Other meanings of Prior probability
Statistics
In Bayesian inference, a prior probability is the initial probability estimate assigned to a hypothesis before new evidence is considered. It represents a degree of belief based on existing knowledge, and it is updated via Bayes' theorem to yield a posterior probability. The choice of prior is central to Bayesian analysis, and its influence diminishes as data accumulate.
The prior probability, denoted P(H), is the probability assigned to a hypothesis H before observing new data. In Bayes' theorem, P(H|E) = P(E|H)P(H)/P(E), the prior is multiplied by the likelihood P(E|H) to yield the posterior probability. This framework allows for sequential updating: the posterior from one analysis becomes the prior for the next. Priors can be informative, based on previous studies or expert opinion, or non-informative, intended to exert minimal influence. The choice of prior is subjective in the sense that it reflects available knowledge, but it is a formal requirement for Bayesian inference.
Priors range from informative to weakly informative to non-informative. Informative priors incorporate strong existing evidence, such as historical clinical trial data. Non-informative priors, like the uniform distribution, attempt to represent ignorance, but they are not always invariant under transformation; Jeffreys priors address this by being invariant to reparameterization. Conjugate priors, such as the Beta prior for a binomial likelihood, yield posterior distributions in the same family, simplifying computation. Empirical Bayes methods estimate prior parameters from the data, blurring the line between Bayesian and frequentist approaches. Hierarchical priors model variability across groups, enabling partial pooling in multilevel models.
The concept originates with Thomas Bayes, whose essay was published posthumously in 1763, and was independently developed by Pierre-Simon Laplace, who used it for astronomical and legal problems. In the 20th century, the subjective interpretation, championed by Bruno de Finetti and Leonard J. Savage, framed priors as personal degrees of belief. The objective Bayesian school, led by Harold Jeffreys, sought priors that represent ignorance. The debate between subjective and objective priors remains active, with implications for scientific objectivity and reproducibility.
Prior probability has surprising applications beyond statistics. In machine learning, priors encode regularization, such as in ridge regression where a Gaussian prior on coefficients shrinks estimates. In cosmology, the prior on the cosmological constant influences model comparison. The principle of maximum entropy, introduced by E.T. Jaynes, provides a method for constructing priors from partial information. In legal settings, Bayesian priors have been used to interpret forensic evidence, but their subjectivity has sparked controversy. The reference prior, developed by José-Miguel Bernardo, aims to maximize the information from data, offering a formal objective choice. Even in philosophy, priors are central to Bayesian epistemology, where they are seen as rational constraints on belief.
The prior probability is a foundational concept in Bayesian statistics, with philosophical and practical implications across disciplines.
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