here is a
hypothesis that may be true (state
)
or false (state
).
The decision
is the assertion of the hypothesis. The penalty function
has the property
Such property enforces greater penalty for the false decision then for the
correct one for every prior state.
Given the observation
we choose the decision
if
and choose
otherwise.
Equivalently,
Note
that
Hence,
where the
stands for ''threshold''. This result is called ''likelihood ratio test''.
Note, that if
and
then the rule simply says ''choose
if
''.
With the decision rule
known one may compute an apriori probability of the correct judgement
as
