
4522:)
4520:B
4518:(
4516:P
4512:/
4509:)
4507:A
4503:B
4501:(
4499:P
4488:B
4480:B
4476:A
4474:(
4472:P
4468:.
4466:A
4458:A
4456:(
4454:P
4447:B
4443:A
4407:Y
4403:C
4399:P
4395:C
4392:X
4390:(
4388:P
4365:5
4360:=
4339:=
4333:)
4330:Y
4327:(
4324:P
4319:)
4314:C
4310:X
4306:(
4303:P
4300:)
4295:C
4291:X
4286:|
4282:Y
4279:(
4276:P
4270:=
4267:)
4264:Y
4260:|
4254:C
4250:X
4246:(
4243:P
4230:C
4227:X
4223:Y
4200:=
4197:)
4191:(
4188:)
4182:(
4179:+
4176:)
4170:(
4167:)
4161:(
4158:+
4155:)
4149:(
4146:)
4140:(
4137:=
4134:)
4129:i
4125:X
4121:(
4118:P
4115:)
4110:i
4106:X
4101:|
4097:Y
4094:(
4091:P
4086:i
4078:=
4075:)
4072:Y
4069:(
4066:P
4053:P
4033:=
4030:)
4025:C
4021:X
4016:|
4012:Y
4009:(
4006:P
4002:,
3996:=
3993:)
3988:B
3984:X
3979:|
3975:Y
3972:(
3969:P
3965:,
3959:=
3956:)
3951:A
3947:X
3942:|
3938:Y
3935:(
3932:P
3919:A
3916:X
3912:Y
3910:(
3908:P
3888:=
3885:)
3880:C
3876:X
3872:(
3869:P
3865:,
3859:=
3856:)
3851:B
3847:X
3843:(
3840:P
3836:,
3830:=
3827:)
3822:A
3818:X
3814:(
3811:P
3798:Y
3794:i
3790:i
3785:i
3783:X
3740:5
3645:1
3640:=
3628:)
3621:/
3614:(
3611:+
3602:1
3591:1
3585:=
3572:)
3564:(
3561:P
3558:)
3549:|
3540:(
3537:P
3534:+
3531:)
3523:(
3520:P
3517:)
3508:|
3499:(
3496:P
3491:)
3483:(
3480:P
3477:)
3468:|
3459:(
3456:P
3450:=
3437:)
3429:(
3426:P
3421:)
3413:(
3410:P
3407:)
3398:|
3389:(
3386:P
3380:=
3373:)
3364:|
3355:(
3352:P
2971:)
2963:|
2955:(
2952:P
2909:)
2901:(
2898:P
2895:)
2887:|
2879:(
2876:P
2873:+
2870:)
2862:(
2859:P
2856:)
2848:|
2840:(
2837:P
2834:=
2831:)
2823:(
2820:P
2778:+
2766:=
2751:+
2725:=
2712:)
2704:(
2701:P
2698:)
2690:|
2682:(
2679:P
2676:+
2673:)
2665:(
2662:P
2659:)
2651:|
2643:(
2640:P
2635:)
2627:(
2624:P
2621:)
2613:|
2605:(
2602:P
2596:=
2583:)
2575:(
2572:P
2567:)
2559:(
2556:P
2553:)
2545:|
2537:(
2534:P
2528:=
2521:)
2513:|
2505:(
2502:P
2475:)
2467:|
2459:(
2456:P
2327:)
2324:y
2321:=
2318:Y
2314:|
2310:)
2307:X
2304:(
2299:A
2295:1
2291:(
2288:E
2285:=
2282:)
2279:A
2276:(
2271:y
2266:X
2262:P
2239:y
2236:=
2233:Y
2213:X
2191:y
2186:X
2182:P
2161:)
2158:x
2155:d
2152:(
2147:X
2143:P
2139:)
2136:y
2133:d
2130:(
2125:x
2120:Y
2116:P
2112:=
2109:)
2106:y
2103:d
2100:,
2097:x
2094:d
2091:(
2086:Y
2083:,
2080:X
2076:P
2055:X
2033:X
2029:P
2008:x
2005:=
2002:X
1982:Y
1960:x
1955:Y
1951:P
1922:.
1916:)
1913:y
1910:(
1905:Y
1901:f
1895:)
1892:x
1889:(
1884:X
1880:f
1876:)
1873:y
1870:(
1865:x
1862:=
1859:X
1856:|
1853:Y
1849:f
1842:=
1839:)
1836:x
1833:(
1828:y
1825:=
1822:Y
1819:|
1816:X
1812:f
1782:)
1779:x
1776:(
1771:X
1767:f
1761:)
1758:y
1755:,
1752:x
1749:(
1744:Y
1741:,
1738:X
1734:f
1727:=
1724:)
1721:y
1718:(
1713:x
1710:=
1707:X
1704:|
1701:Y
1697:f
1672:)
1669:y
1666:(
1661:Y
1657:f
1651:)
1648:y
1645:,
1642:x
1639:(
1634:Y
1631:,
1628:X
1624:f
1617:=
1614:)
1611:x
1608:(
1603:y
1600:=
1597:Y
1594:|
1591:X
1587:f
1569:Y
1565:X
1534:)
1531:B
1528:(
1525:P
1517:,
1511:)
1508:B
1505:(
1502:P
1497:)
1494:A
1491:(
1488:P
1485:)
1482:A
1479:|
1476:B
1473:(
1470:P
1464:=
1461:)
1458:B
1455:|
1452:A
1449:(
1446:P
1423:)
1420:B
1417:|
1414:A
1411:(
1408:P
1388:)
1385:B
1379:A
1376:(
1373:P
1344:)
1341:A
1338:(
1335:P
1327:,
1321:)
1318:A
1315:(
1312:P
1307:)
1304:B
1298:A
1295:(
1292:P
1286:=
1283:)
1280:A
1277:|
1274:B
1271:(
1268:P
1245:)
1242:B
1236:A
1233:(
1230:P
1207:,
1204:0
1198:)
1195:B
1192:(
1189:P
1181:,
1175:)
1172:B
1169:(
1166:P
1161:)
1158:B
1152:A
1149:(
1146:P
1140:=
1137:)
1134:B
1131:|
1128:A
1125:(
1122:P
1082:.
1062:B
1042:A
1022:)
1019:B
1016:(
1013:P
993:)
990:A
987:(
984:P
974:.
962:)
959:B
956:|
953:A
950:(
947:L
944:=
941:)
938:A
935:|
932:B
929:(
926:P
906:B
886:A
862:A
842:B
822:)
819:A
816:|
813:B
810:(
807:P
797:.
785:B
765:A
741:B
721:A
697:)
694:B
691:|
688:A
685:(
682:P
659:0
653:)
650:B
647:(
644:P
620:B
600:A
574:)
571:B
568:(
565:P
560:)
557:A
554:(
551:P
548:)
545:A
542:|
539:B
536:(
533:P
527:=
524:)
521:B
518:|
515:A
512:(
509:P
394:/
391:z
385:b
382:/
378:(
307:e
300:t
293:v
23:.
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