Conjugate Prior

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 conjugate prior is a prior distribution that belongs to the same family of probability distributions as the posterior distribution that is obtained after observing the data. This means that the prior and posterior distributions are mathematically compatible and can be easily combined to update the degree of belief about the parameter. Conjugate priors are particularly useful in Bayesian inference because they allow for analytical solutions to the posterior distribution, which simplifies the computation of the posterior mean, variance, and other statistics. Common examples of conjugate priors include the normal distribution as a conjugate prior for the mean of a normal distribution, the gamma distribution as a conjugate prior for the precision of a normal distribution, and the beta distribution as a conjugate prior for the probability of a binomial distribution.
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