We can generalize this to define another probability measure with respect to any
given event C with positive probability. For any event B 2 A, define PBjC
PB\ C
PC
, which is the conditional probability of B given C. The measure P :jCis a
probability measure on the measurable space (O,A).
If we have a decomposition of O into disjoint subsets {B
1
, B
2
, …} with P(B
1
)>0,
then the probability of any event C can be retrieved by the sum of probabilities with
respect to the decomposition, i.e.,
PC
X
i
PCjB
i
P B
i
: (3-61)
Conversely, if P(C) > 0, the probability for each B
i
conditioned on C can be calcu-
lated by Bayes’ rule, i.e.
PB
i
jC
PCjB
i
PB
i
P
j
PCjB
j
PB
j
: (3-62)
In the Bayesian setup, the probabilities P(B
i
) are called a priori probabilities.These
describe the probability of the events with no additional information. If we consider
now an event C with positive probability, one can ask about the a posteriori probabil-
ities P(B
i
|C) of the events in light of event C. In practice, Eq. (3-62) is very important
since many problems do not allow a direct calculation of the probability of an event
but rather the probability of the event conditioned on another event or series of other
events.
Example 3-16: Power of diagnostics
Consider a specific disease affecting 0.5% of the population. A diagnostic test
with a false positive rate of 5% and a true positive rate of 90% is conducted with a
randomly picked person. The test result is positive. What is the probability that
this person has the disease? Let B
1
be the event that a person has the disease (B
c
1
is the complementary event). Let B
2
be the event that the test is positive. Thus, we
are asking for the conditional probability that the person has the disease given
that the test is positive, i.e., P(B
1
|B
2
). From Eq. (3-62) we get
PB
1
jB
2
PB
2
jB
1
PB
1
PB
2
jB
1
PB
1
PB
2
jB
c
1
PB
c
1
0.9 0.005
0.9 0.005 0.05 0.995
0.083:
That means that only 8% of persons with a positive test result will actually have
the disease!
The above effect is due to the fact that the disease is rare and thus that a randomly
picked person will have a low chance a priori of having the disease. The diagnostic
test, however, will produce a high number of false positives in this sample. The di-
agnostic power of the test can be improved by decreasing the error rate. For exam-
ple, decreasing the false positive rate to 1% would give a predictive success rate of
31.142% (0.5% would give 47.493%).
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3 Mathematics in a Nutshell