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## 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

and that a prior distribution is given by

Then we obtain the posterior density

Thus, the family of is conjugate to , and the parameter of prior distribution is called the hyperparameter.

Bernoulli trials. Consider independent Bernoulli trials. Let

be a prior density of beta distribution. Given the data , the posterior density is calculated as

The expected value of posterior density becomes