3.1
Bayesian approach
In the Bayesian approach we assign costs to our decisions; in
particular we introduce positive numbers
,
, where
is the cost incurred by choosing hypothesis
when hypothesis
is true. We define the
conditional risk
of a decision rule
for each hypothesis as
where
is the probability distribution of the data when hypothesis
is true. Next we assign probabilities
and
to the occurrences of hypothesis
and
, respectively. These probabilities are called
a priori probabilities
or
priors
. We define the
Bayes risk
as the overall average cost incurred by the decision rule
:
Finally we define the
Bayes rule
as the rule that minimizes the Bayes risk
.