## Bayesian Approach

Let be a density function with parameter . In a Bayesian model the paramter space has a distribution , called a prior distribution. Furthermore, is viewed as the conditional distribution of given . By the Bayes' rule the conditional density can be derived from

Posterior distribution. The distribution is called the posterior distribution. Whether is discrete or continuous, the posterior distribution is ``proportional'' to up to the constant. Thus, we write

It is often the case that both the prior density function and the posterior density function belong to the same family of density function with parameter . Then is called conjugate to .

Exponential conjugate family. Suppose that the pdf has the form

Bernoulli trials. Consider independent Bernoulli trials. Let

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