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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.
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.
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.
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".
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 [a reference here would be useful] based on Lie groupIn mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical anas that 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 one's state of ignorance is the uniform distributionIn mathematics, the uniform distributions are simple probability distributions. The distribution can be either discrete or continuous. In the discrete case, they can be characterized by saying that all possible values are equally probable. In the continuo on the interval [0, 1].