
D. SCHUM
GEORGE MASON UNIVERSITY
FAIRFAX, VIRGINIA
USA
UNIVERSITY COLLEGE LONDON
LEVERHULME FOUNDATION STUDIES OF EVIDENCE
MAY 4, 2004

1.O EVIDENCE ABOUT A SOURCE OF EVIDENCE: SOME INITIAL QUESTIONS
What we learn about the credibility of a human source of testimonial evidence certainly influences our beliefs about the inferential force or weight of this person's testimony. On some occasions we may feel justified in believing that what we know about a human source of testimonial evidence is at least as valuable inferentially as the event(s) thIs source has reported to us. The issues addressed in this report concern a formal justification for having such beliefs about the importance of evidence about a source. Also of interest are the conditions under which such justification will hold.
An Example: Some Obvious Questions
Here is an example of a situation in which what we know about a human source of evidence might be at least as valuable as knowing for certain that the event he/she reports to us has occurred. Troubled by chronic congestion, fever and general malaise, you consult your physician. After her analysis of your symptoms and the results of some preliminary tests she has ordered, she tells you that pneumonia and tuberculosis are among the possibilities she is considering. She further orders you to have a series of chest X-rays or other radiological tests to help her make a determination about whether you should be treated for tuberculosis or pneumonia. Suppose there is an anomaly, call it event E, which, if it appears on your X-ray, is strong but inconclusive evidence that you have tuberculosis. That is, this anomaly E has some probability of occurring for patients who have tuberculosis and also for those who have pneumonia, but more often for patients who have tuberculosis. Tuberculosis is something you would rather not have; pneumonia may be far easier to cure. You undergo these radiological tests and receive the depressing news from the radiologist that his examination revealed the occurrence of this anomaly on your X-ray. In symbols, the radiologist reports evidence E* that your examination revealed event E, the existence of this anomaly. You understand that events E* and E are not the same. Evidence E* about event E does not entail that E occurred or is true.
How might you think about this evidence E*?
Your first thoughts might concern the authenticity of this X-ray test result. Can you be sure that the radiologist examined X-rays taken from you and not from some other person. You know that there is always the possibility that medical tests or other records can be mixed up regarding the patients to whom they belong. But the X-ray laboratory where your tests were taken has an excellent record concerning the handling of its X-ray images and is able to verify the chain of custody of your X-rays from the time they were taken until the time your radiologist examined them. So, you believe that the radiologist's report to you is authentic and concerns the results of
X-rays that were in fact taken of your chest area.
But you have other natural questions concerning the credibility of the test result saying that you have the anomaly E. For a start, a natural question concerns the ability of the radiologist to detect the occurrence of anomaly E on X-rays. If appropriate records were available, you would like to have some indication of this radiologist's hit probability and false-positive probability as far as the detection of anomaly E is concerned. A hit occurs when the radiologist reports the existence of the anomaly, given that it is present. A false-positive occurs when the radiologist reports the existence of the anomaly, given that it is not present. As far as reports E* of the existence of anomaly E are concerned, we require estimates of the following two probabilities: h =
"Hit" = P(E*|E) and f = "False--Positive" = P(E*|E c ), where E c = the non-existence of anomaly E.
Your radiologist is Board-Certified and so you believe that his hits have substantially exceeded his false-positives in his previous examinations of X-rays for the occurrence of anomaly E. Should you be confident, just on this assumption, that the radiologist's report indicates that you do have this anomaly and that it may favor your having tuberculosis over pneumonia? There other, not quite so obvious, credibility-related questions you should ask. Where do these additional questions come from?
2

2.0 FURTHER QUESTIONS GENERATED BY BAYES' RULE.
One of the frequently-overlooked virtues of probabilistic analyses of situations like the one described in the example above is that, when carefully performed, they suggest important questions to us that we might not have entertained in the absence of such analyses. In short, probabilistic analyses can be heuristically valuable in generating new lines of inquiry and new important evidence. To make the example just given fairly simple, so as not to obscure the importance of additional credibility-related questions, I will first suppose that your physician is only entertaining two hypotheses at this point: H
1
= You have tuberculosis, and H
2
= You have pneumonia. In fact, of course, there may be other possible explanations for the pattern of symptoms your physicians have observed. It might be true, in addition, that you have both tuberculosis and pneumonia, but I will leave this very depressing possibility aside in this example.
Finally, I understand that a complete diagnosis of your condition would require many other possible tests. However, I wish to concentrate just on the anomaly evidence E* that has been provided by the radiologist.
From a Bayesian point of view in the present situation, we need to determine the two posterior probabilities P(H
1
|E*) and P(H
2
|E*), based on evidence E*. As just noted, I am assuming in this example that H
1
and H
2 are mutually exclusive, but not exhaustive. I first focus on the posterior odds of H
1
to H
2
; these odds are given by the ratio: Ω* = P(H
1
|E*)/P(H
2
|E*). Using the oddslikelihood ratio form of Bayes' rule, we can determine that: Ω* = ΩL
E*
, where Ω =
P(H
1
)/P(H
2
), the prior odds on H
1
relative to H
2
; the term L
E*
= P(E*|H
1
)/P(E*|H
2
) is the likelihood ratio for evidence E*. In Bayesian formulations it is the likelihood ratio for an item or body of evidence that grades the inferential force or weight of the evidence. This is true since L
E*
= Ω*/ Ω measures the possible change of your prior to posterior probabilistic beliefs about the odds of H
1 to H
2
, given evidence E*. Our current interests involve this likelihood ratio term for evidence E*.
Following is the first of two structural representations for our inference involving H
1
and
H
2
, based just on evidence E*. This structure seems quite obvious but, as we will observe when we examine L
E* carefully, we will see that something is left out.
H
1
= You have tuberculosis
H
2
= You have pneumonia.
Stage 2: The importance of the anomaly.
E = Anomaly E is present.
E c = Anomaly E is not present.
Stage 1: The radiologist's credibility.
E* = Radiologist's Report that
Anomaly E is Present.
Figure 1. An Initial Posing of the Inference Problem
As you see in Figure 1, there are two stages in an inference regarding H
1 and H
2
, based on evidence E*. The first stage involves the credibility of the radiologist. To what extent can we infer that the anomaly is present in your chest, based on the radiologist's report that it is there? At the second stage, supposing that the anomaly is present (event E is true), how strongly would this anomaly point toward H
1
and toward H
2
?. Recall our assumption that anomaly E, if it is there in your chest, is just inconclusive evidence that you have tuberculosis; it might also be there if you had pneumonia. The relevant probabilities at Stage 2 are the two likelihoods: P(E|H
1
) and
P(E|H
2
). Their ratio L
E
= P(E|H
1
)/P(E|H
2
) shows how strongly event E [the known presence of the anomaly] favors H
1
or H
2
. In this example, we are supposing that P(E|H
1
) > P(E|H
2
). This means that the actual occurrence of anomaly E would favor your having tuberculosis over your having
3

 pneumonia. We must keep L
E in mind as we observe what else Bayes' rule tells us about evidence E*.
We can now form an expression for L
E*
that will take account of both of these sources of uncertainty. In the process, we will observe that we must take account of another possible probabilistic linkage that is missing in Figure 1. This additional linkage suggests another very important question we should ask about the radiologist's credibility and will expose the major evidential subtlety that is the subject of this report. It will also expose a very important consideration involving the anomaly E that we have not yet considered, namely, how rare or improbable is this anomaly. Following is an expansion of L
E*
= P(E*|H
1
)/P(E*|H
2
) that takes account of both sources of uncertainty shown in Figure 1. The derivation of L
E*
in this case is described in detail elsewhere [Schum, 1994, 2001, 293 - 294].
L
E *

P ( E | H
1
)[ P ( E * | EH
1
)  P ( E * | E
P ( E | H
2
)[ P ( E * | EH
2
)  P ( E * | E c H c H
1
)]  P ( E * | E
2
)]  P ( E * | E c H c H
1
)
2
)
.........
Equation 1
Anomaly Rareness: The first thing Equation 1 tells us is that we must have the exact values of P(E|H
1
) and P(E|H
2
) and not simply their ratio L
E
= P(E|H
1
)/ P(E|H
2
). The reason is that it makes a great difference in applying Bayes' rule how rare or improbable is the anomaly, given either tuberculosis or given pneumonia. In its wisdom, Bayes' rule responds to differences as well as to ratios involving the likelihood ingredients we incorporate when using this rule. By such means we take account of rareness issues in the following way. Suppose, for a start, that we believe that L
E
= P(E|H
1
)/P(E|H
2
) = 5; i.e. the known occurrence of the anomaly is five times more probable, given tuberculosis than given pneumonia. But there is an infinity of different combinations of the two likelihoods P(E|H
1
) and P(E|H
2
) whose ratio is 5. Following are three pairs of likelihoods, each of which has a ratio L
E
= 5.

Case 1: P(E|H
1
) = 0.20; P(E|H
2
) = 0.04

Case 2: P(E|H
1
) = 0.020; P(E|H
2
) = 0.004

Case 3: P(E|H
1
) = 0.0020; P(E|H
2
) = 0.0004
Going from Case 1 to Case 3 we are making the known occurrence of anomaly E a successively rarer event, given either H
1
or H
2
. Though the ratios remain the same in each case, the differences in these two likelihoods decrease as we make event E rarer. It is easily shown that
Equation 1 will give us different values of L
E* depending upon which of the three cases listed above we choose to insert in this equation; an example comes a bit later.
As an aside, even the earliest probabilists suspected that the force of testimonial evidence depends not only on the credibility of the source, but also on how rare or improbable is the event being reported by the source [e.g. Hume, 1777, 113; Laplace, 1796, 114; Daston, 1988;
Schum, 1994, 306 - 308]. But the early probablists were never able to show the precise relation between event rareness and source credibility in their influence on the force of evidence;
Equation 1 makes this relation very clear.
Conditional Nonindependence and State-Dependent Credibility
I come finally to a characteristic of Equation 1 that suggests a question about the radiologist's credibility that we have not yet asked. This characteristic involves a feature of
Bayesian analysis that is remarkably suited to the capture of a very wide array of evidential and inferential subtleties; it is called conditional nonindependence. To see how it occurs in Equation 1, observe the four remaining likelihoods, all of which involve our evidence E*. First observe that these four likelihoods are of the form h = P(E*|E --) and f = P(E*|E c
--); i.e. they are hit and falsepositive probabilities. But also observe the addition of events H
1
or H
2
in these expressions. I will use a hybrid notation, introduced by Kolmogorov [1933, 6], for representing these conditional probabilities. Take the common expression P(A|B) = P(AB)/P(B), for P(B) > 0. Kolmogorov noted
4

 that P(A|B) could also be represented by P
B
(A). This emphasizes that we have a new probability measure P
B
, which we apply to event E, when we are given event B.
To apply Kolmogorov's hybrid notation to Equation 1, take P(E*|EH
1
) as an example.
Using this hybrid notation, we can express P(E*|EH
1
) as P
H1
(E*|E). This expression asks: Does the source's hit probability depend upon H
1
being true? If this hit probability is not conditional on
H
1
, then we have P(E*|EH
1
) = P(E*|E). We must ask the same question for each one of the four likelihoods involving E*; the questions are:
Does P(E*|EH
1
) = P(E*|E) ?
Does P(E*|E c H
1
) = P(E*|E c ) ?
Does P(E*|EH
2
) = P(E*|E) ?
Does P(E*|E c H
2
) = P(E*|E c ) ?
These are all questions concerning the possible nonindependence of hit or of false-positive probabilities, conditional on either H
1
or on H
2
. Answering these questions suggests that we need to add a link to the chain of reasoning shown above in Figure 1. It is important to note that a "no" answer to one of these probabilities does not entail a "no" answer to any of the others; some may exhibit conditional nonindependence and others not. When some are conditionally nonindependent, given H
1 or H
2
, we can say that these probabilities are state-dependent; this term comes from the field of engineering. Here is the added link suggested by our Bayesian formulation.
?
H
1
= You have tuberculosis
H
2
= You have pneumonia.
E = Anomaly E is present.
E c = Anomaly E is not present.
E* = Radiologist's Report that
Anomaly E is Present.
Figure 2. A New Link Suggested by Bayes' Rule
The question-mark attached to the added link in Figure 2 is answered when we answer the four questions posed above. When all of these questions are answered "yes" [i.e. there are no conditional dependencies of either hits or false-positives under either H
1
or under H
2
], then we can sever this added link and return to the representation shown in Figure 1. I will return to this situation in a moment.
Now suppose that all of the four questions posed above are answered "no"; that is the source's hit and false-positive probabilities are conditional on both H
1
and on H
2
. In this case we can express L
E*
as:
L
E *

P ( E | H
1
)[ h
1
P ( E | H
2
)[ h
2
 f
1
 f
2
]  f
1
]  f
2
..........
Equation 2 where h
1
= P(E*|EH
1
); f
1
= P(E*|E c H
1
); h
2
= P(E*|EH
2
); f
2
= P(E*|E c H
2
).
5






In the case in which all four of the questions above are answered "yes" [i.e. none of the hit and false positive probabilities are conditional on either H
1
or H
2
], then we can express L
E* as follows:
L
E *

P ( E | H
1
)[ h  f ]  f
P ( E | H
2
)[ h  f ]  f
..........
Equation 3 where h = P(E*|E) and f = P(E*|E c ).
It is an easy matter to show that L
E*
in Equation 3 is bounded above by L
E
=
P(E|H
1
)/P(E|H
2
) and bounded below by L
E c = P(E c |H
1
)/P(E c |H
2
). To see this, first suppose that f =
0. In this case, L
E*
= L
E
= P(E|H
1
)/P(E|H
2
). Next, suppose h = 0. Then:
L
E *

P ( E | H
1
)(  f )  f
P ( E | H
2
)(  f )  f
 f [1  P ( E | H
1 f [1  P ( E | H
2
]
]

P
P (
( E
E c c | H
| H
1
2
)
)
 L
E c
But the interesting thing about L
E*
in Equation 2 is that it has no such bounds; it can take on any value greater than zero.
Here's the bottom line as far as what Bayes' rule is telling us about your question concerning the credibility of the radiologist. Applying Bayes' rule, another question is suggested, namely, is the credibility of the radiologist, in terms of his hit and false-positive probabilities, dependent upon whether you have tuberculosis or have pneumonia? Either his hit probability or his false-positive probability might depend upon which one of these diseases you actually have.
Equation 2 let's us explore the consequences of situations in which the radiologist's hit and/or false-positive probabilities depend upon which of these diseases you might have. Depending on how we answer this question determines whether the following inequality in Equation 4 below holds and L
E*
> L
E
.
3.0 CONDITIONS UNDER WHICH THE FORCE OF A REPORT OF AN EVENT EXCEEDS
THE FORCE OF KNOWING FOR SURE THAT THIS EVENT OCCURRED
The original problem initially posed is to determine the conditions under which testimony about some event from might have more inferential force than knowing for sure that this reported event occurred. In symbols, here is what we wish to determine; when is it the case that:
L
E *

P ( E | H
1
)[ h
1
P ( E | H
2
)[ h
2
 f
1
]  f
1
 f
2
]  f
2

P ( E | H
1
)
P ( E | H
2
)
 L
E
?
To simplify notation a bit, let p = P(E|H
1
) and q = P(E|H
2
), what we then have is: q p [ h
1
[ h
2
 f
1
 f
2
]
 f
1
]
 f
2
 p q
Cross multiplying, we have q { p [ h
1 pq [ h
1
 f
1
 f
1
]
 f
]
 qf
1
1
}
 p { q [ h
2
 pq [ h
2
 f

2
] f
2

]
 pf
2 f
2
}

6



Dividing both sides of this inequality by pq, we have;
[ h
1
 f
1
]  f
1 p
 [ h
2
 f
2
]  f
2 q
, or
[ h
1
 f
1
]  f
1  [ h
2
 f
2
]  f
2 ......(4)
P ( E | H
1
) P ( E | H
2
)
To make sense out of this result, I will first note again what L
E*
> L
E
means. In words, this expression says that there’s more inferential force in evidence about event E than there would be if we knew for sure that event E occurred. There’s another formal result that we can draw upon to see what’s happening. Let’s take a situation in which, for some source and any hypothesis H,
P(E*|EH) > P(E*|E).. In words, this expression says that a source's hit probability is conditional upon hypothesis H. In fact this hit probability is greater, in light of H, than it would be if we did not consider H. When this is so, we have the following:
P ( E * | EH )

P ( E * | E )
P ( E * EH )

P ( E * E )
P ( EH ) P ( E )
P ( E * EH )

P ( EH )
P ( E * E ) P ( E )
P ( H | E * E )

P ( H | E ).
What this says is that, when P(E*|EH) > P(E*|E) for a certain source, the posterior probability of H, if we knew that event E occurred and also had evidence of event E from this source, is greater than the posterior probability of H just knowing for sure that event E occurred.
In short, there’s something extra about hearing about event E from this source. This is the major reason why we can say that what we know about some source can be valuable in addition to what the source tells us. The fact that this particular source, rather than another source, gave evidence about event E may be very important. What this says is that there can be inferential value in the source’s behavior. Bayes’ rule allows us to capture this evidential subtlety.
Let’s now return to the inequality in Equation 4 above to see what it means. The first thing we note is that whether L
E*
> L
E
depends not only upon the credibility values h
1
, f
1
, h
2
and f
2
, but also upon P(E|H
1
) and P(E|H
2
). This is really no surprise since saying that there’s more force in evidence about some event than there is in knowing that this event occurred requires us to consider how much force there is in knowing this event for sure. This is what P(E|H
1
) and P(E|H
2
) tell us. Here’s one very important fact to remember and it is that likelihoods such as P(E|H
P(E|H
2
) don’t have to sum to one. Neither do h
1
) and
1
and f
1
or h
2
and f
2
. These likelihood pairs can take on any appropriate probability values. They may both be very small, but have a very large ratio.
Looking at expression (4) above we see that there is a difference and a ratio term on each side of the inequality. As noted above, Bayes’ rule responds to both differences and ratios in its ingredients. Expression (4) is further evidence of this fact. What’s interesting about these differences and ratios is that the ratios can play a greater role than the differences in determining whether the inequality in (4 ) holds. The reason is that both [h
1
– f
1
] and [h
2
– f
2
] are bounded. Let either one of these differences be represented by D. It is true that -1.0 ≤ D ≤ 1, since the differences involve probabilities. But the two ratios f
1
/P(E|H
1
) and f
2
/P(E|H
2
) can take any positive value since the ingredients of either ratio are probabilities. So, these two ratios can play a greater role than the two differences in determining whether the inequality in (4) holds. In fact, the most
7

important ingredients in this inequality happen to be the false-positives f
1
and f
2
, as I will explain in a minute. Many persons may believe that the role played by these ingredients is counterintuitive.
What we have in Equation 2 is a very good example of how nonlinear expressions capture complexity and always produce surprises. There are six variables in Equation 2 and we can set any of them between zero and one. Even these simple expressions produce surprises and illustrate interesting interactions that appear on both sides of the inequality in (4). For example, suppose we wish to make the left-hand side large by making the difference [h
1
– f
1
] large. So, we set h
1
= 0.9 and f
1
= 0.01. In this case [h
1
– f
1
] = 0.89. But, doing so requires that
P(E|H
1
) be very small [less than 0.01] for the ratio f
1
/P(E|H
1
) to be > 1.0. In short, setting the difference terms influences the ratio terms and vice versa.
On the Importance of False-Positives
Before I provide some examples of the workings of Equation 2 and the inequality in (4), I need to dwell a bit longer on the inferential importance of the false-positive probabilities f
1
=
P(E*|E c H
1
) and f
2
= P(E*|E c H
2
). For many years radiologists, as well as other persons such as polygraphers who use various sensing devices, only talked about "hits", but rarely about "falsepositives". Recognition of the importance of false-positives came only slowly in radiology; it does not yet appear to be of importance in the field of polygraphy [a possible reason why polygraph evidence is not admissible in our courts]. A high hit rate is of course desirable in influencing the force of evidence, but it loses its influence when a corresponding false-positive rate is also large.
Equation 3 makes this quite clear. Observe that when h = f in this equation, L
E*
= 1.0; i.e. the report E* has no value at all in discriminating between H
1
and H
2
. But in determining conditions under which L
E*
> L
E
, the major problem addressed in this report, observe that it makes all the difference how probable is the actual occurrence of the event reported. The reason concerns the relative size of the two ratios: f
1
/P(E|H
1
) and f
2
/P(E|H
2
). As noted above, these two ratio terms have a greater influence on whether L
E*
> L
E
, than do the difference terms [h
1
- f
1
] and [h
2
- f
2
].
Notice especially that either of these two ratios would increase in size as we make event E rarer or more improbable under H
1
and/or H
2
.
But I have not yet answered the question concerning why the false-positive probabilities are so important in determining whether the inequality in (4) will hold and L
E*
> L
E
. Answering this question requires attention to the two linkages involving evidence E* shown in Figure 2. First consider the linkage involving E* and the events {E, E c }. What this credibility-related linkage says is that the radiologist's report E* depends on whether E occurred or not. The strength of this linkage, by itself, is given by h = P(E*|E) and f = P(E*|E c ). What I have shown in other places is that this linkage can be decomposed to reveal three major attributes of the credibility of human sources of testimony: veracity, objectivity and observational sensitivity [Schum, 1989; 1991; 1994,
100-108]. In considering veracity we ask whether the human source believes what he/she reported to us. This source is untruthful if this person reports an event he/she does not believe to have occurred. In considering objectivity, we ask whether the source's beliefs about what was observed were based on sensory evidence rather than upon what this person either expected or desired to observe. In considering observational sensitivity we ask how good might the sensory evidence have been if the source believed his/her senses. What happens is that deficiencies in any of these attributes gives rise to increases in the false-positive probability f = P(E*|E c ), regardless of what h = P(E*|E) might be. This is a major reason why just considering hit probabilities and not false-positive probabilities can be so devastating in inferences based on testimonial evidence.
Now consider the additional linkage in Figure 2 between evidence E* and hypotheses
{H
1
, H
2
}. Even though I have not decomposed the linkage between E* and {E, E c } in this figure and in Equation 2, issues of the source's veracity, objectivity and observational sensitivity are still there and exist in the background, so to speak. What this additional linkage allows us to capture are situations in which a source's veracity, objectivity and observational sensitivity may in fact
8

depend on whether H
1
or H
2 is true. remember that L
E*
in Equation 2 concerns the evidence given by the source. In the abstract, we can all think of many reasons why a person's veracity, objectivity or observational sensitivity might be dependent on whether H
1
or H
2
is true. In the present example, however, we might not ordinarily have any reason to question the veracity of the radiologist; but we might have reason to question his objectivity and observational sensitivity.
Suppose it happens that the detection of the anomaly is more difficult for patients who have tuberculosis than for patients who have pneumonia. Or, suppose that the radiologist has more often expects to see the anomaly for patients who he expects to have tuberculosis than for patients who he expects to have have pneumonia.
In its further wisdom, Bayes' rule gives us the opportunity to capture these additional
"state-dependent" characteristics of the credibility of sources of testimonial evidence. As argued above, the state-dependence of false-positive probabilities is additional information about the source of testimony and, as (4) shows, when these false positives are large in comparison with likelihoods for the event reported, then there is extra inferential force associated with the source's testimony. What may be counterintuitive to many persons is the greater importance of falsepositive probabilities than of hit probabilities in determining whether the inequality in Equation 4 holds. It is always advisable to note that formal expressions such as Bayes' rule know only numbers and how to combine them. Interpretations of the results these expressions provide are left up to us. There is no formal expression in any area that is guaranteed to provide results that will always match what we expect intuitively.
4.0 SOME EXAMPLES IN WHICH L
E*
> L
E
.
It is not easy to see any general patterns that emerge in nonlinear expressions, such as
Equation 2, that have six variables. At least I have not been successful in seeing how this might be done. However, we can always resort to sensitivity analyses in which we hold certain variables constant and see how changes in other variables affect the result. Another description for this approach is to say that we are telling alternative stories or presenting different scenarios by inserting different combinations of probabilities. The virtue of having formal expressions such as
Equations 2 and 4 above is that we can determine how each story we tell should end. So, here is a collection of ten stories that tell us a variety of different things about conditions under which L
E*
> L
E
. In each of these cases it will be true that L
E
= 5, for the known occurrence of event E.
Case P(E|H
1
) P(E|H
2
) h
1
1 0.20 0.04 0.90
f
1
0.01
h
2
0.90
f
2
0.01
L
E*
4.123
(4) LH
0.94
(4) RH
1.14
2
3
0.02
0.20
0.004
0.04
0.90
0.90
0.01
0.40
0.90
0.90
0.01
0.01
2.050
9.174
1.39
2.50
3.39
1.09
4
5
6
7
0.02
0.80
0.08
0.20
0.004 0.90 0.40 0.90 0.01 28.374 20.5
0.16 0.90 0.40 0.90 0.01 5.249 1.14
0.016 0.90 0.40 0.90 0.01 18.152 5.50
0.04 0.90 0.01 0.90 0.40 0.448* 0.94
2.89
0.95
1.52
10.50
8
9
0.02
0.20
0.004 0.90 0.01 0.90 0.40 0.069* 1.39
0.04 0.90 0.01 0.50 0.48 0.391* 0.94
100.5
12.02
10 0.02 0.004 0.90 0.01 0.50 0.48 0.058* 1.39 24.02
Following is an explanation of each of these ten cases. All but the last two headings of the columns are self-explanatory, they are the ingredients of Equation 2 and its result L
E*.
The last two columns on the right show the left-hand and right-hand results of the inequality in (4).
Case 1 . This is just a "base-line" case showing what happens when hits and false-positives are not conditional upon either H
1
or H
2
, i.e. Equation 3 applies. As you see, L
E*
= 4.123 < L
E
= 5.0 as we expect since L
E*
is bounded above by L
E
in this situation.
9

Case 2 . Here is the example promised that shows how event rareness interacts aith source credibility. First observe that I have made event E ten times less probable under H
1
and H
2
than it was in Case 1. In this case, L
E*
= 2.050 is smaller in value from what it was in Case 1. For fixed values of h and f, the rarer we make an event the smaller is the likelihood ratio for evidence about this event.
Case 3 . The only change made in this case is to increase the value of f
1
to 0.4; all other ingredients remain the same as in the first two cases. In this case, some combination of defects in the credibility attributes of the source is conditional on H
1
being true. This causes L
E*
= 9.174 to have nearly twice the force of knowing event E for sure. As the last two columns verify, the inequality in Expression 4 holds in this case.
Case 4 . This case is Case 3 repeated with just one change; I have made event E ten times rarer under both H
1
and H
2
. As I noted above, this will act to intensify the false positive probability in its conditionally nonindependent effects on L
E*
. In this case, L
E*
= 28.374 is now considerably greater than its value in Case 3. All such examples show the value of Equation 2 in capturing the interaction between event rareness and credibility-related ingredients. Also observe in the last two columns the increased value of the left-hand size of the inequality in (4) above.
Case 5 . Here I examine a case in which the event being reported has larger values of both
P(E|H
1
) and P(E|H
2
) but when their ratio is still L
E
= 5.0. In this case, the ratio f
1
/P(E|H
1
) in (4) will be smaller because of the increase in the size of P(E|H
1
). It is still true in this case that L
E*
> L
E
, but not by much.
Case 6. The same as in Case 5 except that I have made event E ten times rarer under both H1 and H
2
. This enhances the value of L
E*
over L
E
, as you see.
Case 7 . This case is the same as Case 3 except that I have conditioned the high false-positive probability on H
2
rather than upon H
1
. In this case L
E*
= 0.448 = 1/2.232, meaning that the source's report that event E occurred actually favors H
2
rather than H
1
. In all of may examples I have had the known occurrence of E favoring H
1
over H
2
with the result that the known nonoccurrence of E favors H
2
over H
1
. Notice here that the value of knowing E c for sure is L
E c =
(1 - 0.2)/(1 - 0.04) = 0.8/0.96 = 0.833 = 1/1.2. So, the source's report E* favors H
2
over H
1
in a ratio that is greater then it would be if we knew for sure that E did not happen. In times past, I have referred to such instances as "impact reversals", cases in which the event reported favors one hypothesis and the report of the event favors another hypothesis. All the asterisked L
E* values in Cases 7 - 10 have this characteristic.
Case 8 . The result in Case 7 is enhanced here as I make event E ten time less probable under both hypotheses than it is in Case 7. The result is that the source's report E* favors H
2
over H
1
in the ratio L
E*
= 0.069 = 1/14.493. This case would be very pleasing to the person in my example
[you], who has been told that the radiologist said he detected an anomaly that favored your having tuberculosis by a factor of 5. His report in this case favors your having pneumonia over tuberculosis in the ratio 14.493:1.
Case 9 . In this case I have reduced the source's hit probability and increased his false-positive probability, conditional on H
2
[pneumonia, in the example]. In this case, the report favors H2 over
H1 in the ratio L
E*
= 0.391 = 1/2.558. The source's report here is again stronger evidence of H
2 than knowing E c for sure.
Case 10 . This is Case 9 over again except that I have made event E rare under both H
1
and H
2
.
The result is that the source's saying E* that event E occurred favors H
2
over H
1
in the ratio L
E*
=
0.058 = 1/17.241. This would also be very pleasing to you to learn that the radiologists report of
E* actually favors your having pneumonia over tuberculosis in the ratio in excess of 17:1.
10

I could go on contriving interesting cases in which there is more inferential force in a source's report of an event than there is in the known occurrence of this event. These cases are probably sufficient to illustrate the importance of false-positive probabilities and the rareness of events reported in determining whether or not this will happen.
5.0 FINAL WORDS :
Bayes' rule contains within it a remarkable device, conditional nonindependence, for capturing evidential and inferential subtleties. Many of these subtleties lurk just below the surface of even the simplest of inferential situations. The situation I have posed in this report is about as simple as one could hope for. Many years ago at Rice University my esteemed faculty mentor
Paul Pfeiffer reminded me that simple things are not necessarily trivial things. I see this every day in my studies of evidence when I apply Bayesian ideas in attempts to capture evidential and inferential subtleties for study and analysis.
REFERENCES
Daston, L. Classical Probability in the Enlightenment . Princeton University Press, 1988
Hume, D. [1777]. Enquiries Concerning Human Understanding . Ed. P. Nidditch, 3rd ed.
Clarendon Press, Oxford, 1989
Kolmogorov, A. N. [1933]. Foundations of a Theory of Probability. 2nd English ed. Chelsea
Publishing Co.,NY. 1956
Schum, D. Evidential Foundations of Probabilistic Reasoning . Wiley & Sons, NY, 1994;
Northwestern University Press, 2001
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