Prior probability 472877 224701708 2008-07-10T01:02:57Z Classicalecon 4831766 {{Nofootnotes|article|date=February 2008}} A '''prior probability''' is a [[conditional probability|marginal probability]], interpreted as a description of what is known about a variable in the absence of some [[Marginal likelihood|evidence]]. The ''[[posterior probability]]'' is then the [[conditional probability]] of the variable taking the evidence into account. The posterior probability is computed from the prior and the [[likelihood function]] via [[Bayes' theorem]]. As ''prior'' and ''posterior'' are not terms used in [[frequency probability|frequentist]] analyses, this article uses the vocabulary of [[Bayesian probability]] and [[Bayesian inference]]. Throughout this article, for the sake of brevity the term ''variable'' encompasses observable variables, latent (unobserved) variables, parameters, and hypotheses. == Prior probability distribution == In [[Bayesian probability|Bayesian]] [[statistical inference]], a '''prior probability distribution''', often called simply the '''prior''', of an uncertain quantity ''p'' (for example, suppose ''p'' is the proportion of voters who will vote for the politician named Smith in a future election) is the [[probability distribution]] that would express one's uncertainty about ''p'' before the "data" (for example, an opinion poll) are taken into account. It is meant to attribute uncertainty rather than randomness to the uncertain quantity. One applies [[Bayes' theorem]], multiplying the prior by the [[likelihood function]] and then normalizing, to get the ''posterior probability distribution'', which is the conditional distribution of the uncertain quantity given the data. A prior is often the purely subjective assessment of an experienced expert. Some will choose a ''[[conjugate prior]]'' when they can, to make calculation of the posterior distribution easier. == Informative priors == An ''informative prior'' expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach is to make the prior a [[normal distribution]] with [[expected value]] equal to today's noontime temperature, with [[variance]] equal to the day-to-day variance of atmospheric temperature. This example has a property in common with many priors, namely, that the posterior from one problem (today's temperature) becomes the prior for another problem (tomorrow's temperature); pre-existing evidence which has already been taken into account is part of the prior and as more evidence accumulates the prior is determined largely by the evidence rather than any original assumption, provided that the original assumption admitted the possibility of what the evidence is suggesting. The terms "prior" and "posterior" are generally relative to a specific datum or observation. == Uninformative priors ==<!-- This section is linked from [[Non-informative prior]] --> An ''uninformative prior'' expresses vague or general information about a variable. The term "uninformative prior" may be somewhat of a misnomer; often, such a prior might be called a ''not very informative prior'', or an ''objective prior'', i.e. one that's not subjectively elicited. Uninformative priors can express "objective" information such as "the variable is positive" or "the variable is less than some limit".. In parameter estimation problems, the use of an uninformative prior typically yields results which are not too different from conventional statistical analysis, as the likelihood function often yields more information than the uninformative prior. Some attempts have been made at finding probability distributions in some sense logically required by the nature of one's state of uncertainty; these are a subject of philosophical controversy. For example, [[Edwin T. Jaynes]] has published an argument (Jaynes 1968) based on [[Lie group]]s that suggests that the prior for the proportion ''p'' of voters voting for a candidate, given no other information, should be the Haldane prior ''p''<sup>&minus;1</sup>(1&nbsp;&minus;&nbsp;''p'')<sup>&minus;1</sup>. If one is so uncertain about the value of the aforementioned proportion ''p'' that one knows only that at least one voter will vote for Smith and at least one will not, then the conditional probability distribution of ''p'' given this information alone is the [[uniform distribution]] on the interval [0, 1], which is obtained by applying [[Bayes' theorem]] to the data set consisting of one vote for Smith and one vote against, using the above prior. The Haldane prior has been criticized on the grounds that it yields an improper posterior distribution that puts 100% of the probability content at either ''p'' = 0 or at ''p'' = 1 if a finite sample of voters all favor the same candidate, even though mathematically the posterior probability is simply not defined and thus we cannot even speak of a probability content. The Jeffreys prior ''p''<sup>&minus;1/2</sup>(1&nbsp;&minus;&nbsp;''p'')<sup>&minus;1/2</sup> is therefore preferred (see below). Priors can be constructed which are proportional to the [[Haar measure]] if the parameter space ''X'' carries a [[transformation group|natural group structure]] which leaves invariant our relevant data. For example, in physics we might expect that an experiment will give the same results regardless of our choice of the origin of a coordinate system. This induces the group structure of the [[translation group]] on ''X'', and the resulting prior is a constant [[improper prior]]. Similarly, some measurements are naturally invariant to the choice of an arbitrary scale (i.e., it doesn't matter if we use centimeters or inches, we should get results that are physically the same). In such a case, the scale group is the natural group structure, and the corresponding prior on ''X'' is proportional to 1/''x''. It sometimes matters whether we use the left-invariant or right-invariant Haar measure. For example, the left and right invariant Haar measures on the [[affine group]] are not equal. Berger (1985, p. 413) argues that the right-invariant Haar measure is the correct choice. Another idea, championed by Edwin T. Jaynes, is to use the principle of ''[[maximum entropy]]'' (MAXENT). The motivation is that the [[Shannon entropy]] of a probability distribution measures the amount of information contained the distribution. The larger the entropy, the less information is provided by the distribution. Thus, by maximizing the entropy over a suitable set of probability distributions on ''X'', one finds that distribution that is least informative in the sense that it contains the least amount of information consistent with the constraints that define the set. For example, the maximum entropy prior on a discrete space, given only that the probability is normalized to 1, is the prior that assigns equal probability to each state. And in the continuous case, the maximum entropy prior given that the density is normalized with mean zero and variance unity is the standard [[normal distribution]]. The principle of ''[[minxent|minimum cross-entropy]]'' generalizes MAXENT to the case of "updating" an arbitrary prior distribution with suitability constraints in the maximum-entropy sense. A related idea, [[reference prior]]s, was introduced by Jose M. Bernardo. Here, the idea is to maximize the expected [[Kullback-Leibler divergence]] of the posterior distribution relative to the prior. This maximizes the expected posterior information about ''X'' when the prior density is ''p''(''x''); thus, in some sense, ''p''(''x'') is the "least informative" prior about X. The reference prior is defined in the asymptotic limit, i.e., one considers the limit of the priors so obtained as the number of data points goes to infinity. Reference priors are often the objective prior of choice in multivariate problems, since other rules (e.g., [[Jeffreys prior| Jeffreys' rule]]) may result in priors with problematic behavior. Objective prior distributions may also be derived from other principles, such as [[information theory|information]] or [[coding theory]] (see e.g. [[minimum description length]]) or [[frequentist statistics]] (see [[frequentist matching]]). Philosophical problems associated with uninformative priors are associated with the choice of an appropriate metric, or measurement scale. Suppose we want a prior for the running speed of a runner who is unknown to us. We could specify, say, a normal distribution as the prior for his speed, but alternatively we could specify a normal prior for the time he takes to complete 100 metres, which is proportional to the reciprocal of the first prior. These are very different priors, but it is not clear which is to be preferred. Similarly, if asked to estimate an unknown proportion between 0 and 1, we might say that all proportions are equally likely and use a uniform prior. Alternatively, we might say that all orders of magnitude for the proportion are equally likely, which gives a prior proportional to the logarithm. The [[Jeffreys prior]] attempts to solve this problem by computing a prior which expresses the same belief no matter which metric is used. The Jeffreys prior for an unknown proportion ''p'' is ''p''<sup>&minus;1/2</sup>(1&nbsp;&minus;&nbsp;''p'')<sup>&minus;1/2</sup>, which differs from Jaynes' recommendation. Practical problems associated with uninformative priors include the requirement that the posterior distribution be proper. The usual uninformative priors on continuous, unbounded variables are improper. This need not be a problem if the posterior distribution is proper. Another issue of importance is that if an uninformative prior is to be used ''routinely'', i.e., with many different data sets, it should have good [[frequentist]] properties. Normally a [[Bayesian]] would not be concerned with such issues, but it can be important in this situation. For example, one would want any [[decision theory|decision rule]] based on the posterior distribution to be [[admissible decision rule|admissible]] under the adopted loss function. Unfortunately, admissibility is often difficult to check, although some results are known (e.g., Berger and Strawderman 1996). The issue is particularly acute with [[hierarchical Bayes model]]s; the usual priors (e.g., Jeffreys' prior) may give badly inadmissible decision rules if employed at the higher levels of the hierarchy. ==Improper priors== If Bayes' theorem is written as :<math>P(A_i|B) = \frac{P(B | A_i) P(A_i)}{\sum_j P(B|A_j)P(A_j)}\, ,</math> then it is clear that it would remain true if all the prior probabilities ''P''(''A''<sub>''i''</sub>) and ''P''(''A''<sub>''j''</sub>) were multiplied by a given constant; the same would be true for a [[continuous random variable]]. The posterior probabilities will still sum (or integrate) to 1 even if the prior values do not, and so the priors only need be specified in the correct proportion. Taking this idea further, in many cases the sum or integral of the prior values may not even need to be finite to get sensible answers for the posterior probabilities. When this is the case, the prior is called an '''improper prior'''. Some statisticians use improper priors as uninformative priors. For example, if they need a prior distribution for the mean and variance of a random variable, they may assume ''p''(''m'',&nbsp;''v'')&nbsp;~&nbsp;1/''v'' (for ''v''&nbsp;>&nbsp;0) which would suggest that any value for the mean is "equally likely" and that a value for the positive variance becomes "less likely" in inverse proportion to its value. Many authors (Lindley, 1973; De Groot, 1937; Kass and Wasserman, 1996) warn against the danger of over-interpreting those priors since they are not probability densities. The only relevance they have is found in the corresponding posterior, as long as it is well-defined for all observations. (The Haldane prior is a typical counterexample.) == References == * [[Andrew Gelman]], [[John B. Carlin]], [[Hal S. Stern]], and [[Donald B. Rubin]]. ''Bayesian Data Analysis'', 2nd edition. [[CRC Press]], 2003. ISBN 1-58488-388-X * [[James O. Berger]], ''Statistical Decision Theory and Bayesian Analysis'', Second Edition. [[Springer-Verlag]], 1985. ISBN 0-387-96098-8 * [[James O. Berger]] and [[William E. Strawderman]], [http://links.jstor.org/sici?sici=0090-5364%28199606%2924%3A3%3C931%3ACOHPAI%3E2.0.CO%3B2-5 ''Choice of hierarchical priors: admissibility in estimation of normal means], ''[[Annals of Statistics]]'', '''24''', pp. 931-95, 1996. * [[Jose M. Bernardo]], [http://links.jstor.org/sici?sici=0035-9246%281979%2941%3A2%3C113%3ARPDFBI%3E2.0.CO%3B2-P ''Reference Posterior Distributions for Bayesian Inference],'' [[Journal of the Royal Statististical Society, Series B]]'', '''41''', 113-147, 1979. * [[Edwin T. Jaynes]], "Prior Probabilities," ''[[IEEE Transactions on Systems Science and Cybernetics]]'', '''SSC-4''', 227-241, Sept. 1968. Reprinted in Roger D. Rosenkrantz, Compiler, ''E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics''. Dordrecht, Holland: Reidel Publishing Company, pp. 116-130, 1983. ISBN 90-277-1448-7 [[Category:Bayesian statistics]] [[Category:Probability and statistics]] [[de:A-priori-Wahrscheinlichkeit]] [[ja:事前確率]]