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1. It simplifies the combination of a variety of different kinds of
evidence, lab tests, animal experiments, and clinical trials, and it
serves as an effective aid to decision making.
2. It permits evaluating evidence in favor of a null hypothesis. And
with very large samples, a null hypothesis is not automatically
rejected.
3. It provides flexibility during the conduct of an experiment; sample
sizes can be modified, measuring devices altered, subject popula-
tions changed, and endpoints redefined.
Suppose we have in hand a set of evidence
E
= {
E
1
,
E
2
,...,
E
n
} and
thus have determined the conditional probability Pr{
A
|
E
} that some
event
A
is true. “
A
” might be the event that O.J. killed his ex-wife, that
the Captain of the Valdez behaved recklessly, or some other incident
whose truth or falsehood we wish to establish. An additional piece of
evidence
E
n
+1
now comes to light. Bayes' Theorem tell us that
{
}
=
{
Pr
AE
,...,
E E
,
1
nn
+
1
}
{
}
Pr
EA AE
Pr
,...,
E
n
+
1
1
n
{
}
{
}
+
{
}
{
}
Pr
EA AE
Pr
,...,
E
Pr
E A
~
Pr
AE
,...,
E
n
+
1
1
n
n
+
1
1
n
where ~
A
, read not
A
, is the event that A did not occur. Recall that Pr{
A
}
+ Pr{~
A
} = 1. Pr{
A
|
E
1
,...,
E
n
} is the
prior
probability of
A
, and Pr{
A
|
E
1
,...,
E
n
,
E
n
+1
} the
posterior
probability of A once the item of evidence
E
n
+1
is in hand. Gather sufficient evidence and we shall have an automatic
verdict.
The problem with the application of Bayes' Theorem in practice comes
at the beginning when we have no evidence in hand, and
n
= 0. What is
the prior probability of
A
then?
Applications in the Courtroom
6
Bayes' Theorem has seen little use in criminal trials as ultimately the
theorem relies on unproven estimates rather than known facts.
7
Tribe
[1971] states several objections including the argument that a jury might
actually use the evidence twice, once in its initial assessment of guilt—that
is, to determine a prior probability—and a second time when the jury
applies Bayes' Theorem. A further objection to the theorem's application
is that if a man is innocent until proven guilty, the prior probability of his
guilt must be zero; by Bayes' Theorem the posterior probability of his
6
The majority of this section is reprinted with permission from
Applying Statistics in the
Courtroom,
by Phillip Good, Copyright 2001 by CRC Press, Inc.
7
See, for example, People v. Collins, 68 Cal .2d 319, 36 ALR3d 1176 (1968).
