_THE APPLICATION OF PROBABILITY TO TESTIMONY._
1. On the principles which have been adopted in this
work, it becomes questionable whether several classes of
problems which may seem to have acquired a prescriptive
right to admission, will not have to be excluded from the
science of Probability. The most important, perhaps, of
these refer to what is commonly called the credibility of
testimony, estimated either at first hand and directly, or as
influencing a juryman, and so reaching us through his
sagacity and trustworthiness. Almost every treatise upon
the science contains a discussion of the principles according
to which credit is to be attached to combinations of the
reports of witnesses of various degrees of trustworthiness, or
the verdicts of juries consisting of larger or smaller numbers.
A great modern mathematician, Poisson, has written an
elaborate treatise expressly upon this subject; whilst a considerable
portion of the works of Laplace, De Morgan, and
others, is devoted to an examination of similar enquiries. It
would be presumptuous to differ from such authorities as
these, except upon the strongest grounds; but I confess that
the extraordinary ingenuity and mathematical ability which
have been devoted to these problems, considered as questions
in Probability, fails to convince me that they ought to have
been so considered. The following are the principal grounds
for this opinion.
2. It will be remembered that in the course of the
chapter on Induction we entered into a detailed investigation
of the process demanded of us when, instead of the
appropriate propositions from which the inference was to be
made being set before us, the _individual_ presented himself,
and the task was imposed upon us of selecting the requisite
groups or series to which to refer him. In other words, instead
of calculating the chance of an event from determinate
conditions of frequency of its occurrence (these being either
obtained by direct experience, or deductively inferred) we
have to _select_ the conditions of frequency out of a plurality
of more or less suitable ones. When the problem is presented
to us at such a stage as this, we may of course assume
that the preliminary process of obtaining the statistics
which are extended into the proportional propositions has
been already performed; we may suppose therefore that we
are already in possession of a quantity of such propositions,
our principal remaining doubt being as to which of them
we should then employ. This selection was shown to be
to a certain extent arbitrary; for, owing to the fact of the
individual possessing a large number of different properties,
he became in consequence a member of different series or
groups, which might present different averages. We must
now examine, somewhat more fully than we did before,
the practical conditions under which any difficulty arising
from this source ceases to be of importance.
3. One condition of this kind is very simple and obvious.
It is that the different statistics with which we are
presented should not in reality offer materially different
results, If, for instance, we were enquiring into the probability
of a man aged forty dying within the year, we might
if we pleased take into account the fact of his having red
hair, or his having been born in a certain county or town.
Each of these circumstances would serve to specialize the
individual, and therefore to restrict the limits of the statistics
which were applicable to his case. But the consideration of
such qualities as these would either leave the average precisely
as it was, or produce such an unimportant alteration
in it as no one would think of taking into account. Though
we could hardly say with certainty of any conceivable
characteristic that it has absolutely no bearing on the
result, we may still feel very confident that the bearing of
such characteristics as these is utterly insignificant. Of
course in the extreme case of the things most perfectly
suited to the Calculus of Probability, viz. games of pure
chance, these subsidiary characteristics are quite irrelevant.
Any further particulars about the characteristics of the cards
in a really fair pack, beyond those which are familiar to all
the players, would convey no information whatever about the
result.
Or again; although the different sets of statistics may
not as above give almost identical results, yet they may do
what practically comes to very much the same thing, that
is, arrange themselves into a small number of groups, all of
the statistics in any one group practically coinciding in their
results. If for example a consumptive man desired to insure
his life, there would be a marked difference in the statistics
according as we took his peculiar state of health into account
or not. We should here have two sets of statistics, so clearly
marked off from one another that they might almost rank
with the distinctions of natural kinds, and which would in
consequence offer decidedly different results. If we were
to specialize still further, by taking into account insignificant
qualities like those mentioned in the last paragraph, we
might indeed get more limited sets of statistics applicable
to persons still more closely resembling the individual in
question, but these would not differ sufficiently in their
results to make it worth our while to do so. In other words,
the different propositions which are applicable to the case in
point arrange themselves into a limited number of groups,
which, and which only, need be taken into account; whence
the range of choice amongst them is very much diminished
in practice.
4. The reasons for the conditions above described are
not difficult to detect. Where these conditions exist the
process of selecting a series or class to which to refer any
individual is very simple, and the selection is, for the particular
purposes of inference, final. In any case of insurance,
for example, the question we have to decide is of the very
simple kind; Is A. B. a man of a certain age? If so one in
fifty in his circumstances will die in the course of the year.
If any further questions have to be decided they would be of
the following description. Is A. B. a healthy man? Does he
follow a dangerous trade? But here too the classes in
question are but few, and the limits by which they are
bounded are tolerably precise; so that the reference of an
individual to one or other of them is easy. And when we
have once chosen our class we remain untroubled by any
further considerations; for since no other statistics are supposed
to offer a materially different average, we have no
occasion to take account of any other properties than those
already noticed.
The case of games of chance, already referred to, offers
of course an instance of these conditions in an almost ideal
state of perfection; the same circumstances which fit them
so eminently for the purposes of fair gambling, fitting them
equally to become examples in Probability. When a die is
to be thrown, all persons alike stand on precisely the same
footing of knowledge and of ignorance about the result; the
only data to which any one could appeal being that each
face turns up on an average once in six times.
5. Let us now examine how far the above conditions
are fulfilled in the case of problems which discuss what is
called the credibility of testimony. The following would be
a fair specimen of one of the elementary enquiries out of
which these problems are composed;--Here is a statement
made by a witness who lies once in ten times, what am I to
conclude about its truth? Objections might fairly be raised
against the possibility of thus assigning a man his place
upon a graduated scale of mendacity. This however we will
pass over, and will assume that the witness goes about the
world bearing stamped somehow on his face the appropriate
class to which he belongs, and consequently, the degree of
credit to which he has a claim on such general grounds.
But there are other and stronger reasons against the admissibility
of this class of problems.
6. That which has been described in the previous
sections as the 'individual' which had to be assigned to an
appropriate class or series of statistics is, of course, in this
case, _a statement_. In the particular instance in question this
individual statement is already assigned to a class, that
namely of statements made by a witness of a given degree of
veracity; but it is clearly optional with us whether or not we
choose to confine our attention to this class in forming our
judgment; at least it would be optional whenever we were
practically called on to form an opinion. But in the case of
this statement, as in that of the mortality of the man whose
insurance we were discussing, there are a multitude of other
properties observable, besides the one which is supposed to
mark the given class. Just as in the latter there were
(besides his age), the place of his birth, the nature of his
occupation, and so on; so in the former there are (besides its
being a statement by a certain kind of witness), the fact of its
being uttered at a certain time and place and under certain
circumstances. At the time the statement is made all these
qualities or attributes of the statement are present to us, and
we clearly have a right to take into account as many of them
as we please. Now the question at present before us seems to
be simply this;--Are the considerations, which we might thus
introduce, as immaterial to the result in the case of the truth
of a statement of a witness, as the corresponding considerations
are in the case of the insurance of a life? There can
surely be no hesitation in the reply to such a question.
Under ordinary circumstances we soon know all that we can
know about the conditions which determine us in judging of
the prospect of a man's death, and we therefore rest content
with general statistics of mortality; but no one who heard a
witness speak would think of simply appealing to his figure
of veracity, even supposing that this had been authoritatively
communicated to us. The circumstances under which the
statement is made instead of being insignificant, are of overwhelming
importance. The appearance of the witness, the
tone of his voice, the fact of his having objects to gain,
together with a countless multitude of other circumstances
which would gradually come to light as we reflect upon the
matter, would make any sensible man discard the assigned
average from his consideration. He would, in fact, no more
think of judging in this way than he would of appealing to
the Carlisle or Northampton tables of mortality to determine
the probable length of life of a soldier who was already in
the midst of a battle.
7. It cannot be replied that under these circumstances
we still refer the witness to a class, and judge of his veracity
by an average of a more limited kind; that we infer, for example,
that of men who look and act like him under such
circumstances, a much larger proportion, say nine-tenths,
are found to lie. There is no appeal to a class in this way at
all, there is no immediate reference to statistics of any kind
whatever; at least none which we are conscious of using at
the time, or to which we should think of resorting for justification
afterwards. The decision seems to depend upon the
quickness of the observer's senses and of his apprehension
generally.
Statistics about the veracity of witnesses seem in fact to
be permanently as inappropriate as all other statistics occasionally
may be. We may know accurately the percentage
of recoveries after amputation of the leg; but what surgeon
would think of forming his judgment solely by such tables
when he had a case before him? We need not deny, of
course, that the opinion he might form about the patient's
prospects of recovery might ultimately rest upon the proportions
of deaths and recoveries he might have previously witnessed.
But if this were the case, these data are lying, as
one may say, obscurely in the background. He does not
appeal to them directly and immediately in forming his
judgment. There has been a far more important intermediate
process of apprehension and estimation of what is
essential to the case and what is not. Sharp senses, memory,
judgment, and practical sagacity have had to be called into
play, and there is not therefore the same direct conscious
and sole appeal to statistics that there was before. The
surgeon may have in his mind two or three instances in
which the operation performed was equally severe, but in
which the patient's constitution was different; the latter
element therefore has to be properly allowed for. There may
be other instances in which the constitution was similar, but
the operation more severe; and so on. Hence, although the
ultimate appeal may be to the statistics, it is not so directly;
their value has to be estimated through the somewhat hazy
medium of our judgment and memory, which places them
under a very different aspect.
8. Any one who knows anything of the game of whist
may supply an apposite example of the distinction here
insisted on, by recalling to mind the alteration in the nature
of our inferences as the game progresses. At the commencement
of the game our sole appeal is rightfully made to the
theory of Probability. All the rules upon which each player
acts, and therefore upon which he infers that the others will
act, rest upon the observed frequency (or rather upon the
frequency which calculation assures us will be observed) with
which such and such combinations of cards are found to
occur. Why are we told, if we have more than four trumps,
to lead them out at once? Because we are convinced, on
pure grounds of probability, capable of being stated in the
strictest statistical form, that in a majority of instances we
shall draw our opponent's trumps, and therefore be left with
the command. Similarly with every other rule which is
recognized in the early part of the play.
But as the play progresses all this is changed, and
towards its conclusion there is but little reliance upon any
rules which either we or others could base upon statistical
frequency of occurrence, observed or inferred. A multitude
of other considerations have come in; we begin to be influenced
partly by our knowledge of the character and
practice of our partner and opponents; partly by a rapid
combination of a multitude of judgments, founded upon
our observation of the actual course of play, the grounds
of which we could hardly realize or describe at the time
and which may have been forgotten since. That is, the
particular combination of cards, now before us, does not
readily fall into any well-marked class to which alone it can
reasonably be referred by every one who has the facts before
him.
9. A criticism somewhat resembling the above has
been given by Mill (_Logic,_ Bk. III. Chap. XVIII. §3) upon
the applicability of the theory of Probability to the credibility
of witnesses. But he has added other reasons which
do not appear to me to be equally valid; he says "common
sense would dictate that it is impossible to strike a general
average of the veracity, and other qualifications for true
testimony, of mankind or any class of them; and if it were
possible, such an average would be no guide, the credibility of
almost every witness being either below or above the average,"
The latter objection would however apply with equal force
to estimating the length of a man's life from tables of mortality;
for the credibility of different witnesses can scarcely
have a wider range of variation than the length of different
lives. If statistics of credibility could be obtained, and
could be conveniently appealed to when they were obtained,
they might furnish us in the long run with as accurate
inferences as any other statistics of the same general description.
These statistics would however in practice naturally
and rightly be neglected, because there can hardly fail
to be circumstances in each individual statement which would
more appropriately refer it to some new class depending on
different statistics, and affording a far better chance of our
being right in that particular case. In most instances of
the kind in question, indeed, such a change is thus produced
in the mode of formation of our opinion, that, as already
pointed out, the mental operation ceases to be in any proper
sense founded on appeal to statistics.[1]
10. The Chance problems which are concerned with testimony
are not altogether confined to such instances as those
hitherto referred to. Though we must, as it appears to me,
reject all attempts to estimate the credibility of any particular
witness, or to refer him to any assigned class in
respect of his trustworthiness, and consequently abandon as
unsuitable any of the numerous problems which start from
such data as 'a witness who is wrong once in ten times,'
yet it does not follow that testimony may not to a slight
extent be treated by our science in a somewhat different
manner. We may be quite unable to estimate, except in
the roughest possible way, the veracity of any particular
witness, and yet it may be possible to form some kind of
opinion upon the veracity of certain classes of witnesses;
to say, for instance, that Europeans are superior in this way
to Orientals. So we might attempt to explain why, and to
what extent, an opinion in which the judgments of ten persons,
say jurors, concur, is superior to one in which five only
concur. Something may also be done towards laying down
the principles in accordance with which we are to decide
whether, and why, extraordinary stories deserve less credence
than ordinary ones, even if we cannot arrive at any precise
and definite decision upon the point. This last question is
further discussed in the course of the next chapter.
11. The change of view in accordance with which it
follows that questions of the kind just mentioned need not
be entirely rejected from scientific consideration, presents itself
in other directions also. It has, for instance, been already
pointed out that the individual characteristics of any sick
man's disease would be quite sufficiently important in most
cases to prevent any surgeon from judging about his recovery
by a genuine and direct appeal to statistics, however such
considerations might indirectly operate upon his judgment.
But if an opinion had to be formed about a considerable
number of cases, say in a large hospital, statistics might
again come prominently into play, and be rightly recognized
as the principal source of appeal. We should feel able to
compare one hospital, or one method of treatment, with
another. The ground of the difference is obvious. It arises
from the fact that the characteristics of the individuals,
which made us so ready to desert the average when we had
to judge of them separately, do not produce the same disturbance
when were have to judge about a group of cases.
The averages then become the most secure and available
ground on which to form an opinion, and therefore Probability
again becomes applicable.
But although some resort to Probability may be admitted
in such cases as these, it nevertheless does not
appear to me that they can ever be regarded as particularly
appropriate examples to illustrate the methods and resources
of the theory. Indeed it is scarcely possible to resist the
conviction that the refinements of mathematical calculation
have here been pushed to lengths utterly unjustifiable, when
we bear in mind the impossibility of obtaining any corresponding
degree of accuracy and precision in the data from
which we have to start. To cite but one instance. It would
be hard to find a case in which love of consistency has prevailed
over common sense to such an extent as in the admission
of the conclusion that it is unimportant what are
the numbers for and against a particular statement, provided
the actual majority is the same. That is, the unanimous
judgment of a jury of eight is to count for the same as a
majority of ten to two in a jury of twelve. And yet this
conclusion is admitted by Poisson. The assumptions under
which it follows will be indicated in the course of the next
chapter.
Again, perfect independence amongst the witnesses or
jurors is an almost necessary postulate. But where can this
be secured? To say nothing of direct collusion, human
beings are in almost all instances greatly under the influence
of sympathy in forming their opinions. This influence, under
the various names of political bias, class prejudice, local
feeling, and so on, always exists to a sufficient degree to
induce a cautious person to make many of those individual
corrections which we saw to be necessary when we were
estimating the trustworthiness, in any given case, of a single
witness; that is, they are sufficient to destroy much, if not
all, of the confidence with which we resort to statistics and
averages in forming our judgment. Since then this Essay is
mainly devoted to explaining and establishing the general
principles of the science of Probability, we may very fairly
be excused from any further treatment of this subject, beyond
the brief discussions which are given in the next chapter.
1. It may be remarked also that there is another reason which tends to
dissuade us from appealing to principles of Probability in the
majority of the cases where testimony has to be estimated. It often,
perhaps usually happens, that we are not absolutely forced to come
to a decision; at least so far as the acquitting of an accused
person may be considered as avoiding a decision. It may be of much
greater importance to us to attain not merely truth on the average,
but truth in each individual instance, so that we had rather not
form an opinion at all than form one of which we can only say in its
justification that it will tend to lead us right in the long run.