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OCR for page 35
APPENDI X ~
CRITICISM OF PROBABILITY CALCULATIONS
Criticism of BRSW Probabilities 0. Be, 0. 88, 0. 50, and 0.75
35
This criticism refers to the calculation by BRSW of probabilities of
0. S8, O. S8, O. 50, and 0.75 for the identification of 4 impulse patterns of
the DPD tapes as representing four shots. The third pattern is associated
with the con Lectured shot from the grassy knoll . These claimed
probabilities are, at the very least, larger than the BRSW reasoning should
permit, if that line of reasoning were to be accepted.
In Append ix C, pages C 1 -C2, of the BRSW reports, it is calculated
that the 432 x 4 correlation matches of the 432 echo patterns ~ derived from
the test shots ~ with the four impulse patterns ~ on the DPD tapes that were
suspected of being patterns from shots ~ should give an expected 13 fal se
alarms. BRSW found 15 matches, which is within reason . (See Section 5.3
and Figure 22 on pages 60-63 of that report ~ . BRSW applied a 2 x 2
contingency table test based on the data in their figure 22 to decide that
the matches are not randomly located and consequently at least two of
those matches must be real. (See page 66 of BRSW).
We shall comment on this conclusion later, but let us grant this
conclusion for the time being. BRSW observe that six of the matches are
clearly fal se alarms . This leaves nine points of which at least two are
true matches. We quote BRSW on page 66.
"However, the expected number of false alarms to
be found when testing four different impulse patterns
is 13 (see Appendix C), and only six have been found.
Therefore, it is not unreasonable to expect that there
are seven more, although that would be the largest
number possible since at least two of the remaining nine
OCR for page 36
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are probably detections. The best that can be safely
assumed is that each of the nine remaining correlations
is equally likely to represent a detection or a false
alarm . t' 1
The intended meaning of that last sentence is that each of the nine
candidates has as large a probability of being a true match as any other
candidate. We would suggest that a conservative estimate of this
probability based on the BRSW reasoning would be 2/9. Instead, BRSW
apparently misinterpreted their own words, reading "equally likely" to mean
that the probability of a false alarm for each detection is 1/2, an
interpretation consistent with the somewhat ambiguous language but not with
the reasoning.
From their interpretation they proceed to calculate the probabilities
0.88 = 1-(1/2)3, 0.88 = 1_(1/2)3, 0.5 = 1-1/2, and 0.75 = 1_(1/2)2 on
p. 67, using the probability 0.5, the questionable assumption of
independence and the fact that of the remaining nine matches which are not
obviously false alarms three correspond to the first conjectured shot,
three to the second, one to the third (grassy knoll) and two to the fourth
(see Figure 22 of BRSW report).
Using 2/9 instead of 1/2 would give probabilities 1-(7/9)3 = 0.53,
1-(7/9)3 = 0.53, 1-(7/9) = 0.22, and 1_(7/9)2 = 0.40. These are
considerably smaller than the BRSW probabilities.
One may argue that the above calculations are too conservative and
that the probability 2/9 should be replaced by a larger number, say 3/9 or
4/9. Certainly the use of 4/9 would be unduly "optimistic" since there are
at most four shots.
OCR for page 37
A-2. Criticism of BRSW Certainty that Microphone Detected Sound of
Gunfire (p. 64)
37
One must question the reasoning that led to the inference that at
least two of the alarms were "true". The 2x2 chi-square contingency table
analysis used depends on the assumption of 15 independently located alarms.
Those alarms corresponded to microphone and rifle locations closely grouped
in space. Hence the signals are similar and one should expect that the
high correlation coefficients for these alarms are highly dependent events.
An informal study of these locations and Figure 22 would suggest that these
15 alarms are effectively equivalent to fewer, say about 7 or 8,
independent points. In that case the significance of the layout is
considerably reduced and one may question the conclusion on page 64 of the
BRSW report that "the motorcycle was moving through Dealey Plaza and did,
in fact, detect the sounds of gunfire." (To be specific, a two by two
table with entries 1,3,4,0 yields a significance value P = 0.07, rather
than the P<0.01 claimed by BRSW.)
OCR for page 38
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A-3.
Criticism of BRSW/WA Probability of 0.95 for Shot from Grassy Knoll
The BRSW/WA conclusion of a 95% probability of a shot or loud noise
from the grassy knoll fails to be convincing because (i) the use of
subjective procedures which easily lead one to unconsciously biased
reporting and make it difficult to reproduce the results by independent
observers, (ii) serious errors in statistical reasoning which render the
calculated probabilities meaningless, and (iii) the failure to apply the WA
methodology to the other suspected shots which leaves the method
insufficently tested and calibrated.
We elaborate here on (ii). First the BRSW calculations on page 75,
using the claim of coincidence of 10 out of 12 predicted echoes with 10 of
the 14 impulses on the DPD tape that exceeded a threshold, should conclude
that the probability of observing 10 or more coincidences is less than
0.053 under a Poisson randomness hypothesis. It does not follow that
the Poisson randomness hypothesis has probability less than 0.053. Such a
conclusion is no more valid than it would be to conclude that the dealer of
a bridge hand who deals himself 3 aces on the first deal is dishonest with
probability 0.96. This first inference would require the application of
Ba yes Theorem using prior probabilities for all plausible hypotheses, and
the probabilities of the observation of 10 matches under each of these
plausible hypotheses. In the case of the bridge dealer, a probability
assignment to the hypothesis of honesty would depend on one's prior belief
in the dealer's honesty and would also involve the calculation of the
probability of his dealing himself 3 aces on the first deal if he were
dishonest.
Second, the calculation of 0.053, which is typically called the
significance level or P value, should have included an adjustment to allow
for the fact that the hypothesis of a shot from the grassy knoll involved 7
"free" parameters that were adjusted to maximize the number of
coincidences. These parameters were time of shot, position of shooter (2),
initial position of motorcycle (2), velocity of motorcycle and velocity of
sound combined with tape recorder speed. The adjustment used by BR SW/WA is
somewhat ad hoc, depends mainly on adjusting three of these parameters
OCR for page 39
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(initial position of motorcycle and time of shot), and requires review.
That adjustment consists approximately of reducing the number of matches by
one and multiplying the computed probability (3.13 x 10~4) of as good a
match by a factor of 180. The latter factor was based on the number of
plausible initial positions of the motorcycle that could have led to
different results in matching predicted echoes against observed impulses.
But if they had chosen to consider different plausible values for the other
parameters, their reasoning would have produced a greater factor. For
example, the 50-foot range of plausible shooter positions along the fence
could contribute an additional factor of 5 to 10, since movements of more
than +5 feet could change the relative positions of the predicted
echoes by substantially more than could be compensated for by readjusting
the initial position of the microphone (see page 29 of WA). A more
traditional and possibly overconservative adjustment would consist of
subtracting one match for each free parameter. This adjustment would lead
to a less significant result (high P value).
Third, alternative hypotheses to the two primarily considered
(gunshots or random locations of impulses according to a Poisson process)
should have been considered, such as non-white (non-Poisson) noise and
static. Such distributions could increase the likelihood of the BRSW and
WA results having been obtained by chance.
Fourth, the calculation of 0.053 involved some further errors. On
page 75, the BRSW calculations claim that 12 of 22 predicted echoes were
loud enough to exceed a threshold. (It seems that 22 should be 26 if Table
4 on p. 27 of WA is the source.) Then 10 of these 12 predicted echoes
occurred within +1 msec. of the occurrence of 10 of 14 impulses on the
DPD tape that were loud enough to exceed a threshold. Later there is
reference to two time intervals of 90 msec total duration, representing
forty-five 2-msec. windows.
The time should be 180 msec representing 90 windows. In two cases a
pair of impulses correspond to a single window. (See Figure 7, page 28 of
WA.) These are marked (19,20) and (23,24~. For the BRSW hypergeometric
probability calculations to be appropriate, it is necessary to use
OCR for page 40
40
non-overlapping windows and to count as coincidences the number of windows
in which there are at least one predicted echo and at least one observed
impulse. Moreover, two of the predicted echoes appear in one window. Thus
10 coincidences among 12 predicted echoes and 14 impulses out of 45 windows
should be adjusted to ~ coincidences among 11 predicted echoes and 12
impulses out of 90 windows. If we now reduce the number of concidences by
7 to make a conservative adjustment for the "free" parameters, we would
have 1 coincidence among 4 predicted echoes and 5 impulses out of 83
windows. The hypergeometric probability calculation would then yield a
significance level of P = 0.223, which is not at all impressive in contrast
to the claim that P = 0.053. However, this adjustment may be unduly
conservative.
(4)
In summary,
(1) The BRSW/WA conclusion of a probability of 0.5 of a shot from the
grassy knoll on the basis of the BRSW analysis is invalid as is
also the conclusion of a probability of 0.95 for such a shot on
the basis of the WA analysis.
(2) There are several inaccuracies.
(3) Except for the rather conservative analysis above, the data do tend
to cast doubt on the hypothesis of random impulse locations according
to a Poisson process.
Alternative hypotheses to a random Poissson process and a shot should
have been examined as possible explanations of the coincidences.
These might invoke the nature of the bursts of noise prevalent during
the period under study and a consideration of other possible
non-Poisson distributions.
Representative terms from entire chapter:
grassy knoll
