be a density function with parameter
In a Bayesian model
the paramter space
called a prior distribution
is viewed as the conditional
By the Bayes' rule the conditional
can be derived from
is called the
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 .
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
prior distribution is called
Consider independent Bernoulli trials.
be a prior density of beta distribution.
Given the data
the posterior density is calculated as
The expected value of posterior density becomes
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