Review of Probability Theory - Sorry

• If P( 1 ) is very much greater than P( 2 ), our decision in favour of 1 will be right
most of the time.
• If P( 1 ) = P( 2 ), we have only a fifty-fifty chance of being right.
In general, the probability of error is the smaller of P( 1 ) and P( 2 ).
In most circumstances, we have not to make decisions with so little evidence. In our example
example, we can use the brightness measurement x as evidence, as Thassos marble is lighter
than Carrara marble. Different samples of marble will yield different brightness readings, and
it is natural to express this variability in probabilistic terms; we consider x as a continuous
random variable whose distribution depends on the state of nature.
Let p(x | j ) be the state-conditional probability density function for x, i.e. the probability
density function for x given that the state of nature is j . Then, the difference between
p(x | 1 ) and p(x | 2 ) describes the difference in brightness between Carrara and Thassos
marble (see figure 16)
Figure 16: Hypothetical class-conditional probability density functions.
Suppose that we know both the a priori probabilities P( j ) and the conditional densities
p(x | j ). Suppose further that we measure the brightness of a piece of marble and discover
the value of x. How does this measurement influence our attitude concerning the true state of
nature? The answer to this question is provided by Bayes‘ rule:
(3)