Bayesian prior

Bayesian Prior

A Bayesian prior, also known as a prior probability distribution, is a fundamental concept in Bayesian statistics. It represents the probability distribution that would express one's beliefs about this quantity before some evidence is taken into account.

Definition

In the context of Bayesian inference, a prior probability distribution, often simply called the "prior", of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity 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.

Role in Bayesian Inference

The Bayesian prior plays a crucial role in Bayesian inference, a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. The prior represents the initial state of knowledge before evidence is accounted for. It is then updated with the evidence using Bayes' theorem to produce the posterior probability.

Types of Priors

There are several types of priors that can be used in Bayesian inference, including but not limited to:

• Informative priors: These reflect specific, definite information about the variable in question. An example might be a prior distribution for the height of an adult male, for which we have plenty of historical data.

• Uninformative priors: These reflect a lack of specific knowledge about the variable in question. They are often flat or uniform distributions over the range of possible values.

• Conjugate priors: In Bayesian probability theory, if the posterior distributions p(θ|x) are in the same family as the prior probability distribution p(θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function.

Criticisms and Controversies

The use of Bayesian priors has been subject to criticism and controversy. Some critics argue that the use of priors introduces subjectivity into the analysis, as different analysts may use different priors. Others argue that the use of priors allows for the incorporation of relevant background information, and that this is in fact a strength of the Bayesian approach.

See Also

• Bayesian statistics
• Bayesian inference
• Bayes' theorem
• Posterior probability