Conjugate Priors

In Bayesian probability theory, a prior distribution is a probability distribution that expresses the degree of belief or uncertainty about the value of a parameter before observing the data. A prior distribution is said to be conjugate for a likelihood function if the posterior distribution is in the same family of distributions as the prior. Conjugate priors are useful because they allow for the posterior distribution to be calculated analytically, which can simplify the computation of Bayesian inference. For example, if the likelihood function is a binomial distribution, a beta distribution can be used as a conjugate prior. Similarly, if the likelihood function is a normal distribution with known variance, a normal distribution can be used as a conjugate prior for the mean. Conjugate priors are often used in machine learning and data science applications, particularly in Bayesian hierarchical modeling.
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