620:
5407:
sensible generalisation. The full derivation can be found in Jaynes' book, but it does admit an easier to understand alternative derivation, once the solution is known. Another point to emphasise is that the prior state of knowledge described by the rule of succession is given as an enumeration of the possibilities, with the additional information that it is possible to observe each category. This can be equivalently stated as observing each category once prior to gathering the data. To denote that this is the knowledge used, an
50:
5815:
4915:
6263:
not some "universal" set. In fact Larry
Bretthorst shows that including the possibility of "something else" into the hypothesis space makes no difference to the relative probabilities of the other hypothesis—it simply renormalises them to add up to a value less than 1. Until "something else" is specified, the likelihood function conditional on this "something else" is indeterminate, for how is one to determine
5488:
4110:
4451:
2624:. This puts the information contained in the rule of succession in greater light: it can be thought of as expressing the prior assumption that if sampling was continued indefinitely, we would eventually observe at least one success, and at least one failure in the sample. The prior expressing total ignorance does not assume this knowledge.
5201:
6262:
no account of an observation previously believed to have zero probability—it is still declared impossible. However, only considering a fixed set of the possibilities is an acceptable route, one just needs to remember that the results are conditional on (or restricted to) the set being considered, and
595:
Laplace knew this well, and he wrote to conclude the sunrise example: "But this number is far greater for him who, seeing in the totality of phenomena the principle regulating the days and seasons, realizes that nothing at the present moment can arrest the course of it." Yet
Laplace was ridiculed for
6389:
may need to be very large. They are not always small, and thereby soon outweighed by actual observations, as is often assumed. However, although a last resort, for everyday purposes, prior knowledge is usually vital. So most decisions must be subjective to some extent (dependent upon the analyst and
1602:
6385:
region. Also important when there are many observations, where it is believed that the expectation should be heavily weighted towards the prior estimates, in spite of many observations to the contrary, such as for a roulette wheel in a well-respected casino. In the latter case, at least some of the
6211:. This indicates that mere knowledge of more than two outcomes we know are possible is relevant information when collapsing these categories down to just two. This illustrates the subtlety in describing the prior information, and why it is important to specify which prior information one is using.
6384:
Prior probabilities are only worth spending significant effort estimating when likely to have significant effect. They may be important when there are few observations — especially when so few that there have been few, if any, observations of some possibilities – such as a rare animal, in a given
2165:
819:
Note: The actual probability needs to use the length of blue arcs divided by the length of all arcs. However, when k points are uniformly randomly distributed on a circle, the distance from a point to the next point is 1/k. So on average each arc is of the same length and ratio of lengths becomes
5406:
The rule of succession has many different intuitive interpretations, and depending on which intuition one uses, the generalisation may be different. Thus, the way to proceed from here is very carefully, and to re-derive the results from first principles, rather than to introduce an intuitively
587:
Laplace used the rule of succession to calculate the probability that the Sun will rise tomorrow, given that it has risen every day for the past 5000 years. One obtains a very large factor of approximately 5000 × 365.25, which gives odds of about 1,826,200 to 1 in favour of the Sun rising
2424:
6246:
known possible outcomes prior to observing any data, only then does the rule of succession apply. If the rule of succession is applied in problems where this does not accurately describe the prior state of knowledge, then it may give counter-intuitive results. This is not because the rule of
6316:
However, it is sometimes debatable whether prior knowledge should affect the relative probabilities, or also the total weight of the prior knowledge compared to actual observations. This does not have a clear cut answer, for it depends on what prior knowledge one is considering. In fact, an
5810:{\displaystyle f(p_{1},\ldots ,p_{m}\mid n_{1},\ldots ,n_{m},I)={\begin{cases}{\displaystyle {\frac {\Gamma \left(\sum _{i=1}^{m}(n_{i}+1)\right)}{\prod _{i=1}^{m}\Gamma (n_{i}+1)}}p_{1}^{n_{1}}\cdots p_{m}^{n_{m}}},\quad &\sum _{i=1}^{m}p_{i}=1\\\\0&{\text{otherwise.}}\end{cases}}}
4440:
3782:
351:+ 1 successes. Although this may seem the simplest and most reasonable assumption, which also happens to be true, it still requires a proof. Indeed, assuming a pseudocount of one per possibility is one way to generalise the binary result, but has unexpected consequences — see
5419:
The rule of succession comes from setting a binomial likelihood, and a uniform prior distribution. Thus a straightforward generalisation is just the multivariate extensions of these two distributions: 1) Setting a uniform prior over the initial m categories, and 2) using the
1055:
591:
However, as the mathematical details below show, the basic assumption for using the rule of succession would be that we have no prior knowledge about the question whether the Sun will or will not rise tomorrow, except that it can do either. This is not the case for sunrises.
4910:{\displaystyle {\begin{aligned}\sum _{R=1}^{N-n}{\prod _{j=1}^{n-1}(N-R-j) \over R}&\approx \int _{1}^{N-n}{(N-R)^{n-1} \over R}\,dR\\&=N\int _{1}^{N-n}{(N-R)^{n-2} \over R}\,dR-\int _{1}^{N-n}(N-R)^{n-2}\,dR\\&=N^{n-1}\left\approx N^{n-1}\ln(N)\end{aligned}}}
3332:
1287:
6321:
potential categories, but I am sure that only one of them is possible prior to observing the data. However, I do not know which particular category this is." A mathematical way to describe this prior is the
Dirichlet distribution with all parameters equal to
3738:
5354:(for the simpler analytic properties) we are "throwing away" a piece of very important information. Note that this ignorance relationship only holds as long as only no successes are observed. It is correspondingly revised back to the observed frequency rule
6234:
One of the most difficult aspects of the rule of succession is not the mathematical formulas, but answering the question: When does the rule of succession apply? In the generalisation section, it was noted very explicitly by adding the prior information
338:
Since we have the prior knowledge that we are looking at an experiment for which both success and failure are possible, our estimate is as if we had observed one success and one failure for sure before we even started the experiments. In a sense we made
3531:
6254:), no possibility should have its probability (or its pseudocount) set to zero, since nothing in the physical world should be assumed strictly impossible (though it may be)—even if contrary to all observations and current theories. Indeed,
1936:
759:
defines the first non-blue/failure arc. Since the next point is a uniformly random point, if it falls in any of the blue arcs then the trial succeeds while if it falls in any of the other arcs, then it fails. So the probability of success
2571:
675:
The rule of succession can be interpreted in an intuitive manner by considering points randomly distributed on a circle rather than counting the number "success"/"failures" in an experiment. To mimic the behavior of the proportion
5310:
4930:
1330:
1956:
6174:
2227:
1742:
470:
4227:
6006:
328:
723:
points uniformly distributed on the circle; any point in the "success" fraction is a success and a failure otherwise. This provides an exact mapping from success/failure experiments with probability of success
6230:
Given a good model, it is best to make as many observations as practicable, depending on the expected reliability of prior knowledge, cost of observations, time and resources available, and accuracy required.
4105:{\displaystyle E\left({S \over N}|n,s=0,N\right)={1 \over N}\sum _{S=1}^{N-n}SP(S|N,n=1,s=0)={1 \over N}{\sum _{S=1}^{N-n}\prod _{j=1}^{n-1}(N-S-j) \over \sum _{R=1}^{N-n}{\prod _{j=1}^{n-1}(N-R-j) \over R}}}
4216:
942:
6311:
6227:: Although we have a huge number of samples of the sun rising, there are far better models of the sun than assuming it has a certain probability of rising each day, e.g., simply having a half-life.
2576:
Thus, with the prior specifying total ignorance, the probability of success is governed by the observed frequency of success. However, the posterior distribution that led to this result is the Beta(
4456:
3091:
162:. The formula is still used, particularly to estimate underlying probabilities when there are few observations or events that have not been observed to occur at all in (finite) sample data.
1073:
3542:
647:
is precisely the number of blue arcs divided by the total number of arcs. If we let the first clockwise point of an arc define it, then every point on the circle defines one arc with
2698:
2781:
2749:
6379:
6504:
6461:
2945:
2901:
2983:
3536:
Adding in the normalising constant, which is always finite (because there are no singularities in the range of the posterior, and there are a finite number of terms) gives:
789:
5385:
3774:
2814:
833:
is assigned a uniform distribution to describe the uncertainty about its true value. (This proportion is not random, but uncertain. We assign a probability distribution to
5424:
as the likelihood function (which is the multivariate generalisation of the binomial distribution). It can be shown that the uniform distribution is a special case of the
3359:
1760:
577:
5331: = 5 (tens of thousands), the expected proportion rises to approximately 0.86%, and so on. Similarly, if the number of observations is smaller, so that
529:
503:
5196:{\displaystyle E\left({S \over N}|n,s=0,N\right)\approx {1 \over N}{{N^{n} \over n} \over N^{n-1}\ln(N)}={1 \over n}={\log _{10}(e) \over n}={0.434294 \over n}}
3083:
3063:
3043:
3023:
3003:
2854:
2834:
552:
1597:{\displaystyle f(p\mid X_{1}=x_{1},\ldots ,X_{n}=x_{n})={L(p)f(p) \over \int _{0}^{1}L(r)f(r)\,dr}={p^{s}(1-p)^{n-s} \over \int _{0}^{1}r^{s}(1-r)^{n-s}\,dr}}
5220:
728:
to uniformly random points on the circle. In the figure the success fraction is colored blue to differentiate it from the rest of the circle and the points
2160:{\displaystyle \operatorname {E} (p\mid X_{i}=x_{i}{\text{ for }}i=1,\dots ,n)=\int _{0}^{1}pf(p\mid X_{1}=x_{1},\ldots ,X_{n}=x_{n})\,dp={s+1 \over n+2}.}
2455:
2419:{\displaystyle P(X_{n+1}=1\mid X_{i}=x_{i}{\text{ for }}i=1,\dots ,n)=\operatorname {E} (p\mid X_{i}=x_{i}{\text{ for }}i=1,\dots ,n)={s+1 \over n+2}.}
37:"Laplace–Bayes estimator" redirects here. For statistical estimators that maximize posterior expected utility or minimize posterior expected loss, see
5323: = 10 results without success, then the expected proportion in the population is approximately 0.43%. If the population is smaller, so that
6069:
4435:{\displaystyle \sum _{S=1}^{N-n}\prod _{j=1}^{n-1}(N-S-j)\approx \int _{1}^{N-n}(N-S)^{n-1}\,dS={(N-1)^{n}-n^{n} \over n}\approx {N^{n} \over n}}
1617:
751:
points such that the portion from a point on the circle to the next point (moving clockwise) is one arc associated with the first point. Thus,
579:
get more and more similar, which is intuitively clear: the more data we have, the less importance should be assigned to our prior information.
5432:
for the multinomial distribution, which means that the posterior distribution is also a
Dirichlet distribution with different parameters. Let
2612:). This means that we cannot use this form of the posterior distribution to calculate the probability of the next observation succeeding when
5889:
215:
368:
17:
5350:. This means that the probability depends on the size of the population from which one is sampling. In passing to the limit of infinite
6011:
This solution reduces to the probability that would be assigned using the principle of indifference before any observations made (i.e.
1050:{\displaystyle f(p)={\begin{cases}0&{\text{for }}p\leq 0\\1&{\text{for }}0<p<1\\0&{\text{for }}p\geq 1\end{cases}}}
680:
on the circle, we will color the circle in two colors and the fraction of the circle colored in the "success" color will be equal to
6015: = 0), consistent with the original rule of succession. It also contains the rule of succession as a special case, when
4125:
607:, and developed measures of degree of confirmation, which he considered as alternatives to Laplace's rule of succession. See also
6266:
114:
4445:
The same procedure is followed for the denominator, but the process is a bit more tricky, as the integral is harder to evaluate
6547:
5428:
with all of its parameters equal to 1 (just as the uniform is Beta(1,1) in the binary case). The
Dirichlet distribution is the
86:
30:
This article is about the rule of succession in probability theory. For monarchical and presidential rules of succession, see
6448:
2437:, including ignorance with regard to the question whether the experiment can succeed, or can fail. This improper prior is 1/(
3327:{\displaystyle P(S|N,n,s)\propto {1 \over S(N-S)}{S \choose s}{N-S \choose n-s}\propto {S!(N-S)! \over S(N-S)(S-s)!(N-S-)!}}
6247:
succession is defective, but that it is effectively answering a different question, based on different prior information.
93:
67:
1282:{\displaystyle L(p)=P(X_{1}=x_{1},\ldots ,X_{n}=x_{n}\mid p)=\prod _{i=1}^{n}p^{x_{i}}(1-p)^{1-x_{i}}=p^{s}(1-p)^{n-s}}
6404:
930:
133:
3733:{\displaystyle P(S|N,n,s=0)={\prod _{j=1}^{n-1}(N-S-j) \over S\sum _{R=1}^{N-n}{\prod _{j=1}^{n-1}(N-R-j) \over R}}}
3349: = 0, then one of the factorials in the numerator cancels exactly with one in the denominator. Taking the
100:
6179:
which is different from the original rule of succession. But note that the original rule of succession is based on
6622:
5342:
This probability has no positive lower bound, and can be made arbitrarily small for larger and larger choices of
6313:? Thus no updating of the prior probability for "something else" can occur until it is more accurately defined.
82:
71:
6044:
probabilities that correspond to "success" to get the probability of success. Supposing that this aggregates
5883:
be the total number of observations made. The result, using the properties of the
Dirichlet distribution is:
2646:
2754:
6627:
2703:
6341:
2640:
608:
2196:
tells us that the expected probability of success in the next experiment is just the expected value of
596:
this calculation; his opponents gave no heed to that sentence, or failed to understand its importance.
2906:
2862:
6409:
6219:
A good model is essential (i.e., a good compromise between accuracy and practicality). To paraphrase
5210:
has been used in the final answer for ease of calculation. For instance if the population is of size
2950:
2430:
5576:
966:
767:
5421:
5357:
3746:
2786:
2193:
2177:
886:
203:
3526:{\displaystyle P(S|N,n,s=0)\propto {(N-S-1)! \over S(N-S-n)!}={\prod _{j=1}^{n-1}(N-S-j) \over S}}
362:
known from the start that both success and failure are possible, then we would have had to assign
1931:{\displaystyle f(p\mid X_{1}=x_{1},\ldots ,X_{n}=x_{n})={(n+1)! \over s!(n-s)!}p^{s}(1-p)^{n-s}.}
60:
6063:
values that have been termed "success". The probability of "success" at the next trial is then:
107:
5425:
878:
2449: ≤ 1 and 0 otherwise. If the calculation above is repeated with this prior, we get
691:
A fraction is chosen by selecting two uniformly random points on the circle. The first point
5820:
To get the generalised rule of succession, note that the probability of observing category
1608:
155:
6572:
8:
6242:
into the calculations. Thus, when all that is known about a phenomenon is that there are
1324:
as the data actually observed). Putting it all together, we can calculate the posterior:
1065:
604:
508:
482:
31:
6427:
Laplace, Pierre-Simon (1814). Essai philosophique sur les probabilités. Paris: Courcier.
557:
6527:
6484:
6399:
6251:
5305:{\displaystyle E\left({S \over N}\mid n,s=0,N=10^{k}\right)\approx {0.434294 \over nk}}
3068:
3048:
3028:
3008:
2988:
2839:
2819:
815:
points. Substituting the values with number of successes gives the rule of succession.
537:
147:
6444:
4921:
1942:
922:
897:
6488:
6338:
to each category. This gives a slightly different probability in the binary case of
5466:) actually was observed. Then the joint posterior distribution of the probabilities
2566:{\displaystyle P'(X_{n+1}=1\mid X_{i}=x_{i}{\text{ for }}i=1,\dots ,n)={s \over n}.}
619:
6519:
6476:
6224:
5429:
2189:
862:
209:
that each can assume the value 0 or 1, then, if we know nothing more about them,
206:
159:
38:
5387:
as soon as one success is observed. The corresponding results are found for the
6598:
5315:
So for example, if the population be on the order of tens of billions, so that
1946:
6169:{\displaystyle P({\text{success}}|n_{1},\ldots ,n_{m},I_{m})={s+c \over n+m},}
2176:
tells us the probability of success in any experiment, and each experiment is
182:
failures, then what is the probability that the next repetition will succeed?
6616:
1748:
600:
479:, below, for an analysis of its validity. In particular it is not valid when
6317:
alternative prior state of knowledge could be of the form "I have specified
5391:
case by switching labels, and then subtracting the probability from 1.
170:
If we repeat an experiment that we know can result in a success or failure,
799:
is the total number of arcs. Note that there is one more blue arc (that of
6386:
2180:, the conditional probability for success in the next experiment is just
1737:{\displaystyle \int _{0}^{1}r^{s}(1-r)^{n-s}\,dr={s!(n-s)! \over (n+1)!}}
344:
6531:
6255:
6001:{\displaystyle P(A_{i}|n_{1},\ldots ,n_{m},I_{m})={n_{i}+1 \over n+m}.}
323:{\displaystyle P(X_{n+1}=1\mid X_{1}+\cdots +X_{n}=s)={s+1 \over n+2}.}
6439:
Part II Section 18.6 of Jaynes, E. T. & Bretthorst, G. L. (2003).
5207:
6523:
465:{\displaystyle P'(X_{n+1}=1\mid X_{1}+\cdots +X_{n}=s)={s \over n}.}
49:
6480:
5339: = 10, the proportion rise to approximately 0.86% again.
4221:
and then replacing the summation in the numerator with an integral
671:) more than total number of trials which is the rule of succession.
841:. But this amounts, mathematically, to the same thing as treating
352:
6220:
5416:
is put as part of the conditions in the probability assignments.
2600: = 0 (i.e. the normalisation constant is infinite when
5394:
4119:
is given by first making the approximation to the product term:
837:
to express our uncertainty, not to attribute randomness to
703:
within . In terms of the circle the fraction of the circle from
582:
695:
corresponds to the zero in the interval and the second point
5399:
This section gives a heuristic derivation similar to that in
5214:
then probability of success on the next sample is given by:
4924:
plugging in these approximations into the expectation gives
4211:{\displaystyle \prod _{j=1}^{n-1}(N-R-j)\approx (N-R)^{n-1}}
2836:
as the number of successes in the total population, of size
2700:. This is the approach taken in Jaynes (2003). The binomial
743:
is the fraction colored blue. Let us divide the circle into
6306:{\displaystyle Pr({\text{data}}|{\text{something else}},I)}
5803:
1043:
5851:
denote the event that the next observation is in category
1754:
The posterior probability density function is therefore
6573:
http://www.stats.org.uk/priors/noninformative/Smith.pdf
6031:
are mutually exclusive, it is possible to collapse the
631:
is the point such that the fraction of the circle from
165:
6344:
6269:
6072:
5892:
5580:
5491:
5360:
5223:
4933:
4454:
4230:
4128:
3785:
3749:
3545:
3362:
3094:
3071:
3051:
3031:
3011:
2991:
2953:
2909:
2865:
2842:
2822:
2789:
2757:
2706:
2649:
2458:
2230:
1959:
1763:
1620:
1333:
1076:
945:
770:
560:
540:
511:
485:
371:
218:
900:
to find the conditional probability distribution of
6600:
Bayesian
Spectrum Analysis and parameter estimation
74:. Unsourced material may be challenged and removed.
6548:"An elegant proof of Laplace's rule of succession"
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6168:
6000:
5809:
5379:
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5195:
4909:
4434:
4210:
4104:
3768:
3732:
3525:
3326:
3077:
3057:
3037:
3017:
2997:
2977:
2939:
2895:
2848:
2828:
2808:
2775:
2743:
2692:
2565:
2418:
2159:
1930:
1736:
1596:
1281:
1049:
783:
571:
546:
523:
497:
464:
322:
3211:
3182:
3173:
3160:
803:) than success point and two more arcs (those of
6614:
684:. To express the uncertainty about the value of
6195:. This means that the information contained in
4115:An approximate analytical expression for large
2627:To evaluate the "complete ignorance" case when
2429:The same calculation can be performed with the
688:, we need to select a fraction of the circle.
154:is a formula introduced in the 18th century by
1316:is the number of trials (we are using capital
6502:
6459:
5842:, we simply require its expectation. Letting
5395:Generalization to any number of possibilities
583:Historical application to the sunrise problem
353:Generalization to any number of possibilities
6587:, Cambridge, UK, Cambridge University Press.
6019: = 2, as a generalisation should.
5824:on the next observation, conditional on the
2639:can be dealt with, we first go back to the
1320:to denote a random variable and lower-case
603:investigated a probability-based theory of
6035:categories into 2. Simply add up the
925:" (i.e., marginal) probability measure of
655:defining a non-blue arc. The estimate for
6596:
6441:Probability Theory: The Logic of Science.
6186:, whereas the generalisation is based on
4745:
4686:
4608:
4358:
2751:can be derived as a limiting form, where
2588:) distribution, which is not proper when
2118:
1674:
1584:
1475:
534:If the number of observations increases,
476:
134:Learn how and when to remove this message
6585:Probability Theory: The Logic of Science
6512:Philosophy and Phenomenological Research
6435:
6433:
5401:Probability Theory: The Logic of Science
663:) more than blue points divided by two (
618:
6505:"On the Application of Inductive Logic"
2693:{\displaystyle \mathrm {Hyp} (s|N,n,S)}
824:
14:
6615:
2776:{\displaystyle N,S\rightarrow \infty }
857:be 1 if we observe a "success" on the
343: + 2 observations (known as
6430:
5441:denote the probability that category
3045:, and then dividing this estimate by
2744:{\displaystyle \mathrm {Bin} (r|n,p)}
1751:for more on integrals of this form).
6374:{\displaystyle {\frac {s+0.5}{n+1}}}
6326:, which then gives a pseudocount of
6204:is different from that contained in
5454:denote the number of times category
2204:is conditional on the observed data
869:of success on each trial. Thus each
72:adding citations to reliable sources
43:
6405:Krichevsky–Trofimov estimator
6214:
6022:Because the propositions or events
1064:under our observations, we use the
739:Given this circle, the estimate of
166:Statement of the rule of succession
24:
6545:
5663:
5584:
3186:
3164:
2770:
2714:
2711:
2708:
2657:
2654:
2651:
2433:that expresses total ignorance of
2317:
1960:
711:moving clockwise will be equal to
25:
6639:
3743:So the posterior expectation for
1312:is the number of "successes" and
755:defines the first blue arc while
333:
5319: = 10, and we observe
2940:{\displaystyle {1 \over S(N-S)}}
2896:{\displaystyle {1 \over p(1-p)}}
2816:remains fixed. One can think of
865:, otherwise 0, with probability
48:
6056:denote the sum of the relevant
5859: = 1, ...,
5743:
5462: = 1, ...,
2978:{\displaystyle 1\leq S\leq N-1}
2783:in such a way that their ratio
2188:is being treated as if it is a
795:is the number of blue arcs and
59:needs additional citations for
6590:
6577:
6566:
6539:
6496:
6453:
6421:
6330:to the denominator instead of
6300:
6285:
6276:
6131:
6085:
6076:
5956:
5910:
5896:
5685:
5666:
5632:
5613:
5565:
5495:
5187:
5184:
5178:
5162:
5144:
5141:
5135:
5119:
5111:
5105:
5080:
5077:
5071:
5062:
5044:
5038:
4953:
4900:
4894:
4730:
4717:
4665:
4652:
4587:
4574:
4534:
4516:
4384:
4371:
4343:
4330:
4303:
4285:
4193:
4180:
4174:
4156:
4090:
4072:
4010:
3992:
3919:
3888:
3881:
3805:
3718:
3700:
3635:
3617:
3581:
3556:
3549:
3514:
3496:
3454:
3436:
3425:
3407:
3398:
3373:
3366:
3353: = 0 case, we have:
3315:
3312:
3300:
3285:
3279:
3267:
3264:
3252:
3241:
3229:
3151:
3139:
3124:
3105:
3098:
2931:
2919:
2887:
2875:
2767:
2738:
2725:
2718:
2687:
2668:
2661:
2544:
2467:
2381:
2323:
2311:
2234:
2115:
2051:
2024:
1966:
1910:
1897:
1878:
1866:
1852:
1840:
1831:
1767:
1725:
1713:
1705:
1693:
1659:
1646:
1569:
1556:
1514:
1501:
1472:
1466:
1460:
1454:
1431:
1425:
1419:
1413:
1401:
1337:
1264:
1251:
1216:
1203:
1159:
1095:
1086:
1080:
1060:For the likelihood of a given
955:
949:
784:{\displaystyle {\frac {b}{t}}}
609:New riddle of induction#Carnap
443:
380:
285:
222:
158:in the course of treating the
13:
1:
6415:
6052:categories as "failure". Let
5874: + ... +
5380:{\displaystyle p={s \over n}}
3769:{\displaystyle p={S \over N}}
2809:{\displaystyle p={S \over N}}
1303: + ... +
719:trials can be interpreted as
174:times independently, and get
27:Formula in probability theory
6597:Bretthost, G. Larry (1988).
6443:Cambridge University Press.
6334:, and adds a pseudocount of
6048:categories as "success" and
3337:And it can be seen that, if
3025:is equivalent to estimating
614:
7:
6393:
2641:hypergeometric distribution
10:
6644:
5445:will be observed, and let
747:arcs corresponding to the
36:
29:
6606:(PhD thesis). p. 55.
6410:Principle of indifference
2985:. Working conditional to
2178:conditionally independent
887:conditionally independent
204:conditionally independent
5422:multinomial distribution
2859:The equivalent prior to
2194:law of total probability
736:are highlighted in red.
651:defining a blue arc and
358:Nevertheless, if we had
933:over the open interval
18:Laplace–Bayes estimator
6623:Probability assessment
6503:Rudolf Carnap (1947).
6460:Rudolf Carnap (1945).
6375:
6307:
6170:
6002:
5811:
5766:
5662:
5612:
5426:Dirichlet distribution
5381:
5306:
5197:
4911:
4515:
4485:
4436:
4284:
4257:
4212:
4155:
4106:
4071:
4041:
3991:
3964:
3874:
3770:
3734:
3699:
3669:
3616:
3527:
3495:
3328:
3079:
3059:
3039:
3019:
3005:means that estimating
2999:
2979:
2941:
2897:
2850:
2830:
2810:
2777:
2745:
2694:
2567:
2420:
2161:
1932:
1738:
1598:
1283:
1185:
1051:
879:Bernoulli distribution
785:
672:
639:(in blue) is equal to
573:
548:
525:
499:
466:
324:
6583:Jaynes, E.T. (2003),
6469:Philosophy of Science
6376:
6308:
6171:
6003:
5812:
5746:
5642:
5592:
5382:
5307:
5198:
4912:
4489:
4459:
4437:
4258:
4231:
4213:
4129:
4107:
4045:
4015:
3965:
3938:
3848:
3771:
3735:
3673:
3643:
3590:
3528:
3469:
3329:
3080:
3060:
3040:
3020:
3000:
2980:
2942:
2898:
2851:
2831:
2811:
2778:
2746:
2695:
2568:
2445:)) for 0 ≤
2441:(1 −
2421:
2162:
1933:
1739:
1599:
1284:
1165:
1052:
786:
622:
574:
549:
526:
500:
467:
325:
6462:"On Inductive Logic"
6342:
6267:
6070:
5890:
5489:
5358:
5221:
4931:
4452:
4228:
4126:
3783:
3747:
3543:
3360:
3092:
3069:
3065:. The posterior for
3049:
3029:
3009:
2989:
2951:
2907:
2863:
2840:
2820:
2787:
2755:
2704:
2647:
2456:
2228:
1957:
1761:
1618:
1609:normalizing constant
1331:
1074:
943:
931:uniform distribution
825:Mathematical details
768:
558:
538:
509:
483:
477:Mathematical details
369:
216:
185:More abstractly: If
156:Pierre-Simon Laplace
83:"Rule of succession"
68:improve this article
6628:Inductive reasoning
5737:
5712:
4803:
4716:
4648:
4570:
4329:
2947:, with a domain of
2584: −
2044:
1635:
1545:
1450:
1066:likelihood function
627:is the zero point,
605:inductive reasoning
524:{\displaystyle s=n}
498:{\displaystyle s=0}
32:Order of succession
6400:Additive smoothing
6371:
6303:
6250:In principle (see
6166:
5998:
5807:
5802:
5738:
5716:
5691:
5377:
5302:
5206:where the base 10
5193:
4907:
4905:
4783:
4696:
4628:
4550:
4432:
4309:
4208:
4102:
3766:
3730:
3523:
3324:
3075:
3055:
3035:
3015:
2995:
2975:
2937:
2893:
2846:
2826:
2806:
2773:
2741:
2690:
2631: = 0 or
2616: = 0 or
2604: = 0 or
2563:
2416:
2157:
2030:
1928:
1734:
1621:
1594:
1531:
1436:
1279:
1047:
1042:
781:
673:
572:{\displaystyle P'}
569:
544:
521:
495:
462:
320:
152:rule of succession
148:probability theory
6552:Unexpected Values
6449:978-0-521-59271-0
6369:
6292:
6282:
6161:
6082:
5993:
5798:
5689:
5473:, ...,
5375:
5327: = 10,
5300:
5240:
5191:
5148:
5084:
5048:
5013:
4994:
4950:
4922:natural logarithm
4858:
4838:
4817:
4684:
4606:
4541:
4430:
4410:
4100:
4097:
3933:
3846:
3802:
3764:
3728:
3725:
3521:
3461:
3322:
3209:
3171:
3155:
3085:can be given as:
3078:{\displaystyle S}
3058:{\displaystyle N}
3038:{\displaystyle S}
3018:{\displaystyle p}
2998:{\displaystyle N}
2935:
2891:
2849:{\displaystyle N}
2829:{\displaystyle S}
2804:
2558:
2521:
2411:
2358:
2288:
2152:
2001:
1943:beta distribution
1885:
1732:
1592:
1483:
1029:
1000:
977:
845:it were random).
779:
547:{\displaystyle P}
457:
315:
144:
143:
136:
118:
16:(Redirected from
6635:
6608:
6607:
6605:
6594:
6588:
6581:
6575:
6570:
6564:
6563:
6561:
6559:
6543:
6537:
6535:
6509:
6500:
6494:
6493:; here: p.86, 97
6492:
6466:
6457:
6451:
6437:
6428:
6425:
6390:analysis used).
6380:
6378:
6377:
6372:
6370:
6368:
6357:
6346:
6312:
6310:
6309:
6304:
6293:
6290:
6288:
6283:
6280:
6215:Further analysis
6175:
6173:
6172:
6167:
6162:
6160:
6149:
6138:
6130:
6129:
6117:
6116:
6098:
6097:
6088:
6083:
6080:
6007:
6005:
6004:
5999:
5994:
5992:
5981:
5974:
5973:
5963:
5955:
5954:
5942:
5941:
5923:
5922:
5913:
5908:
5907:
5816:
5814:
5813:
5808:
5806:
5805:
5799:
5796:
5786:
5776:
5775:
5765:
5760:
5739:
5736:
5735:
5734:
5724:
5711:
5710:
5709:
5699:
5690:
5688:
5678:
5677:
5661:
5656:
5640:
5639:
5635:
5625:
5624:
5611:
5606:
5582:
5558:
5557:
5539:
5538:
5526:
5525:
5507:
5506:
5386:
5384:
5383:
5378:
5376:
5368:
5335: = 5,
5311:
5309:
5308:
5303:
5301:
5299:
5288:
5283:
5279:
5278:
5277:
5241:
5233:
5202:
5200:
5199:
5194:
5192:
5190:
5174:
5173:
5154:
5149:
5147:
5131:
5130:
5114:
5101:
5100:
5090:
5085:
5083:
5054:
5049:
5047:
5031:
5030:
5014:
5009:
5008:
4999:
4997:
4995:
4987:
4982:
4978:
4956:
4951:
4943:
4920:where ln is the
4916:
4914:
4913:
4908:
4906:
4887:
4886:
4868:
4864:
4863:
4859:
4851:
4839:
4837:
4823:
4818:
4813:
4805:
4802:
4791:
4777:
4776:
4755:
4744:
4743:
4715:
4704:
4685:
4680:
4679:
4678:
4650:
4647:
4636:
4618:
4607:
4602:
4601:
4600:
4572:
4569:
4558:
4542:
4537:
4514:
4503:
4487:
4484:
4473:
4441:
4439:
4438:
4433:
4431:
4426:
4425:
4416:
4411:
4406:
4405:
4404:
4392:
4391:
4369:
4357:
4356:
4328:
4317:
4283:
4272:
4256:
4245:
4217:
4215:
4214:
4209:
4207:
4206:
4154:
4143:
4111:
4109:
4108:
4103:
4101:
4099:
4098:
4093:
4070:
4059:
4043:
4040:
4029:
4013:
3990:
3979:
3963:
3952:
3936:
3934:
3926:
3891:
3873:
3862:
3847:
3839:
3834:
3830:
3808:
3803:
3795:
3775:
3773:
3772:
3767:
3765:
3757:
3739:
3737:
3736:
3731:
3729:
3727:
3726:
3721:
3698:
3687:
3671:
3668:
3657:
3638:
3615:
3604:
3588:
3559:
3532:
3530:
3529:
3524:
3522:
3517:
3494:
3483:
3467:
3462:
3460:
3431:
3405:
3376:
3333:
3331:
3330:
3325:
3323:
3321:
3247:
3221:
3216:
3215:
3214:
3208:
3197:
3185:
3178:
3177:
3176:
3163:
3156:
3154:
3131:
3108:
3084:
3082:
3081:
3076:
3064:
3062:
3061:
3056:
3044:
3042:
3041:
3036:
3024:
3022:
3021:
3016:
3004:
3002:
3001:
2996:
2984:
2982:
2981:
2976:
2946:
2944:
2943:
2938:
2936:
2934:
2911:
2902:
2900:
2899:
2894:
2892:
2890:
2867:
2855:
2853:
2852:
2847:
2835:
2833:
2832:
2827:
2815:
2813:
2812:
2807:
2805:
2797:
2782:
2780:
2779:
2774:
2750:
2748:
2747:
2742:
2728:
2717:
2699:
2697:
2696:
2691:
2671:
2660:
2572:
2570:
2569:
2564:
2559:
2551:
2522:
2519:
2517:
2516:
2504:
2503:
2485:
2484:
2466:
2431:(improper) prior
2425:
2423:
2422:
2417:
2412:
2410:
2399:
2388:
2359:
2356:
2354:
2353:
2341:
2340:
2289:
2286:
2284:
2283:
2271:
2270:
2252:
2251:
2166:
2164:
2163:
2158:
2153:
2151:
2140:
2129:
2114:
2113:
2101:
2100:
2082:
2081:
2069:
2068:
2043:
2038:
2002:
1999:
1997:
1996:
1984:
1983:
1937:
1935:
1934:
1929:
1924:
1923:
1896:
1895:
1886:
1884:
1858:
1838:
1830:
1829:
1817:
1816:
1798:
1797:
1785:
1784:
1743:
1741:
1740:
1735:
1733:
1731:
1711:
1685:
1673:
1672:
1645:
1644:
1634:
1629:
1603:
1601:
1600:
1595:
1593:
1591:
1583:
1582:
1555:
1554:
1544:
1539:
1529:
1528:
1527:
1500:
1499:
1489:
1484:
1482:
1449:
1444:
1434:
1408:
1400:
1399:
1387:
1386:
1368:
1367:
1355:
1354:
1288:
1286:
1285:
1280:
1278:
1277:
1250:
1249:
1237:
1236:
1235:
1234:
1202:
1201:
1200:
1199:
1184:
1179:
1152:
1151:
1139:
1138:
1120:
1119:
1107:
1106:
1056:
1054:
1053:
1048:
1046:
1045:
1030:
1027:
1001:
998:
978:
975:
881:. Suppose these
873:is 0 or 1; each
820:ratio of counts.
790:
788:
787:
782:
780:
772:
578:
576:
575:
570:
568:
553:
551:
550:
545:
530:
528:
527:
522:
504:
502:
501:
496:
471:
469:
468:
463:
458:
450:
436:
435:
417:
416:
398:
397:
379:
329:
327:
326:
321:
316:
314:
303:
292:
278:
277:
259:
258:
240:
239:
207:random variables
139:
132:
128:
125:
119:
117:
76:
52:
44:
21:
6643:
6642:
6638:
6637:
6636:
6634:
6633:
6632:
6613:
6612:
6611:
6603:
6595:
6591:
6582:
6578:
6571:
6567:
6557:
6555:
6544:
6540:
6524:10.2307/2102920
6507:
6501:
6497:
6464:
6458:
6454:
6438:
6431:
6426:
6422:
6418:
6396:
6358:
6347:
6345:
6343:
6340:
6339:
6289:
6284:
6279:
6268:
6265:
6264:
6252:Cromwell's rule
6241:
6225:sunrise problem
6217:
6210:
6203:
6194:
6185:
6150:
6139:
6137:
6125:
6121:
6112:
6108:
6093:
6089:
6084:
6079:
6071:
6068:
6067:
6062:
6043:
6030:
5982:
5969:
5965:
5964:
5962:
5950:
5946:
5937:
5933:
5918:
5914:
5909:
5903:
5899:
5891:
5888:
5887:
5882:
5873:
5850:
5841:
5832:
5801:
5800:
5795:
5793:
5787:
5784:
5783:
5771:
5767:
5761:
5750:
5744:
5730:
5726:
5725:
5720:
5705:
5701:
5700:
5695:
5673:
5669:
5657:
5646:
5641:
5620:
5616:
5607:
5596:
5591:
5587:
5583:
5581:
5579:
5572:
5571:
5553:
5549:
5534:
5530:
5521:
5517:
5502:
5498:
5490:
5487:
5486:
5481:
5472:
5453:
5440:
5430:conjugate prior
5415:
5397:
5367:
5359:
5356:
5355:
5292:
5287:
5273:
5269:
5232:
5231:
5227:
5222:
5219:
5218:
5169:
5165:
5158:
5153:
5126:
5122:
5115:
5096:
5092:
5091:
5089:
5058:
5053:
5020:
5016:
5015:
5004:
5000:
4998:
4996:
4986:
4952:
4942:
4941:
4937:
4932:
4929:
4928:
4904:
4903:
4876:
4872:
4850:
4846:
4827:
4822:
4806:
4804:
4792:
4787:
4782:
4778:
4766:
4762:
4753:
4752:
4733:
4729:
4705:
4700:
4668:
4664:
4651:
4649:
4637:
4632:
4616:
4615:
4590:
4586:
4573:
4571:
4559:
4554:
4543:
4504:
4493:
4488:
4486:
4474:
4463:
4455:
4453:
4450:
4449:
4421:
4417:
4415:
4400:
4396:
4387:
4383:
4370:
4368:
4346:
4342:
4318:
4313:
4273:
4262:
4246:
4235:
4229:
4226:
4225:
4196:
4192:
4144:
4133:
4127:
4124:
4123:
4060:
4049:
4044:
4042:
4030:
4019:
4014:
3980:
3969:
3953:
3942:
3937:
3935:
3925:
3887:
3863:
3852:
3838:
3804:
3794:
3793:
3789:
3784:
3781:
3780:
3756:
3748:
3745:
3744:
3688:
3677:
3672:
3670:
3658:
3647:
3639:
3605:
3594:
3589:
3587:
3555:
3544:
3541:
3540:
3484:
3473:
3468:
3466:
3432:
3406:
3404:
3372:
3361:
3358:
3357:
3248:
3222:
3220:
3210:
3198:
3187:
3181:
3180:
3179:
3172:
3159:
3158:
3157:
3135:
3130:
3104:
3093:
3090:
3089:
3070:
3067:
3066:
3050:
3047:
3046:
3030:
3027:
3026:
3010:
3007:
3006:
2990:
2987:
2986:
2952:
2949:
2948:
2915:
2910:
2908:
2905:
2904:
2871:
2866:
2864:
2861:
2860:
2841:
2838:
2837:
2821:
2818:
2817:
2796:
2788:
2785:
2784:
2756:
2753:
2752:
2724:
2707:
2705:
2702:
2701:
2667:
2650:
2648:
2645:
2644:
2550:
2520: for
2518:
2512:
2508:
2499:
2495:
2474:
2470:
2459:
2457:
2454:
2453:
2400:
2389:
2387:
2357: for
2355:
2349:
2345:
2336:
2332:
2287: for
2285:
2279:
2275:
2266:
2262:
2241:
2237:
2229:
2226:
2225:
2212:
2190:random variable
2141:
2130:
2128:
2109:
2105:
2096:
2092:
2077:
2073:
2064:
2060:
2039:
2034:
2000: for
1998:
1992:
1988:
1979:
1975:
1958:
1955:
1954:
1913:
1909:
1891:
1887:
1859:
1839:
1837:
1825:
1821:
1812:
1808:
1793:
1789:
1780:
1776:
1762:
1759:
1758:
1712:
1686:
1684:
1662:
1658:
1640:
1636:
1630:
1625:
1619:
1616:
1615:
1572:
1568:
1550:
1546:
1540:
1535:
1530:
1517:
1513:
1495:
1491:
1490:
1488:
1445:
1440:
1435:
1409:
1407:
1395:
1391:
1382:
1378:
1363:
1359:
1350:
1346:
1332:
1329:
1328:
1311:
1302:
1267:
1263:
1245:
1241:
1230:
1226:
1219:
1215:
1195:
1191:
1190:
1186:
1180:
1169:
1147:
1143:
1134:
1130:
1115:
1111:
1102:
1098:
1075:
1072:
1071:
1041:
1040:
1026:
1024:
1018:
1017:
997:
995:
989:
988:
974:
972:
962:
961:
944:
941:
940:
912:
904:given the data
856:
829:The proportion
827:
771:
769:
766:
765:
699:corresponds to
643:. The value of
617:
585:
561:
559:
556:
555:
539:
536:
535:
510:
507:
506:
484:
481:
480:
449:
431:
427:
412:
408:
387:
383:
372:
370:
367:
366:
336:
304:
293:
291:
273:
269:
254:
250:
229:
225:
217:
214:
213:
201:
191:
178:successes, and
168:
160:sunrise problem
140:
129:
123:
120:
77:
75:
65:
53:
42:
39:Bayes estimator
35:
28:
23:
22:
15:
12:
11:
5:
6641:
6631:
6630:
6625:
6610:
6609:
6589:
6576:
6565:
6546:Neyman, Eric.
6538:
6518:(1): 133–148.
6495:
6481:10.1086/286851
6452:
6429:
6419:
6417:
6414:
6413:
6412:
6407:
6402:
6395:
6392:
6367:
6364:
6361:
6356:
6353:
6350:
6302:
6299:
6296:
6291:something else
6287:
6278:
6275:
6272:
6239:
6216:
6213:
6208:
6199:
6190:
6183:
6177:
6176:
6165:
6159:
6156:
6153:
6148:
6145:
6142:
6136:
6133:
6128:
6124:
6120:
6115:
6111:
6107:
6104:
6101:
6096:
6092:
6087:
6078:
6075:
6060:
6039:
6026:
6009:
6008:
5997:
5991:
5988:
5985:
5980:
5977:
5972:
5968:
5961:
5958:
5953:
5949:
5945:
5940:
5936:
5932:
5929:
5926:
5921:
5917:
5912:
5906:
5902:
5898:
5895:
5878:
5871:
5846:
5837:
5828:
5818:
5817:
5804:
5794:
5792:
5789:
5788:
5785:
5782:
5779:
5774:
5770:
5764:
5759:
5756:
5753:
5749:
5745:
5742:
5733:
5729:
5723:
5719:
5715:
5708:
5704:
5698:
5694:
5687:
5684:
5681:
5676:
5672:
5668:
5665:
5660:
5655:
5652:
5649:
5645:
5638:
5634:
5631:
5628:
5623:
5619:
5615:
5610:
5605:
5602:
5599:
5595:
5590:
5586:
5578:
5577:
5575:
5570:
5567:
5564:
5561:
5556:
5552:
5548:
5545:
5542:
5537:
5533:
5529:
5524:
5520:
5516:
5513:
5510:
5505:
5501:
5497:
5494:
5477:
5470:
5449:
5436:
5411:
5396:
5393:
5374:
5371:
5366:
5363:
5313:
5312:
5298:
5295:
5291:
5286:
5282:
5276:
5272:
5268:
5265:
5262:
5259:
5256:
5253:
5250:
5247:
5244:
5239:
5236:
5230:
5226:
5204:
5203:
5189:
5186:
5183:
5180:
5177:
5172:
5168:
5164:
5161:
5157:
5152:
5146:
5143:
5140:
5137:
5134:
5129:
5125:
5121:
5118:
5113:
5110:
5107:
5104:
5099:
5095:
5088:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5057:
5052:
5046:
5043:
5040:
5037:
5034:
5029:
5026:
5023:
5019:
5012:
5007:
5003:
4993:
4990:
4985:
4981:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4955:
4949:
4946:
4940:
4936:
4918:
4917:
4902:
4899:
4896:
4893:
4890:
4885:
4882:
4879:
4875:
4871:
4867:
4862:
4857:
4854:
4849:
4845:
4842:
4836:
4833:
4830:
4826:
4821:
4816:
4812:
4809:
4801:
4798:
4795:
4790:
4786:
4781:
4775:
4772:
4769:
4765:
4761:
4758:
4756:
4754:
4751:
4748:
4742:
4739:
4736:
4732:
4728:
4725:
4722:
4719:
4714:
4711:
4708:
4703:
4699:
4695:
4692:
4689:
4683:
4677:
4674:
4671:
4667:
4663:
4660:
4657:
4654:
4646:
4643:
4640:
4635:
4631:
4627:
4624:
4621:
4619:
4617:
4614:
4611:
4605:
4599:
4596:
4593:
4589:
4585:
4582:
4579:
4576:
4568:
4565:
4562:
4557:
4553:
4549:
4546:
4544:
4540:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4513:
4510:
4507:
4502:
4499:
4496:
4492:
4483:
4480:
4477:
4472:
4469:
4466:
4462:
4458:
4457:
4443:
4442:
4429:
4424:
4420:
4414:
4409:
4403:
4399:
4395:
4390:
4386:
4382:
4379:
4376:
4373:
4367:
4364:
4361:
4355:
4352:
4349:
4345:
4341:
4338:
4335:
4332:
4327:
4324:
4321:
4316:
4312:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4282:
4279:
4276:
4271:
4268:
4265:
4261:
4255:
4252:
4249:
4244:
4241:
4238:
4234:
4219:
4218:
4205:
4202:
4199:
4195:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4153:
4150:
4147:
4142:
4139:
4136:
4132:
4113:
4112:
4096:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4069:
4066:
4063:
4058:
4055:
4052:
4048:
4039:
4036:
4033:
4028:
4025:
4022:
4018:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3989:
3986:
3983:
3978:
3975:
3972:
3968:
3962:
3959:
3956:
3951:
3948:
3945:
3941:
3932:
3929:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3890:
3886:
3883:
3880:
3877:
3872:
3869:
3866:
3861:
3858:
3855:
3851:
3845:
3842:
3837:
3833:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3807:
3801:
3798:
3792:
3788:
3763:
3760:
3755:
3752:
3741:
3740:
3724:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3697:
3694:
3691:
3686:
3683:
3680:
3676:
3667:
3664:
3661:
3656:
3653:
3650:
3646:
3642:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3614:
3611:
3608:
3603:
3600:
3597:
3593:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3558:
3554:
3551:
3548:
3534:
3533:
3520:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3493:
3490:
3487:
3482:
3479:
3476:
3472:
3465:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3375:
3371:
3368:
3365:
3335:
3334:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3219:
3213:
3207:
3204:
3201:
3196:
3193:
3190:
3184:
3175:
3170:
3167:
3162:
3153:
3150:
3147:
3144:
3141:
3138:
3134:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3107:
3103:
3100:
3097:
3074:
3054:
3034:
3014:
2994:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2933:
2930:
2927:
2924:
2921:
2918:
2914:
2889:
2886:
2883:
2880:
2877:
2874:
2870:
2845:
2825:
2803:
2800:
2795:
2792:
2772:
2769:
2766:
2763:
2760:
2740:
2737:
2734:
2731:
2727:
2723:
2720:
2716:
2713:
2710:
2689:
2686:
2683:
2680:
2677:
2674:
2670:
2666:
2663:
2659:
2656:
2653:
2574:
2573:
2562:
2557:
2554:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2515:
2511:
2507:
2502:
2498:
2494:
2491:
2488:
2483:
2480:
2477:
2473:
2469:
2465:
2462:
2427:
2426:
2415:
2409:
2406:
2403:
2398:
2395:
2392:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2352:
2348:
2344:
2339:
2335:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2282:
2278:
2274:
2269:
2265:
2261:
2258:
2255:
2250:
2247:
2244:
2240:
2236:
2233:
2208:
2170:
2169:
2168:
2167:
2156:
2150:
2147:
2144:
2139:
2136:
2133:
2127:
2124:
2121:
2117:
2112:
2108:
2104:
2099:
2095:
2091:
2088:
2085:
2080:
2076:
2072:
2067:
2063:
2059:
2056:
2053:
2050:
2047:
2042:
2037:
2033:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
1995:
1991:
1987:
1982:
1978:
1974:
1971:
1968:
1965:
1962:
1947:expected value
1939:
1938:
1927:
1922:
1919:
1916:
1912:
1908:
1905:
1902:
1899:
1894:
1890:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1857:
1854:
1851:
1848:
1845:
1842:
1836:
1833:
1828:
1824:
1820:
1815:
1811:
1807:
1804:
1801:
1796:
1792:
1788:
1783:
1779:
1775:
1772:
1769:
1766:
1745:
1744:
1730:
1727:
1724:
1721:
1718:
1715:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1683:
1680:
1677:
1671:
1668:
1665:
1661:
1657:
1654:
1651:
1648:
1643:
1639:
1633:
1628:
1624:
1605:
1604:
1590:
1587:
1581:
1578:
1575:
1571:
1567:
1564:
1561:
1558:
1553:
1549:
1543:
1538:
1534:
1526:
1523:
1520:
1516:
1512:
1509:
1506:
1503:
1498:
1494:
1487:
1481:
1478:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1448:
1443:
1439:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1406:
1403:
1398:
1394:
1390:
1385:
1381:
1377:
1374:
1371:
1366:
1362:
1358:
1353:
1349:
1345:
1342:
1339:
1336:
1307:
1300:
1290:
1289:
1276:
1273:
1270:
1266:
1262:
1259:
1256:
1253:
1248:
1244:
1240:
1233:
1229:
1225:
1222:
1218:
1214:
1211:
1208:
1205:
1198:
1194:
1189:
1183:
1178:
1175:
1172:
1168:
1164:
1161:
1158:
1155:
1150:
1146:
1142:
1137:
1133:
1129:
1126:
1123:
1118:
1114:
1110:
1105:
1101:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1058:
1057:
1044:
1039:
1036:
1033:
1025:
1023:
1020:
1019:
1016:
1013:
1010:
1007:
1004:
996:
994:
991:
990:
987:
984:
981:
973:
971:
968:
967:
965:
960:
957:
954:
951:
948:
929:we assigned a
908:
898:Bayes' theorem
852:
826:
823:
778:
775:
616:
613:
599:In the 1940s,
584:
581:
567:
564:
543:
520:
517:
514:
494:
491:
488:
473:
472:
461:
456:
453:
448:
445:
442:
439:
434:
430:
426:
423:
420:
415:
411:
407:
404:
401:
396:
393:
390:
386:
382:
378:
375:
335:
334:Interpretation
332:
331:
330:
319:
313:
310:
307:
302:
299:
296:
290:
287:
284:
281:
276:
272:
268:
265:
262:
257:
253:
249:
246:
243:
238:
235:
232:
228:
224:
221:
196:
189:
167:
164:
142:
141:
56:
54:
47:
26:
9:
6:
4:
3:
2:
6640:
6629:
6626:
6624:
6621:
6620:
6618:
6602:
6601:
6593:
6586:
6580:
6574:
6569:
6554:. Eric Neyman
6553:
6549:
6542:
6536:; here: p.145
6533:
6529:
6525:
6521:
6517:
6513:
6506:
6499:
6490:
6486:
6482:
6478:
6474:
6470:
6463:
6456:
6450:
6446:
6442:
6436:
6434:
6424:
6420:
6411:
6408:
6406:
6403:
6401:
6398:
6397:
6391:
6388:
6382:
6365:
6362:
6359:
6354:
6351:
6348:
6337:
6333:
6329:
6325:
6320:
6314:
6297:
6294:
6273:
6270:
6261:
6257:
6253:
6248:
6245:
6238:
6232:
6228:
6226:
6222:
6212:
6207:
6202:
6198:
6193:
6189:
6182:
6163:
6157:
6154:
6151:
6146:
6143:
6140:
6134:
6126:
6122:
6118:
6113:
6109:
6105:
6102:
6099:
6094:
6090:
6073:
6066:
6065:
6064:
6059:
6055:
6051:
6047:
6042:
6038:
6034:
6029:
6025:
6020:
6018:
6014:
5995:
5989:
5986:
5983:
5978:
5975:
5970:
5966:
5959:
5951:
5947:
5943:
5938:
5934:
5930:
5927:
5924:
5919:
5915:
5904:
5900:
5893:
5886:
5885:
5884:
5881:
5877:
5870:
5867: =
5866:
5862:
5858:
5854:
5849:
5845:
5840:
5836:
5831:
5827:
5823:
5790:
5780:
5777:
5772:
5768:
5762:
5757:
5754:
5751:
5747:
5740:
5731:
5727:
5721:
5717:
5713:
5706:
5702:
5696:
5692:
5682:
5679:
5674:
5670:
5658:
5653:
5650:
5647:
5643:
5636:
5629:
5626:
5621:
5617:
5608:
5603:
5600:
5597:
5593:
5588:
5573:
5568:
5562:
5559:
5554:
5550:
5546:
5543:
5540:
5535:
5531:
5527:
5522:
5518:
5514:
5511:
5508:
5503:
5499:
5492:
5485:
5484:
5483:
5482:is given by:
5480:
5476:
5469:
5465:
5461:
5457:
5452:
5448:
5444:
5439:
5435:
5431:
5427:
5423:
5417:
5414:
5410:
5404:
5402:
5392:
5390:
5372:
5369:
5364:
5361:
5353:
5349:
5345:
5340:
5338:
5334:
5330:
5326:
5322:
5318:
5296:
5293:
5289:
5284:
5280:
5274:
5270:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5242:
5237:
5234:
5228:
5224:
5217:
5216:
5215:
5213:
5209:
5181:
5175:
5170:
5166:
5159:
5155:
5150:
5138:
5132:
5127:
5123:
5116:
5108:
5102:
5097:
5093:
5086:
5074:
5068:
5065:
5059:
5055:
5050:
5041:
5035:
5032:
5027:
5024:
5021:
5017:
5010:
5005:
5001:
4991:
4988:
4983:
4979:
4975:
4972:
4969:
4966:
4963:
4960:
4957:
4947:
4944:
4938:
4934:
4927:
4926:
4925:
4923:
4897:
4891:
4888:
4883:
4880:
4877:
4873:
4869:
4865:
4860:
4855:
4852:
4847:
4843:
4840:
4834:
4831:
4828:
4824:
4819:
4814:
4810:
4807:
4799:
4796:
4793:
4788:
4784:
4779:
4773:
4770:
4767:
4763:
4759:
4757:
4749:
4746:
4740:
4737:
4734:
4726:
4723:
4720:
4712:
4709:
4706:
4701:
4697:
4693:
4690:
4687:
4681:
4675:
4672:
4669:
4661:
4658:
4655:
4644:
4641:
4638:
4633:
4629:
4625:
4622:
4620:
4612:
4609:
4603:
4597:
4594:
4591:
4583:
4580:
4577:
4566:
4563:
4560:
4555:
4551:
4547:
4545:
4538:
4531:
4528:
4525:
4522:
4519:
4511:
4508:
4505:
4500:
4497:
4494:
4490:
4481:
4478:
4475:
4470:
4467:
4464:
4460:
4448:
4447:
4446:
4427:
4422:
4418:
4412:
4407:
4401:
4397:
4393:
4388:
4380:
4377:
4374:
4365:
4362:
4359:
4353:
4350:
4347:
4339:
4336:
4333:
4325:
4322:
4319:
4314:
4310:
4306:
4300:
4297:
4294:
4291:
4288:
4280:
4277:
4274:
4269:
4266:
4263:
4259:
4253:
4250:
4247:
4242:
4239:
4236:
4232:
4224:
4223:
4222:
4203:
4200:
4197:
4189:
4186:
4183:
4177:
4171:
4168:
4165:
4162:
4159:
4151:
4148:
4145:
4140:
4137:
4134:
4130:
4122:
4121:
4120:
4118:
4094:
4087:
4084:
4081:
4078:
4075:
4067:
4064:
4061:
4056:
4053:
4050:
4046:
4037:
4034:
4031:
4026:
4023:
4020:
4016:
4007:
4004:
4001:
3998:
3995:
3987:
3984:
3981:
3976:
3973:
3970:
3966:
3960:
3957:
3954:
3949:
3946:
3943:
3939:
3930:
3927:
3922:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3884:
3878:
3875:
3870:
3867:
3864:
3859:
3856:
3853:
3849:
3843:
3840:
3835:
3831:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3799:
3796:
3790:
3786:
3779:
3778:
3777:
3761:
3758:
3753:
3750:
3722:
3715:
3712:
3709:
3706:
3703:
3695:
3692:
3689:
3684:
3681:
3678:
3674:
3665:
3662:
3659:
3654:
3651:
3648:
3644:
3640:
3632:
3629:
3626:
3623:
3620:
3612:
3609:
3606:
3601:
3598:
3595:
3591:
3584:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3552:
3546:
3539:
3538:
3537:
3518:
3511:
3508:
3505:
3502:
3499:
3491:
3488:
3485:
3480:
3477:
3474:
3470:
3463:
3457:
3451:
3448:
3445:
3442:
3439:
3433:
3428:
3422:
3419:
3416:
3413:
3410:
3401:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3369:
3363:
3356:
3355:
3354:
3352:
3348:
3344:
3341: =
3340:
3318:
3309:
3306:
3303:
3297:
3294:
3291:
3288:
3282:
3276:
3273:
3270:
3261:
3258:
3255:
3249:
3244:
3238:
3235:
3232:
3226:
3223:
3217:
3205:
3202:
3199:
3194:
3191:
3188:
3168:
3165:
3148:
3145:
3142:
3136:
3132:
3127:
3121:
3118:
3115:
3112:
3109:
3101:
3095:
3088:
3087:
3086:
3072:
3052:
3032:
3012:
2992:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2928:
2925:
2922:
2916:
2912:
2884:
2881:
2878:
2872:
2868:
2857:
2843:
2823:
2801:
2798:
2793:
2790:
2764:
2761:
2758:
2735:
2732:
2729:
2721:
2684:
2681:
2678:
2675:
2672:
2664:
2643:, denoted by
2642:
2638:
2635: =
2634:
2630:
2625:
2623:
2620: =
2619:
2615:
2611:
2608: =
2607:
2603:
2599:
2595:
2592: =
2591:
2587:
2583:
2579:
2560:
2555:
2552:
2547:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2513:
2509:
2505:
2500:
2496:
2492:
2489:
2486:
2481:
2478:
2475:
2471:
2463:
2460:
2452:
2451:
2450:
2448:
2444:
2440:
2436:
2432:
2413:
2407:
2404:
2401:
2396:
2393:
2390:
2384:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2350:
2346:
2342:
2337:
2333:
2329:
2326:
2320:
2314:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2280:
2276:
2272:
2267:
2263:
2259:
2256:
2253:
2248:
2245:
2242:
2238:
2231:
2224:
2223:
2222:
2220:
2216:
2211:
2207:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2154:
2148:
2145:
2142:
2137:
2134:
2131:
2125:
2122:
2119:
2110:
2106:
2102:
2097:
2093:
2089:
2086:
2083:
2078:
2074:
2070:
2065:
2061:
2057:
2054:
2048:
2045:
2040:
2035:
2031:
2027:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
1993:
1989:
1985:
1980:
1976:
1972:
1969:
1963:
1953:
1952:
1951:
1950:
1949:
1948:
1944:
1925:
1920:
1917:
1914:
1906:
1903:
1900:
1892:
1888:
1881:
1875:
1872:
1869:
1863:
1860:
1855:
1849:
1846:
1843:
1834:
1826:
1822:
1818:
1813:
1809:
1805:
1802:
1799:
1794:
1790:
1786:
1781:
1777:
1773:
1770:
1764:
1757:
1756:
1755:
1752:
1750:
1749:beta function
1728:
1722:
1719:
1716:
1708:
1702:
1699:
1696:
1690:
1687:
1681:
1678:
1675:
1669:
1666:
1663:
1655:
1652:
1649:
1641:
1637:
1631:
1626:
1622:
1614:
1613:
1612:
1610:
1588:
1585:
1579:
1576:
1573:
1565:
1562:
1559:
1551:
1547:
1541:
1536:
1532:
1524:
1521:
1518:
1510:
1507:
1504:
1496:
1492:
1485:
1479:
1476:
1469:
1463:
1457:
1451:
1446:
1441:
1437:
1428:
1422:
1416:
1410:
1404:
1396:
1392:
1388:
1383:
1379:
1375:
1372:
1369:
1364:
1360:
1356:
1351:
1347:
1343:
1340:
1334:
1327:
1326:
1325:
1323:
1319:
1315:
1310:
1306:
1299:
1296: =
1295:
1274:
1271:
1268:
1260:
1257:
1254:
1246:
1242:
1238:
1231:
1227:
1223:
1220:
1212:
1209:
1206:
1196:
1192:
1187:
1181:
1176:
1173:
1170:
1166:
1162:
1156:
1153:
1148:
1144:
1140:
1135:
1131:
1127:
1124:
1121:
1116:
1112:
1108:
1103:
1099:
1092:
1089:
1083:
1077:
1070:
1069:
1068:
1067:
1063:
1037:
1034:
1031:
1021:
1014:
1011:
1008:
1005:
1002:
992:
985:
982:
979:
969:
963:
958:
952:
946:
939:
938:
937:
936:
932:
928:
924:
920:
916:
911:
907:
903:
899:
894:
892:
888:
884:
880:
876:
872:
868:
864:
860:
855:
851:
846:
844:
840:
836:
832:
822:
821:
816:
814:
810:
806:
802:
798:
794:
776:
773:
763:
758:
754:
750:
746:
742:
737:
735:
731:
727:
722:
718:
714:
710:
706:
702:
698:
694:
689:
687:
683:
679:
670:
666:
662:
659:is then one (
658:
654:
650:
646:
642:
638:
634:
630:
626:
621:
612:
610:
606:
602:
601:Rudolf Carnap
597:
593:
589:
580:
565:
562:
541:
532:
518:
515:
512:
492:
489:
486:
478:
459:
454:
451:
446:
440:
437:
432:
428:
424:
421:
418:
413:
409:
405:
402:
399:
394:
391:
388:
384:
376:
373:
365:
364:
363:
361:
356:
354:
350:
346:
342:
317:
311:
308:
305:
300:
297:
294:
288:
282:
279:
274:
270:
266:
263:
260:
255:
251:
247:
244:
241:
236:
233:
230:
226:
219:
212:
211:
210:
208:
205:
199:
195:
188:
183:
181:
177:
173:
163:
161:
157:
153:
149:
138:
135:
127:
124:February 2017
116:
113:
109:
106:
102:
99:
95:
92:
88:
85: –
84:
80:
79:Find sources:
73:
69:
63:
62:
57:This article
55:
51:
46:
45:
40:
33:
19:
6599:
6592:
6584:
6579:
6568:
6556:. Retrieved
6551:
6541:
6515:
6511:
6498:
6475:(2): 72–97.
6472:
6468:
6455:
6440:
6423:
6387:pseudocounts
6383:
6335:
6331:
6327:
6323:
6318:
6315:
6259:
6249:
6243:
6236:
6233:
6229:
6218:
6205:
6200:
6196:
6191:
6187:
6180:
6178:
6057:
6053:
6049:
6045:
6040:
6036:
6032:
6027:
6023:
6021:
6016:
6012:
6010:
5879:
5875:
5868:
5864:
5860:
5856:
5852:
5847:
5843:
5838:
5834:
5829:
5825:
5821:
5819:
5478:
5474:
5467:
5463:
5459:
5455:
5450:
5446:
5442:
5437:
5433:
5418:
5412:
5408:
5405:
5400:
5398:
5388:
5351:
5347:
5343:
5341:
5336:
5332:
5328:
5324:
5320:
5316:
5314:
5211:
5205:
4919:
4444:
4220:
4116:
4114:
3742:
3535:
3350:
3346:
3342:
3338:
3336:
2858:
2636:
2632:
2628:
2626:
2621:
2617:
2613:
2609:
2605:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2575:
2446:
2442:
2438:
2434:
2428:
2218:
2214:
2209:
2205:
2201:
2197:
2185:
2181:
2173:
2171:
1940:
1753:
1746:
1606:
1321:
1317:
1313:
1308:
1304:
1297:
1293:
1291:
1061:
1059:
934:
926:
918:
914:
909:
905:
901:
895:
890:
882:
874:
870:
866:
858:
853:
849:
847:
842:
838:
834:
830:
828:
818:
817:
812:
808:
804:
800:
796:
792:
761:
756:
752:
748:
744:
740:
738:
733:
729:
725:
720:
716:
712:
708:
704:
700:
696:
692:
690:
685:
681:
677:
674:
668:
664:
660:
656:
652:
648:
644:
640:
636:
632:
628:
624:
598:
594:
590:
586:
533:
474:
359:
357:
348:
345:pseudocounts
340:
337:
197:
193:
186:
184:
179:
175:
171:
169:
151:
145:
130:
121:
111:
104:
97:
90:
78:
66:Please help
61:verification
58:
5863:), and let
1607:To get the
896:We can use
6617:Categories
6416:References
6260:absolutely
6256:Bayes rule
5797:otherwise.
2221:, we have
2217:= 1, ...,
1941:This is a
1611:, we find
917:= 1, ...,
623:The point
588:tomorrow.
94:newspapers
6103:…
5928:…
5748:∑
5714:⋯
5664:Γ
5644:∏
5594:∑
5585:Γ
5544:…
5528:∣
5512:…
5285:≈
5243:∣
5208:logarithm
5176:
5133:
5103:
5069:
5036:
5025:−
4984:≈
4892:
4881:−
4870:≈
4832:−
4820:−
4797:−
4785:∫
4771:−
4738:−
4724:−
4710:−
4698:∫
4694:−
4673:−
4659:−
4642:−
4630:∫
4595:−
4581:−
4564:−
4552:∫
4548:≈
4529:−
4523:−
4509:−
4491:∏
4479:−
4461:∑
4413:≈
4394:−
4378:−
4351:−
4337:−
4323:−
4311:∫
4307:≈
4298:−
4292:−
4278:−
4260:∏
4251:−
4233:∑
4201:−
4187:−
4178:≈
4169:−
4163:−
4149:−
4131:∏
4085:−
4079:−
4065:−
4047:∏
4035:−
4017:∑
4005:−
3999:−
3985:−
3967:∏
3958:−
3940:∑
3868:−
3850:∑
3713:−
3707:−
3693:−
3675:∏
3663:−
3645:∑
3630:−
3624:−
3610:−
3592:∏
3509:−
3503:−
3489:−
3471:∏
3449:−
3443:−
3420:−
3414:−
3402:∝
3307:−
3298:−
3292:−
3274:−
3259:−
3236:−
3218:∝
3203:−
3192:−
3146:−
3128:∝
2970:−
2964:≤
2958:≤
2926:−
2882:−
2771:∞
2768:→
2536:…
2493:∣
2373:…
2330:∣
2321:
2303:…
2260:∣
2087:…
2058:∣
2032:∫
2016:…
1973:∣
1964:
1918:−
1904:−
1873:−
1803:…
1774:∣
1700:−
1667:−
1653:−
1623:∫
1577:−
1563:−
1533:∫
1522:−
1508:−
1438:∫
1373:…
1344:∣
1272:−
1258:−
1224:−
1210:−
1167:∏
1154:∣
1125:…
1035:≥
1028:for
999:for
983:≤
976:for
921:For the "
615:Intuition
422:⋯
406:∣
355:, below.
264:⋯
248:∣
6558:13 April
6489:14481246
6394:See also
5833:is just
5290:0.434294
5156:0.434294
2464:′
2200:. Since
566:′
475:But see
377:′
6532:2102920
6223:on the
6221:Laplace
6081:success
843:p as if
811:) than
347:) with
192:, ...,
108:scholar
6530:
6487:
6447:
6258:takes
2192:, the
2172:Since
1292:where
889:given
885:s are
877:has a
791:where
715:. The
150:, the
110:
103:
96:
89:
81:
6604:(PDF)
6528:JSTOR
6508:(PDF)
6485:S2CID
6465:(PDF)
5346:, or
2184:. As
1945:with
1747:(see
935:(0,1)
923:prior
863:trial
505:, or
180:n − s
115:JSTOR
101:books
6560:2023
6445:ISBN
6281:data
3776:is:
2213:for
1012:<
1006:<
848:Let
807:and
732:and
667:and
554:and
202:are
87:news
6520:doi
6477:doi
6355:0.5
6050:m-c
5389:s=n
5167:log
5124:log
5094:log
3345:or
2903:is
2596:or
861:th
764:is
749:n+2
745:n+2
707:to
635:to
360:not
146:In
70:by
6619::
6550:.
6526:.
6514:.
6510:.
6483:.
6473:12
6471:.
6467:.
6432:^
6381:.
5403:.
5271:10
5212:10
5171:10
5128:10
5098:10
5066:ln
5033:ln
4889:ln
2856:.
919:n.
913:,
893:.
611:.
531:.
349:s
200:+1
6562:.
6534:.
6522::
6516:8
6491:.
6479::
6366:1
6363:+
6360:n
6352:+
6349:s
6336:m
6332:m
6328:1
6324:m
6319:m
6301:)
6298:I
6295:,
6286:|
6277:(
6274:r
6271:P
6244:m
6240:m
6237:I
6209:2
6206:I
6201:m
6197:I
6192:m
6188:I
6184:2
6181:I
6164:,
6158:m
6155:+
6152:n
6147:c
6144:+
6141:s
6135:=
6132:)
6127:m
6123:I
6119:,
6114:m
6110:n
6106:,
6100:,
6095:1
6091:n
6086:|
6077:(
6074:P
6061:i
6058:n
6054:s
6046:c
6041:i
6037:A
6033:m
6028:i
6024:A
6017:m
6013:n
5996:.
5990:m
5987:+
5984:n
5979:1
5976:+
5971:i
5967:n
5960:=
5957:)
5952:m
5948:I
5944:,
5939:m
5935:n
5931:,
5925:,
5920:1
5916:n
5911:|
5905:i
5901:A
5897:(
5894:P
5880:m
5876:n
5872:1
5869:n
5865:n
5861:m
5857:i
5855:(
5853:i
5848:i
5844:A
5839:i
5835:p
5830:i
5826:p
5822:i
5791:0
5781:1
5778:=
5773:i
5769:p
5763:m
5758:1
5755:=
5752:i
5741:,
5732:m
5728:n
5722:m
5718:p
5707:1
5703:n
5697:1
5693:p
5686:)
5683:1
5680:+
5675:i
5671:n
5667:(
5659:m
5654:1
5651:=
5648:i
5637:)
5633:)
5630:1
5627:+
5622:i
5618:n
5614:(
5609:m
5604:1
5601:=
5598:i
5589:(
5574:{
5569:=
5566:)
5563:I
5560:,
5555:m
5551:n
5547:,
5541:,
5536:1
5532:n
5523:m
5519:p
5515:,
5509:,
5504:1
5500:p
5496:(
5493:f
5479:m
5475:p
5471:1
5468:p
5464:m
5460:i
5458:(
5456:i
5451:i
5447:n
5443:i
5438:i
5434:p
5413:m
5409:I
5373:n
5370:s
5365:=
5362:p
5352:N
5348:k
5344:N
5337:k
5333:n
5329:k
5325:n
5321:n
5317:k
5297:k
5294:n
5281:)
5275:k
5267:=
5264:N
5261:,
5258:0
5255:=
5252:s
5249:,
5246:n
5238:N
5235:S
5229:(
5225:E
5188:]
5185:)
5182:N
5179:(
5163:[
5160:n
5151:=
5145:]
5142:)
5139:N
5136:(
5120:[
5117:n
5112:)
5109:e
5106:(
5087:=
5081:]
5078:)
5075:N
5072:(
5063:[
5060:n
5056:1
5051:=
5045:)
5042:N
5039:(
5028:1
5022:n
5018:N
5011:n
5006:n
5002:N
4992:N
4989:1
4980:)
4976:N
4973:,
4970:0
4967:=
4964:s
4961:,
4958:n
4954:|
4948:N
4945:S
4939:(
4935:E
4901:)
4898:N
4895:(
4884:1
4878:n
4874:N
4866:]
4861:)
4856:N
4853:1
4848:(
4844:O
4841:+
4835:1
4829:n
4825:1
4815:R
4811:R
4808:d
4800:n
4794:N
4789:1
4780:[
4774:1
4768:n
4764:N
4760:=
4750:R
4747:d
4741:2
4735:n
4731:)
4727:R
4721:N
4718:(
4713:n
4707:N
4702:1
4691:R
4688:d
4682:R
4676:2
4670:n
4666:)
4662:R
4656:N
4653:(
4645:n
4639:N
4634:1
4626:N
4623:=
4613:R
4610:d
4604:R
4598:1
4592:n
4588:)
4584:R
4578:N
4575:(
4567:n
4561:N
4556:1
4539:R
4535:)
4532:j
4526:R
4520:N
4517:(
4512:1
4506:n
4501:1
4498:=
4495:j
4482:n
4476:N
4471:1
4468:=
4465:R
4428:n
4423:n
4419:N
4408:n
4402:n
4398:n
4389:n
4385:)
4381:1
4375:N
4372:(
4366:=
4363:S
4360:d
4354:1
4348:n
4344:)
4340:S
4334:N
4331:(
4326:n
4320:N
4315:1
4304:)
4301:j
4295:S
4289:N
4286:(
4281:1
4275:n
4270:1
4267:=
4264:j
4254:n
4248:N
4243:1
4240:=
4237:S
4204:1
4198:n
4194:)
4190:R
4184:N
4181:(
4175:)
4172:j
4166:R
4160:N
4157:(
4152:1
4146:n
4141:1
4138:=
4135:j
4117:N
4095:R
4091:)
4088:j
4082:R
4076:N
4073:(
4068:1
4062:n
4057:1
4054:=
4051:j
4038:n
4032:N
4027:1
4024:=
4021:R
4011:)
4008:j
4002:S
3996:N
3993:(
3988:1
3982:n
3977:1
3974:=
3971:j
3961:n
3955:N
3950:1
3947:=
3944:S
3931:N
3928:1
3923:=
3920:)
3917:0
3914:=
3911:s
3908:,
3905:1
3902:=
3899:n
3896:,
3893:N
3889:|
3885:S
3882:(
3879:P
3876:S
3871:n
3865:N
3860:1
3857:=
3854:S
3844:N
3841:1
3836:=
3832:)
3828:N
3825:,
3822:0
3819:=
3816:s
3813:,
3810:n
3806:|
3800:N
3797:S
3791:(
3787:E
3762:N
3759:S
3754:=
3751:p
3723:R
3719:)
3716:j
3710:R
3704:N
3701:(
3696:1
3690:n
3685:1
3682:=
3679:j
3666:n
3660:N
3655:1
3652:=
3649:R
3641:S
3636:)
3633:j
3627:S
3621:N
3618:(
3613:1
3607:n
3602:1
3599:=
3596:j
3585:=
3582:)
3579:0
3576:=
3573:s
3570:,
3567:n
3564:,
3561:N
3557:|
3553:S
3550:(
3547:P
3519:S
3515:)
3512:j
3506:S
3500:N
3497:(
3492:1
3486:n
3481:1
3478:=
3475:j
3464:=
3458:!
3455:)
3452:n
3446:S
3440:N
3437:(
3434:S
3429:!
3426:)
3423:1
3417:S
3411:N
3408:(
3399:)
3396:0
3393:=
3390:s
3387:,
3384:n
3381:,
3378:N
3374:|
3370:S
3367:(
3364:P
3351:s
3347:s
3343:n
3339:s
3319:!
3316:)
3313:]
3310:s
3304:n
3301:[
3295:S
3289:N
3286:(
3283:!
3280:)
3277:s
3271:S
3268:(
3265:)
3262:S
3256:N
3253:(
3250:S
3245:!
3242:)
3239:S
3233:N
3230:(
3227:!
3224:S
3212:)
3206:s
3200:n
3195:S
3189:N
3183:(
3174:)
3169:s
3166:S
3161:(
3152:)
3149:S
3143:N
3140:(
3137:S
3133:1
3125:)
3122:s
3119:,
3116:n
3113:,
3110:N
3106:|
3102:S
3099:(
3096:P
3073:S
3053:N
3033:S
3013:p
2993:N
2973:1
2967:N
2961:S
2955:1
2932:)
2929:S
2923:N
2920:(
2917:S
2913:1
2888:)
2885:p
2879:1
2876:(
2873:p
2869:1
2844:N
2824:S
2802:N
2799:S
2794:=
2791:p
2765:S
2762:,
2759:N
2739:)
2736:p
2733:,
2730:n
2726:|
2722:r
2719:(
2715:n
2712:i
2709:B
2688:)
2685:S
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2679:n
2676:,
2673:N
2669:|
2665:s
2662:(
2658:p
2655:y
2652:H
2637:n
2633:s
2629:s
2622:n
2618:s
2614:s
2610:n
2606:s
2602:s
2598:s
2594:n
2590:s
2586:s
2582:n
2580:,
2578:s
2561:.
2556:n
2553:s
2548:=
2545:)
2542:n
2539:,
2533:,
2530:1
2527:=
2524:i
2514:i
2510:x
2506:=
2501:i
2497:X
2490:1
2487:=
2482:1
2479:+
2476:n
2472:X
2468:(
2461:P
2447:p
2443:p
2439:p
2435:p
2414:.
2408:2
2405:+
2402:n
2397:1
2394:+
2391:s
2385:=
2382:)
2379:n
2376:,
2370:,
2367:1
2364:=
2361:i
2351:i
2347:x
2343:=
2338:i
2334:X
2327:p
2324:(
2318:E
2315:=
2312:)
2309:n
2306:,
2300:,
2297:1
2294:=
2291:i
2281:i
2277:x
2273:=
2268:i
2264:X
2257:1
2254:=
2249:1
2246:+
2243:n
2239:X
2235:(
2232:P
2219:n
2215:i
2210:i
2206:X
2202:p
2198:p
2186:p
2182:p
2174:p
2155:.
2149:2
2146:+
2143:n
2138:1
2135:+
2132:s
2126:=
2123:p
2120:d
2116:)
2111:n
2107:x
2103:=
2098:n
2094:X
2090:,
2084:,
2079:1
2075:x
2071:=
2066:1
2062:X
2055:p
2052:(
2049:f
2046:p
2041:1
2036:0
2028:=
2025:)
2022:n
2019:,
2013:,
2010:1
2007:=
2004:i
1994:i
1990:x
1986:=
1981:i
1977:X
1970:p
1967:(
1961:E
1926:.
1921:s
1915:n
1911:)
1907:p
1901:1
1898:(
1893:s
1889:p
1882:!
1879:)
1876:s
1870:n
1867:(
1864:!
1861:s
1856:!
1853:)
1850:1
1847:+
1844:n
1841:(
1835:=
1832:)
1827:n
1823:x
1819:=
1814:n
1810:X
1806:,
1800:,
1795:1
1791:x
1787:=
1782:1
1778:X
1771:p
1768:(
1765:f
1729:!
1726:)
1723:1
1720:+
1717:n
1714:(
1709:!
1706:)
1703:s
1697:n
1694:(
1691:!
1688:s
1682:=
1679:r
1676:d
1670:s
1664:n
1660:)
1656:r
1650:1
1647:(
1642:s
1638:r
1632:1
1627:0
1589:r
1586:d
1580:s
1574:n
1570:)
1566:r
1560:1
1557:(
1552:s
1548:r
1542:1
1537:0
1525:s
1519:n
1515:)
1511:p
1505:1
1502:(
1497:s
1493:p
1486:=
1480:r
1477:d
1473:)
1470:r
1467:(
1464:f
1461:)
1458:r
1455:(
1452:L
1447:1
1442:0
1432:)
1429:p
1426:(
1423:f
1420:)
1417:p
1414:(
1411:L
1405:=
1402:)
1397:n
1393:x
1389:=
1384:n
1380:X
1376:,
1370:,
1365:1
1361:x
1357:=
1352:1
1348:X
1341:p
1338:(
1335:f
1322:x
1318:X
1314:n
1309:n
1305:x
1301:1
1298:x
1294:s
1275:s
1269:n
1265:)
1261:p
1255:1
1252:(
1247:s
1243:p
1239:=
1232:i
1228:x
1221:1
1217:)
1213:p
1207:1
1204:(
1197:i
1193:x
1188:p
1182:n
1177:1
1174:=
1171:i
1163:=
1160:)
1157:p
1149:n
1145:x
1141:=
1136:n
1132:X
1128:,
1122:,
1117:1
1113:x
1109:=
1104:1
1100:X
1096:(
1093:P
1090:=
1087:)
1084:p
1081:(
1078:L
1062:p
1038:1
1032:p
1022:0
1015:1
1009:p
1003:0
993:1
986:0
980:p
970:0
964:{
959:=
956:)
953:p
950:(
947:f
927:p
915:i
910:i
906:X
902:p
891:p
883:X
875:X
871:X
867:p
859:i
854:i
850:X
839:p
835:p
831:p
813:n
809:Z
805:P
801:Z
797:t
793:b
777:t
774:b
762:p
757:P
753:Z
741:p
734:Z
730:P
726:p
721:n
717:n
713:p
709:P
705:Z
701:p
697:P
693:Z
686:p
682:p
678:p
669:P
665:Z
661:Z
657:p
653:P
649:Z
645:p
641:p
637:P
633:Z
629:P
625:Z
563:P
542:P
519:n
516:=
513:s
493:0
490:=
487:s
460:.
455:n
452:s
447:=
444:)
441:s
438:=
433:n
429:X
425:+
419:+
414:1
410:X
403:1
400:=
395:1
392:+
389:n
385:X
381:(
374:P
341:n
318:.
312:2
309:+
306:n
301:1
298:+
295:s
289:=
286:)
283:s
280:=
275:n
271:X
267:+
261:+
256:1
252:X
245:1
242:=
237:1
234:+
231:n
227:X
223:(
220:P
198:n
194:X
190:1
187:X
176:s
172:n
137:)
131:(
126:)
122:(
112:·
105:·
98:·
91:·
64:.
41:.
34:.
20:)
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