### 4.1 Bayesian estimation

We assign a cost function of estimating the true value of as . We then associate with
an estimator a conditional risk or cost averaged over all realizations of data for each value of the
parameter :
where is the set of observations and is the joint probability distribution of data and
parameter . We further assume that there is a certain a priori probability distribution of the
parameter . We then define the Bayes estimator as the estimator that minimizes the average risk defined
as
where E is the expectation value with respect to an a priori distribution , and is the set of
observations of the parameter . It is not difficult to show that for a commonly used cost function
the Bayesian estimator is the conditional mean of the parameter given data , i.e.,
Update where is the conditional probability density of parameter given the data
.