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512:. As with coins, it is assumed that the initial conditions of throwing the dice are not known with enough precision to predict the outcome according to the laws of mechanics. Dice are typically thrown so as to bounce on a table or other surface(s). This interaction makes prediction of the outcome much more difficult.
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The assumption of symmetry is crucial here. Suppose that we are asked to bet for or against the outcome "6". We might reason that there are two relevant outcomes here "6" or "not 6", and that these are mutually exclusive and exhaustive. A common fallacy is assigning the probability 1/2 to each of the
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This example, more than the others, shows the difficulty of actually applying the principle of indifference in real situations. What we really mean by the phrase "arbitrarily ordered" is simply that we don't have any information that would lead us to favor a particular card. In actual practice, this
1558:
These earlier writers, Laplace in particular, naively generalized the principle of indifference to the case of continuous parameters, giving the so-called "uniform prior probability distribution", a function that is constant over all real numbers. He used this function to express a complete lack of
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rather than indifferent between propositions. This still reduces to the ordinary principle of indifference when one considers a permutation of the labels as generating equivalent problems (i.e. using the permutation transformation group). To apply this to the above box example, we have three random
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It is implicit in this analysis that the forces acting on the coin are not known with any precision. If the momentum imparted to the coin as it is launched were known with sufficient accuracy, the flight of the coin could be predicted according to the laws of mechanics. Thus the uncertainty in the
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the cards; this does not destroy the information we have, but instead (hopefully) renders our information practically unusable, although it is still usable in principle. In fact, some expert blackjack players can track aces through the deck; for them, the condition for applying the principle of
1553:
The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is
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Thus we have achieved invariance with respect to volume and length. One can also show the same invariance with respect to surface area being less than 6(4) = 96. However, note that this probability assignment is not necessarily a "correct" one. For the exact distribution of lengths, volume, or
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1586:, and others) objected to the use of the uniform prior for two reasons. The first reason is that the constant function is not normalizable, and thus is not a proper probability distribution. The second reason is its inapplicability to continuous variables, as described above.
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It is impossible for a Die, with such determin'd force and direction, not to fall on such determin'd side, only I don't know the force and direction which makes it fall on such determin'd side, and therefore I call it Chance, which is nothing but the want of
469:. Assuming that the coin must land on one side or the other, the outcomes of a coin toss are mutually exclusive, exhaustive, and interchangeable. According to the principle of indifference, we assign each of the possible outcomes a probability of 1/2.
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sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.
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A standard deck contains 52 cards, each given a unique label in an arbitrary fashion, i.e. arbitrarily ordered. We draw a card from the deck; applying the principle of indifference, we assign each of the possible outcomes a probability of 1/52.
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The information on the label allows us to calculate that the surface area of the cube is between 54 and 150 cm. We don't know the actual surface area, but we might assume that all values are equally likely and simply pick the mid-value of
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for any one of the variables implies a non-uniform distribution for the other two. In general, the principle of indifference does not indicate which variable (e.g. in this case, length, surface area, or volume) is to have a uniform epistemic
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The information on the label allows us to calculate that the volume of the cube is between 27 and 125 cm. We don't know the actual volume, but we might assume that all values are equally likely and simply pick the mid-value of
1506:, is in a sense an example of the principle of indifference. However, when the microstates are described by continuous variables (such as positions and momenta), an additional physical basis is needed in order to explain under
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knowledge as to the value of a parameter. According to
Stigler (page 135), Laplace's assumption of uniform prior probabilities was not a meta-physical assumption. It was an implicit assumption made for the ease of analysis.
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Given enough time and resources, there is no fundamental reason to suppose that suitably precise measurements could not be made, which would enable the prediction of the outcome of coins, dice, and cards with high accuracy:
1483:{\displaystyle Pr(V<64)=\int _{27}^{64}{dV \over 3V\log({5 \over 3})}={\log({64 \over 27}) \over 3\log({5 \over 3})}={3\log({4 \over 3}) \over 3\log({5 \over 3})}={\log({4 \over 3}) \over \log({5 \over 3})}\approx 0.56}
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variables related by geometric equations. If we have no reason to favour one trio of values over another, then our prior probabilities must be related by the rule for changing variables in continuous distributions. Let
385:. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their credence (or "degrees of belief") equally among all the possible outcomes under consideration.
508:. We assume that the die will land with one face or another upward, and there are no other possible outcomes. Applying the principle of indifference, we assign each of the possible outcomes a probability of 1/
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The principle of indifference can be given a deeper logical justification by noting that equivalent states of knowledge should be assigned equivalent epistemic probabilities. This argument was propounded by
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Applying the principle of indifference incorrectly can easily lead to nonsensical results, especially in the case of multivariate, continuous variables. A typical case of misuse is the following example:
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In this example, mutually contradictory estimates of the length, surface area, and volume of the cube arise because we have assumed three mutually contradictory distributions for these parameters: a
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outcome of a coin toss is derived (for the most part) from the uncertainty with respect to initial conditions. This point is discussed at greater length in the article on
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system, at least, it must be assumed that the physical laws that govern the system are not known well enough to predict the outcome. As observed some centuries ago by
1661:: a formula for estimating underlying probabilities when there are few observations, or for events that have not been observed to occur at all in (finite) sample data
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589:, which can yield an epistemic probability distribution for this problem. This generalises the principle of indifference, by saying that one is indifferent between
1537:' principle of "multiple explanations" (pleonachos tropos), according to which "if more than one theory is consistent with the data, keep them all”. The epicurean
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is rarely the case: a new deck of cards is certainly not in arbitrary order, and neither is a deck immediately after a hand of cards. In practice, we therefore
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However, we have now reached the impossible conclusion that the cube has a side length of 4 cm, a surface area of 102 cm, and a volume of 76 cm!
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We don't know the actual side length, but we might assume that all values are equally likely and simply pick the mid-value of 4 cm.
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1983:
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developed this point with an analogy of the multiple causes of death of a corpse. The original writers on probability, primarily
1549:, considered the principle of indifference to be intuitively obvious and did not even bother to give it a name. Laplace wrote:
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1080:{\displaystyle Pr(L<4)=\int _{3}^{4}{dL \over L\log({5 \over 3})}={\log({4 \over 3}) \over \log({5 \over 3})}\approx 0.56}
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For the volume, this should be equal to the probability that the volume is less than 4 = 64. The pdf of the volume is
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Suppose there is a cube hidden in a box. A label on the box says the cube has a side length between 3 and 5 cm.
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To put this "to the test", we ask for the probability that the length is less than 4. This has probability of:
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726:{\displaystyle f_{L}(L)=\left|{\partial V \over \partial L}\right|f_{V}(V)=3L^{2}f_{V}(L^{3})}
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The "principle of insufficient reason" was renamed the "principle of indifference" by
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The textbook examples for the application of the principle of indifference are
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918:{\displaystyle K^{-1}=\int _{3}^{5}{dL \over L}=\log \left({5 \over 3}\right)}
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Jaynes, Edwin
Thompson (2003). "Ignorance Priors and Transformation Groups".
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The
History of Statistics: The Measurement of Uncertainty Before 1900
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782:(pdf) of the stated variables. This equation has a general solution:
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In probability theory, a rule for assigning epistemic probabilities
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465:(many coins have the head of a person portrayed on one side) and
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shows a dilemma with linked variables, and which one to choose.
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Howson, Colin; Urbach, Peter (1989). "Subjective
Probability".
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surface area will depend on how the "experiment" is conducted.
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two outcomes, when "not 6" is five times more likely than "6."
1856:. Cambridge, Mass: Belknap Press of Harvard University Press.
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parameterization the probability density will be uniform.
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Diaconis, Persi; Keller, Joseph B. (1989). "Fair Dice".
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is a normalization constant, determined by the range of
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Another classic example of this kind of misuse is the
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1913:. Vol. 4. Macmillan and Co. pp. 41–64.
1879:Scientific Reasoning : The Bayesian Approach
1791:"A Philosophical Treatise of Universal Induction"
1789:Rathmanner, Samuel; Hutter, Marcus (2011-06-03).
1746:"Epicurean Meteorology, Lucretius, and the Aetna"
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1620:: it leads to two generalizations, namely the
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1907:"Chapter IV. The Principle of Indifference"
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449:machines is a practical example of this.
69:Learn how and when to remove this message
1937:Probability Theory: The Logic of Science
1881:. La Salle: Open Court. pp. 39–76.
461:coin has two sides, arbitrarily labeled
234:Integrated nested Laplace approximations
32:This article includes a list of general
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1708:The American Mathematical Monthly
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817:{\displaystyle f(L)={K \over L}}
379:principle of insufficient reason
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244:Approximate Bayesian computation
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1752:. De Gruyter. pp. 83–102.
1744:Verde, Francesco (2020-07-06).
1679:Eva, Benjamin (30 April 2019).
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1911:A Treatise on Probability
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482:
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454:
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443:Persi Diaconis
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425:John Arbuthnot
401:
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369:
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475:coin flipping
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447:coin-flipping
445:'s work with
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432:
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377:(also called
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157:Cox's theorem
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1691:30 September
1689:. Retrieved
1684:
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1588:
1580:George Boole
1563:
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554:102 cm.
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418:
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387:
378:
374:
372:
307:Bayes factor
171:
65:
56:
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2103:Game theory
1504:equilibrium
558:76 cm.
421:macroscopic
51:introducing
2138:Categories
1687:(Preprint)
1666:References
1628:, and the
1624:as in the
252:Estimators
124:Background
110:Likelihood
59:April 2010
34:references
2000:Decisions
1835:1099-4300
1808:1105.5721
1776:243676846
1584:John Venn
1539:Lucretius
1475:≈
1453:
1429:
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1072:≈
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459:symmetric
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152:Coherence
106:Posterior
2021:Concepts
1905:(1921).
1643:See also
1535:Epicurus
824:, where
778:are the
400:Examples
118:Evidence
2053:Leximin
2048:Minimax
1813:Bibcode
1795:Entropy
1728:2324089
1593: (
1572:Leibniz
1529:History
531:shuffle
436:art....
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737:where
412:, and
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1772:S2CID
1724:JSTOR
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520:Cards
467:tails
463:heads
453:Coins
419:In a
414:cards
406:coins
114:Prior
2063:Risk
1946:ISBN
1915:ISBN
1883:ISBN
1858:ISBN
1831:ISSN
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490:die
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