8127/frame/fm

i.
Likelihood ratios in Bayes’ theorem.
From the foregoing discussion,
we have learned that LRs answer the question of the probability of the
evidence assuming, alternatively, this source or another source. This will be
a good time to step back and consider LRs in the context of Bayes’ theorem.
Bayes’ theorem provides a general model for updating our certainty about
any proposition.
Using words, the expression of Bayes’ theorem looks like this:
Prior odds 
×
LR = Posterior odds
This may be read: “However likely you think a proposition (prior odds),
change your judgment (posterior odds) by this additional evidence (LR).”
Using symbols, the expression takes this form:
Note several important points. The prior odds are simply the 
hypothesis
given
some 
information
. We have some idea about the probability of an event or
hypothesis. Ideally, this is supported by numerical data, but it could also be
speculative. Either way, it must be expressed as a quantity. This initial estimate
is updated by multiplying it by the LR. Calculation of an LR in light of new
evidence
provides a way to modify the prior odds. This results in posterior
odds. The addition of the 
E
term to the expression of posterior odds reflects
the integration of the new evidence.
Analysts are sometimes concerned about expressing their results in terms
of an LR because they believe they will be forced to define prior odds. In
P H
1
I
(
)
P H
2
I
(
)
-------------------
P E H
1
,
I
(
)
P E H
2
,
I
(
)
-------------------------
×
P H
1
E
,
I
(
)
P H
2
E
,
I
(
)
-------------------------
=
8127/frame/ch06 Page 144 Friday, July 21, 2000 11:47 AM

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Classification, Identification, and Individualization
145
fact, at the level of source determination, the analytical results can be
expressed using only the LR term. Any information about the samples that
directly affects analytical results can be incorporated into either the numer-
ator or denominator of the LR, as appropriate. The strength of the inference
about the source of the evidence can be expressed wholly by the LR. Any
interested party can then pose prior odds to be modified by the inference
resulting from the physical evidence examination. In some cases, the crimi-
nalist may wish to assist by pointing out information to be considered as