A prior probability distribution (often simply called the prior probability, prior distribution, or prior) of an uncertain quantity is its assumed probability distribution before evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable. In Bayesian statistics, Bayes' rule prescribes how to update the prior with new information to obtain the posterior probability distribution, which is the conditional distribution of the uncertain quantity given new data. Historically, the choice of priors was often constrained to a conjugate family of a given likelihood function, so that it would result in a tractable posterior of the same family. The widespread availability of Markov chain Monte Carlo methods, however, has made this less of a concern. There are many ways to construct a prior distribution. In some cases, a prior may be determined from past information, such as previous experiments. A prior can also be elicited from the purely subjective assessment of an experienced expert. When no information is available, an uninformative prior may be adopted as justified by the principle of indifference. In modern applications, priors are also often chosen for their mechanical properties, such as regularization and feature selection. The prior distributions of model parameters will often depend on parameters of their own. Uncertainty about these hyperparameters can, in turn, be expressed as hyperprior probability distributions. For example, if one uses a beta distribution to model the distribution of the parameter p of a Bernoulli distribution, then:
p is a parameter of the underlying system (Bernoulli distribution), and α and β are parameters of the prior distribution (beta distribution); hence hyperparameters. In principle, priors can be decomposed into many conditional levels of distributions, so-called hierarchical priors.
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, or a distribution of the temperature for that day of the year. 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 posterior 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.
Strong prior A strong prior is a preceding assumption, theory, concept or idea upon which, after taking account of new information, a current assumption, theory, concept or idea is founded. A strong prior is a type of informative prior in which the information contained in the prior distribution dominates the information contained in the data being analyzed. The Bayesian analysis combines the information contained in the prior with that extracted from the data to produce the posterior distribution which, in the case of a "strong prior", would be little changed from the prior distribution.
Weakly informative priors A weakly informative prior expresses partial information about a variable, steering the analysis toward solutions that align with existing knowledge without overly constraining the results and preventing extreme estimates. An example is, when setting the prior distribution for the temperature at noon tomorrow in St. Louis, to use a normal distribution with mean 50 degrees Fahrenheit and standard deviation 40 degrees, which very loosely constrains the temperature to the range (10 degrees, 90 degrees) with a small chance of being below -30 degrees or above 130 degrees. The purpose of a weakly informative prior is for regularization, that is, to keep inferences in a reasonable range.
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