Prior probability

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> before some 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 <a href="/facts/Parameter/zFvVFMv9">parameter</a> of the model or a <a href="/facts/Latent_variable/ohg99BnW">latent variable</a> rather than an <a href="/facts/Observable_variable/IAfRc11t">observable variable</a>.
In <a href="/facts/Bayesian_statistics/9w7b1Bw4">Bayesian statistics</a>, <a href="/facts/Bayes%2527_rule/wQqFB2Dt">Bayes' rule</a> prescribes how to update the prior with new information to obtain the <a href="/facts/Posterior_probability_distribution/fojsMD0D">posterior probability distribution</a>, which is the conditional distribution of the uncertain quantity given new data. Historically, the choice of priors was often constrained to a <a href="/facts/Conjugate_prior/ScCFcs8b">conjugate family</a> of a given <a href="/facts/Likelihood_function/1L8P4NBX">likelihood function</a>, so that it would result in a tractable posterior of the same family. The widespread availability of <a href="/facts/Markov_chain_Monte_Carlo/eTjk3WQy">Markov chain Monte Carlo</a> 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 <a href="/facts/Principle_of_indifference/F9uBbTtG">principle of indifference</a>. In modern applications, priors are also often chosen for their mechanical properties, such as <a href="/facts/Bayesian_interpretation_of_kernel_regularization/au7th8GK">regularization</a> and <a href="/facts/Feature_selection/gxbuF4VG">feature selection</a>.
The prior distributions of model parameters will often depend on parameters of their own. Uncertainty about these <a href="/facts/Hyperparameter_(Bayesian_statistics)/zqcLPQcb">hyperparameters</a> can, in turn, be expressed as <a href="/facts/Hyperprior/H3uWbNTX">hyperprior</a> probability distributions. For example, if one uses a <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a> to model the distribution of the parameter p of a <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli distribution</a>, then:

<ul><li>p is a parameter of the underlying system (Bernoulli distribution), and</li>
<li>α and β are parameters of the prior distribution (beta distribution); hence hyperparameters.</li></ul>
In principle, priors can be decomposed into many conditional levels of distributions, so-called hierarchical priors.

Prior probability open-in-new

Prior probability