Prior probability
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A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some 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 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.
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Prior probability distribution
In 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 John Kerry) 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 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
An uninformative prior expresses vague or general information about a variable. The term "uninformative prior" is a misnomer; such a prior might be called a not very informative prior. Uninformative priors can express information such as "the variable is positive" or "the variable is less than some limit". Some authorities prefer the term objective prior.
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 groups that suggests that the prior for the proportion <math>p<math> of voters voting for a candidate, given no other information, should be <math>p^{-1}(1-p)^{-1}<math>. 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 Kerry 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 Kerry and one vote against, using the above prior.
Priors can be constructed which are proportional to the Haar measure if the parameter space <math>X<math> carries a natural group structure. 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 <math>X<math>, 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 <math>X<math> is proportional to <math>1/x<math>. 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. 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 <math>X<math>, 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.
A related idea, reference priors, 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 <math>x<math> when the prior density is <math>p(x)<math>.
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. Alternativley, 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 <math>p<math> is <math>p^{1/2}(1-p)^{1/2}<math>, 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 rule based on the posterior distribution to be 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 in hierarchical models; 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(Ai) and P(Aj) 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,v)~1/v (for v>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. Since
- <math>\int_{-\infty}^{\infty} dm\, = \int_{0}^{\infty} \frac{1}{v} \,dv = \infty <math>
this would be an improper prior both for the mean and for the variance.
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, Choice of hierarchical priors: admissibility in estimation of normal means, Annals of Statistics, 24, pp. 931-95, 1996.
- Jose M. Bernardo, Reference Posterior Distributions for Bayesian Inference, J. Royal Statist. Soc. B, 41, 113-147, 1979.
- Edwin T. Jaynes, Prior Probabilities, IEEE Transactions of 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