_THE RULES OF INFERENCE IN PROBABILITY._
1. In the previous chapter, an investigation was made into
what may be called, from the analogy of Logic, Immediate
Inferences. Given that nine men out of ten, of any assigned
age, live to forty, what could be inferred about the prospect
of life of any particular man? It was shown that, although
this step was very far from being so simple as it is frequently
supposed to be, and as the corresponding step really is in
Logic, there was nevertheless an intelligible sense in which
we might speak of the amount of our belief in any one of
these 'proportional propositions,' as they may succinctly be
termed, and justify that amount. We must now proceed to
the consideration of inferences more properly so called, I
mean inferences of the kind analogous to those which form the
staple of ordinary logical treatises. In other words, having
ascertained in what manner particular propositions could be
inferred from the general propositions which included them,
we must now examine in what cases one general proposition
can be inferred from another. By a general proposition here
is meant, of course, a general proposition of the statistical
kind contemplated in Probability. The rules of such inference
being very few and simple, their consideration will not
detain us long. From the data now in our possession we are
able to deduce the rules of probability given in ordinary
treatises upon the science. It would be more correct to say
that we are able to deduce _some_ of these rules, for, as will
appear on examination, they are of two very different kinds,
resting on entirely distinct grounds. They might be divided
into those which are formal, and those which are more or less
experimental. This may be otherwise expressed by saying
that, from the kind of series described in the first chapters,
some rules will follow necessarily by the mere application of
arithmetic; whilst others either depend upon peculiar hypotheses,
or demand for their establishment continually renewed
appeals to experience, and extension by the aid of the
various resources of Induction. We shall confine our attention
at present principally to the former class; the latter can
only be fully understood when we have considered the connection
of our science with Induction.
2. The fundamental rules of Probability strictly so
called, that is the formal rules, may be divided into two
classes,--those obtained by addition or subtraction on the
one hand, corresponding to what are generally termed the
connection of exclusive or incompatible events;[1] and those
obtained by multiplication or division, on the other hand,
corresponding to what are commonly termed dependent
events. We will examine these in order.
(1) We can make inferences by simple addition. If,
for instance, there are two distinct properties observable in
various members of the series, which properties do not occur
in the same individual; it is plain that in any batch the
number that are of one kind or the other will be equal to the
sum of those of the two kinds separately. Thus 36.4 infants
in 100 live to over sixty, 35.4 in 100 die before they are
ten;[2] take a large number, say 10,000, then there will be
about 3640 who live to over sixty, and about 3540 who do
not reach ten; hence the total number who do not die within
the assigned limits will be about 2820 altogether. Of course
if these proportions were accurately assigned, the resultant
sum would be equally accurate: but, as the reader knows, in
Probability this proportion is merely the limit towards which
the numbers tend in the long run, not the precise result
assigned in any particular case. Hence we can only venture
to say that this is the limit towards which we tend as the
numbers become greater and greater.
This rule, in its general algebraic form, would be expressed
in the language of Probability as follows:--If the
chances of two exclusive or incompatible events be respectively
1/m and 1/n the chance of one or other of them
happening will be 1/m + 1/n or (m + n)/mn. Similarly if there were
more than two events of the kind in question. On the principles
adopted in this work, the rule, when thus algebraically
expressed, means precisely the same thing as when it is
expressed in the statistical form. It was shown at the conclusion
of the last chapter that to say, for example, that the
chance of a given event happening in a certain way is 1/6, is
only another way of saying that in the long run it does tend
to happen in that way once in six times.
It is plain that a sort of corollary to this rule might be
obtained, in precisely the same way, by subtraction instead of
addition. Stated generally it would be as follows:--If the
chance of one or other of two incompatible events be 1/m and
the chance of one alone be 1/n, the chance of the remaining
one will be 1/m - 1/n or (n - m)/nm.
For example, if the chance of any one dying in a year is 1/10,
and his chance of dying of some particular disease is 1/100,
his chance of dying of any other disease is 9/100.
The reader will remark here that there are two apparently
different modes of stating this rule, according as we speak
of 'one or other of two or more events happening,' or of 'the
same event happening in one or other of two or more ways.'
But no confusion need arise on this ground; either way of
speaking is legitimate, the difference being merely verbal,
and depending (as was shown in the first chapter, §8) upon
whether the distinctions between the 'ways' are or are not
too deep and numerous to entitle the event to be conventionally
regarded as the same.
We may also here point out the justification for the common
doctrine that certainty is represented by unity, just as
any given degree of probability is represented by its appropriate
fraction. If the statement that an event happens once
in m times, is equivalently expressed by saying that its chance
is 1/m, it follows that to say that it happens m times in m times,
or every time without exception, is equivalent to
saying that its chance is m/m or 1. Now an event that happens
every time is of course one of whose occurrence we are
certain; hence the fraction which represents the 'chance' of
an event which is certain becomes unity.
It will be equally obvious that given that the chance that
an event will happen is 1/m, the chance that it will not happen
is 1 - 1/m or (m - 1)/m.
3. (2) We can also make inferences by multiplication
or division. Suppose that two events instead of being
incompatible, are connected together in the sense that one
is contingent upon the occurrence of the other. Let us be
told that a given proportion of the members of the series
possess a certain property, and a given proportion again of
these possess another property, then the proportion of the
whole which possess both properties will be found by multiplying
together the two fractions which represent the above
two proportions. Of the inhabitants of London, twenty-five
in a thousand, say, will die in the course of the year; we
suppose it to be known also that one death in five is due to
fever; we should then infer that one in 200 of the inhabitants
will die of fever in the course of the year. It would of course
be equally simple, by division, to make a sort of converse
inference. Given the total mortality per cent. of the population
from fever, and the proportion of fever cases to the
aggregate of other cases of mortality, we might have inferred,
by dividing one fraction by the other, what was the total
mortality per cent. from all causes.
The rule as given above is variously expressed in the
language of Probability. Perhaps the simplest and best
statement is that it gives us the rule of dependent events.
That is; if the chance of one event is 1/m, and the chance that
if it happens another will also happen 1/n, then the chance
of the latter is 1/mn. In this case it is assumed that the latter
is so entirely dependent upon the former that though it does
not always happen with it, it certainly will not happen without
it; the necessity of this assumption however may be
obviated by saying that what we are speaking of in the
latter case is the _joint_ event, viz. both together if they are
simultaneous events, or the latter _in consequence of_ the
former, if they are successive.
4. The above inferences are necessary, in the sense in
which arithmetical inferences are necessary, and they do not
demand for their establishment any arbitrary hypothesis.
We assume in them no more than is warranted, and in fact
necessitated by the data actually given to us, and make our
inferences from these data by the help of arithmetic. In the
simple examples given above nothing is required beyond
arithmetic in its most familiar form, but it need hardly be
added that in practice examples may often present themselves
which will require much profounder methods than
these. It may task all the resources of that higher and more
abstract arithmetic known as algebra to extract a solution.
But as the necessity of appeal to such methods as these does
not touch the principles of this part of the subject we need
not enter upon them here.
5. The formula next to be discussed stands upon a
somewhat different footing from the above in respect of its
cogency and freedom from appeal to experience, or to hypothesis.
In the two former instances we considered cases in
which the data were supposed to be given under the conditions
that the properties which distinguished the different kinds of
events whose frequency was discussed, were respectively
known to be disconnected and known to be connected. Let
us now suppose that no such conditions are given to us.
One man in ten, say, has black hair, and one in twelve
is short-sighted; what conclusions could we then draw as
to the chance of any given man having one only of these
two attributes, or neither, or both? It is clearly possible
that the properties in question might be inconsistent with
one another, so as never to be found combined in the same
person; or all the short-sighted might have black hair; or
the properties might be allotted[3] in almost any other proportion
whatever. If we are perfectly ignorant upon these
points, it would seem that no inferences whatever could be
drawn about the required chances.
Inferences however _are_ drawn, and practically, in most
cases, quite justly drawn. An escape from the apparent
indeterminateness of the problem, as above described, is
found by assuming that, not merely will one-tenth of the
whole number of men have black hair (for this was given as
one of the data), but also that one-tenth alike of those who
are and who are not short-sighted have black hair. Let us
take a batch of 1200, as a sample of the whole. Now, from
the data which were originally given to us, it will easily be
seen that in every such batch there will be on the average
120 who have black hair, and therefore 1080 who have not.
And here in strict right we ought to stop, at least until we
have appealed again to experience; but we do not stop here.
From data which we assume, we go on to infer that of the 120,
10 (i.e. one-twelfth of 120) will be short-sighted, and
110 (the remainder) will not. Similarly we infer that of the 1080,
90 are short-sighted, and 990 are not. On the whole,
then, the 1200 are thus divided:--black-haired short-sighted, 10;
short-sighted without black hair, 90; black-haired men
who are not short-sighted, 110; men who are neither short-sighted
nor have black hair, 990.
This rule, expressed in its most general form, in the
language of Probability, would be as follows:--If the chances
of a thing being p and q are respectively 1/m and 1/n, then the
chance of its being both p and q is 1/mn, p and not q is (n - 1)/mn,
q and not p is (m - 1)/mn, not p and not q is ((m - 1)(n - 1))/mn,
where p and q are independent. The sum of these chances
is obviously unity; as it ought to be, since one or other of
the four alternatives must necessarily exist.
6. I have purposely emphasized the distinction between
the inference in this case, and that in the two preceding,
to an extent which to many readers may seem unwarranted.
But it appears to me that where a science makes
use, as Probability does, of two such very distinct sources of
conviction as the necessary rules of arithmetic and the
merely more or less cogent ones of Induction, it is hardly
possible to lay too much stress upon the distinction. Few
will be prepared to deny that very arbitrary assumptions
have been made by many writers on the subject, and none
will deny that in the case of what are called 'inverse probabilities'
assumptions are sometimes made which are at least
decidedly open to question. The best course therefore is to
make a pause and stringent enquiry at the point at which the
possibility of such error and doubtfulness first exhibits itself.
These remarks apply to some of the best writers on the subject;
in the case of inferior writers, or those who appeal to
Probability without having properly mastered its principles,
we may go further. It would really not be asserting too
much to say that they seem to think themselves justified in
assuming that where we know nothing about the distribution
of the properties alluded to we must assume them to be distributed
as above described, and therefore apportion our
belief in the same ratio. This is called 'assuming the events
to be independent,' the supposition being made that the rule
will certainly follow from this independence, and that we
have a right, if we know nothing to the contrary, to assume
that the events are independent.
The validity of this last claim has already been discussed
in the first chapter; it is only another of the attempts to
construct _à priori_ the series which experience will present to
us, and one for which no such strong defence can be made as
for the equality of heads and tails in the throws of a penny.
But the meaning to be assigned to the 'independence' of the
events in question demands a moment's consideration.
The circumstances of the problem are these. There are
two different qualities, by the presence and absence respectively
of each of which, amongst the individuals of a series,
two distinct pairs of classes of these individuals are produced.
For the establishment of the rule under discussion
it was found that one supposition was both necessary and
sufficient, namely, that the division into classes caused by
each of the above distinctions should subdivide each of the
classes created by the other distinction in the same ratio
in which it subdivides the whole. If the independence be
granted and so defined as to mean this, the rule of course
will stand, but, without especial attention being drawn to
the point, it does not seem that the word would naturally
be so understood.
7. The above, then, being the fundamental rules of
inference in probability, the question at once arises, What is
their relation to the great body of formulæ which are made
use of in treatises upon the science, and in practical applications
of it? The reply would be that these formulæ, in so
far as they properly belong to the science, are nothing else
in reality than applications of the above fundamental rules.
Such applications may assume any degree of complexity, for
owing to the difficulty of particular examples, in the form in
which they actually present themselves, recourse must sometimes
be made to the profoundest theorems of mathematics.
Still we ought not to regard these theorems as being anything
else than convenient and necessary abbreviations of
arithmetical processes, which in practice have become too
cumbersome to be otherwise performed.
This explanation will account for some of the rules as
they are ordinarily given, but by no means for all of them.
It will account for those which are demonstrable by the certain
laws of arithmetic, but not for those which in reality
rest only upon inductive generalizations. And it can hardly
be doubted that many rules of the latter description have
become associated with those of the former, so that in popular
estimation they have been blended into one system, of
which all the separate rules are supposed to possess a similar
origin and equal certainty. Hints have already been frequently
given of this tendency, but the subject is one of
such extreme importance that a separate chapter (that on
Induction) must be devoted to its consideration.
8. In establishing the validity of the above rules, we
have taken as the basis of our investigations, in accordance
with the general scheme of this work, the statistical frequency
of the events referred to; but it was also shown that each
formula, when established, might with equal propriety be expressed
in the more familiar form of a fraction representing
the 'chance' of the occurrence of the particular event. The
question may therefore now be raised, Can those writers who
(as described in the last chapter) take as the primary subject
of the science not the degree of statistical frequency, but the
quantity of belief, with equal consistency make this the basis
of their rules, and so also regard the fraction expressive of
the chance as a merely synonymous expression? De Morgan
maintains that whereas in ordinary logic we suppose the
premises to be absolutely true, the province of Probability is
to study 'the effect which partial belief of the premises produces
with respect to the conclusion.' It would appear
therefore as if in strictness we ought on this view to be able
to determine this consequent diminution at first hand, from
introspection of the mind, that is of the conceptions and
beliefs which it entertains; instead of making any recourse
to statistics to tell us how much we ought to believe the
conclusion.
Any readers who have concurred with me in the general
results of the last chapter, will naturally agree in the conclusion
that nothing deserving the name of logical science can
be extracted from any results of appeal to our consciousness
as to the quantity of belief we entertain of this or that proposition.
Suppose, for example, that one person in 100 dies
on the sea passage out to India, and that one in 9 dies during
a 5 years residence there. It would commonly be said
that the chance that any one, who is now going out, has of
living to start homewards 5 years hence, is 88/100; for his chance
of getting there is 99/100; and of his surviving, if he gets
there, 8/9; hence the result or dependent event is got by
multiplying these fractions together, which gives 88/100. Here
the real basis of the reasoning is statistical, and the processes
or results are merely translated afterwards into fractions.
But can we say the same when we look at the belief side of
the question? I quite admit the psychological fact that we
have degrees of belief, more or less corresponding to the
frequency of the events to which they refer. In the above example,
for instance, we should undoubtedly admit on enquiry
that our belief in the man's return was affected by each of
the risks in question, so that we had less expectation of it
than if he were subject to either risk separately; that is, we
should in some way compound the risks. But what I cannot
recognise is that we should be able to perform the process
with any approach to accuracy without appeal to the statistics,
or that, even supposing we could do so, we should
have any guarantee of the correctness of the result without
similar appeal. It appears to me in fact that but little
meaning, and certainly no security, can be attained by so
regarding the process of inference. The probabilities expressed
as degrees of belief, just as those which are expressed
as fractions, must, when we are put upon our justification,
first be translated into their corresponding facts of statistical
frequency of occurrence of the events, and then the inferences
must be drawn and justified there. This part of
the operation, as we have already shown, is mostly carried
on by the ordinary rules of arithmetic. When we have
obtained our conclusion we may, if we please, translate it
back again into the subjective form, just as we can and do
for convenience into the fractional, but I do not see how the
process of inference can be conceived as taking place in that
form, and still less how any proof of it can thus be given. If
therefore the process of inference be so expressed it must be
regarded as a symbolical process, symbolical of such an inference
about things as has been described above, and it
therefore seems to me more advisable to state and expound
it in this latter form.
_On Inverse Probability and the Rules required for it._
9. It has been already stated that the only fundamental
rules of inference in Probability are the two described
in §§2, 3, but there are of course abundance of derivative
rules, the nature and use of which are best obtained from the
study of any manual upon the subject. One class of these
derivative rules, however, is sufficiently distinct in respect of
the questions to which it may give rise, to deserve special
examination. It involves the distinction commonly recognised
as that between Direct and Inverse Probability. It is
thus introduced by De Morgan:--
"In the preceding chapter we have calculated the chances
of an event, knowing the circumstances under which it is to
happen or fail. We are now to place ourselves in an inverted
position: we know the event, and ask what is the probability
which results from the event in favour of any set of circumstances
under which the same might have happened."[4] The
distinction might therefore be summarily described as that
between finding an effect when we are given the causes, and
finding a cause when we are given effects.
On the principles of the science involved in the definition
which was discussed and adopted in the earlier chapters of
this work, the reader will easily infer that no such distinction
as this can be regarded as fundamental. One common feature
was traced in all the objects which were to be referred to
Probability, and from this feature the possible rules of
inference can be immediately derived. All other distinctions
are merely those of arrangement or management.
But although the distinction is not by any means fundamental,
it is nevertheless true that the practical treatment
of such problems as those principally occurring in Inverse
Probability, does correspond to a very serious source of
ambiguity and perplexity. The arbitrary assumptions which
appear in Direct Probability are not by any means serious,
but those which invade us in a large proportion of the problems
offered by Inverse Probability are both serious and
inevitable.
10. This will be best seen by the examination of
special examples; as any, however simple, will serve our
purpose, let us take the two following:--
(1) A ball is drawn from a bag containing nine black
balls and one white: what is the chance of its being the
white ball?
(2) A ball is drawn from a bag containing ten balls, and
is found to be white; what is the chance of there having
been but that one white ball in the bag?
The class of which the first example is a simple instance
has been already abundantly discussed. The interpretation
of it is as follows: If balls be continually drawn and replaced,
the proportion of white ones to the whole number
drawn will tend towards the fraction 1/10. The contemplated
action is a single one, but we view it as one of the above
series; at least our opinion is formed upon that assumption.
We conclude that we are going to take one of a series of
events which may appear individually fortuitous, but in
which, in the long run, those of a given kind are one-tenth of
the whole; this kind (white) is then singled out by anticipation.
By stating that its chance is 1/10, we merely mean to
assert this physical fact, together with such other mental
facts, emotions, inferences, &c., as may be properly associated
with it.
11. Have we to interpret the second example in a
different way? Here also we have a single instance, but the
nature of the question would seem to decide that the only
series to which it can properly be referred is the following:--Balls
are continually drawn from _different_ bags each containing
ten, and are always found to be white; what is ultimately
the proportion of cases in which they will be found to have
been taken from bags with only one white ball in them?
Now it may be readily shown[5] that time has nothing to
do with the question; omitting therefore the consideration
of this element, we have for the two series from which our
opinions in these two examples respectively are to be
formed:--(1) balls of different colours presented to us in a
given ultimate ratio; (2) bags with different contents similarly
presented. From these data respectively we have to
assign their due weight to our anticipations of (1) a white
ball; (2) a bag containing but one white ball. So stated the
problems would appear to be formally identical.
When, however, we begin the practical work of solving
them we perceive a most important distinction. In the first
example there is not much that is arbitrary; balls would
under such circumstance really come out more or less accurately
in the proportion expected. Moreover, in case it
should be objected that it is difficult to prove that they will
do so, it does not seem an unfair demand to say that the balls
are to be 'well-mixed' or 'fairly distributed,' or to introduce
any of the other conditions by which, under the semblance of
judging _à priori_, we take care to secure our prospect of a
series of the desired kind. But we cannot say the same in
the case of the second example.
12. The line of proof by which it is generally attempted
to solve the second example is of this kind;--It is
shown that there being one white ball for certain in the bag,
the only possible antecedents are of ten kinds, viz. bags,
each of which contains ten balls, but in which the white
balls range respectively from one to ten in number. This of
course imposes limits upon the kind of terms to be found
in our series. But we want more than such limitations, we
must know the proportions in which these terms are ultimately
found to arrange themselves in the series. Now this
requires an experience about bags which may not, and indeed
in a large proportion of similar cases, cannot, be given
to us. If therefore we are to solve the question at all we
must make an assumption; let us make the following;--_that
each of the bags described above occurs equally often_,--and see
what follows. The bags being drawn from equally often, it
does not follow that they will each yield equal numbers of
white balls. On the contrary they will, as in the last
example, yield them in direct proportion to the number of
such balls which they contain. The bag with one white
and nine black will yield a white ball once in ten times; that
with two white, twice; and so on. The result of this, it will
be easily seen, is that in 100 drawings there will be obtained
on the average 55 white balls and 45 black. Now with
those drawings that do not yield white balls we have, by the
question, nothing to do, for that question postulated the
drawing of a white ball as an accomplished fact. The series
we want is therefore composed of those which do yield white.
Now what is the additional attribute which is found in some
members, and in some members only, of this series, and
which we mentally anticipate? Clearly it is the attribute of
having been drawn from a bag which only contained one of
these white balls. Of these there is, out of the 55 drawings,
but one. Accordingly the required chance is 1/55. That is to
say, the white ball will have been drawn from the bag containing
only that one white, once in 55 times.
13. Now, with the exception of the passage in italics,
the process here is precisely the same as in the other example;
it is somewhat longer only because we are not able to
appeal immediately to experience, but are forced to try to
deduce what the result will be, though the validity of this
deduction itself rests, of course, ultimately upon experience.
But the above passage is a very important one. It is scarcely
necessary to point out how arbitrary it is.
For _is_ the supposition, that the different specified kinds
of bags are equally likely, the most reasonable supposition
under the circumstances in question? One man may think
it is, another may take a contrary view. In fact in an excellent
manual[6] upon the subject a totally different supposition
is made, at any rate in one example; it is taken for granted
in that instance, not that every possible number of black and
white balls respectively is equally likely, but that every
possible way of getting each number is equally likely, whence
it follows that bags with an intermediate number of black
and white balls are far more likely than those with an extreme
number of either. On this supposition five black
and five white being obtainable in 252 ways against the
ten ways of obtaining one white and nine black, it follows
that the chance that we have drawn from a bag of
the latter description is much less than on the hypothesis
first made. The chance, in fact, becomes now 1/512 instead
of 1/55. In the one case each distinct result is considered
equally likely, in the other every distinct way of getting
each result.
14. Uncertainties of this kind are peculiarly likely to
arise in these inverse probabilities, because when we are
merely given an effect and told to look out for the chance of
some assigned cause, we are often given no clue as to the relative
prevalence of these causes, but are left to determine them
on general principles. Give us either their actual prevalence
in statistics, or the conditions by which such prevalence is
brought about, and we know what to do; but without the
help of such data we are reduced to guessing. In the above
example, if we had been told how the bag had been originally
filled, that is by what process, or under what circumstances,
we should have known what to do. If it had been filled at
random from a box containing equal numbers of black and
white balls, the supposition in Mr Whitworth's example is
the most reasonable; but in the absence of any such information
as this we are entirely in the dark, and the supposition
made in §12 is neither more nor less trustworthy and
reasonable than many others, though it doubtless possesses
the merit of superior simplicity.[7] If the reader will recur to
Ch. V. §§4, 5, he will find this particular difficulty fully
explained. Everybody practically admits that a certain
characteristic arrangement or distribution has to be introduced
at some prior stage; and that, as soon as this stage
has been selected, there are no further theoretic difficulties
to be encountered. But when we come to decide, in examples
of the class in question, at what stage it is most reasonable
to make our postulate, we are often left without any very
definite or rational guidance.
15. When, however, we take what may be called, by
comparison with the above purely artificial examples, instances
presented by nature, much of this uncertainty will disappear,
and then all real distinction between direct and inverse
probability will often vanish. In such cases the causes are
mostly determined by tolerably definite rules, instead of
being a mere cloud-land of capricious guesses. We may
either find their relative frequency of occurrence by reference
to tables, or may be able to infer it by examination of
the circumstances under which they are brought about.
Almost any simple example would then serve to illustrate
the fact that under such circumstances the distinction
between direct and inverse probability disappears altogether,
or merely resolves itself into one of _time_, which, as will be
more fully shown in a future chapter, is entirely foreign to
our subject.
It is not of course intended to imply that difficulties
similar to those mentioned above do not occasionally invade
us here also. As already mentioned, they are, if not inherent
in the subject, at any rate almost unavoidable in comparison
with the simpler and more direct procedure of determining
what is likely to follow from assigned conditions. What is
meant is that so long as we confine ourselves within the
comparatively regular and uniform field of natural sequences
and co-existences, statistics of causes may be just as readily
available as those of effects. There will not be much more
that is arbitrary in the one than in the other. But of course
this security is lost when, as will be almost immediately
noticed, what may be called metaphysical rather than natural
causes are introduced into the enquiry.
For instance, it is known that in London about 20 people
die per thousand each year. Suppose it also known that of
every 100 deaths there are about 4 attributable to bronchitis.
The odds therefore against any unknown person dying of
bronchitis in a given year are 1249 to 1. Exactly the same
statistics are available to solve the inverse problem:--A man
is dead, what is the chance that he died of bronchitis? Here,
since the man's death is taken for granted, we do not require
to know the general average mortality. All that we want is
the proportional mortality from the disease in question as
given above. If Probability dealt only with inferences
founded in this way upon actual statistics, and these tolerably
extensive, it is scarcely likely that any distinction such
as this between direct and inverse problems would ever have
been drawn.
16. Considered therefore as a contribution to the theory
of the subject, the distinction between Direct and Inverse Probability
must be abandoned. When the appropriate statistics
are at hand the two classes of problems become identical
in method of treatment, and when they are not we have no
more right to extract a solution in one case than in the other.
The discussion however may serve to direct renewed attention
to another and far more important distinction. It will
remind us that there is one class of examples to which the
calculus of Probability is rightfully applied, because statistical
data are all we have to judge by; whereas there are other
examples in regard to which, if we will insist upon making
use of these rules, we may either be deliberately abandoning
the opportunity of getting far more trustworthy information
by other means, or we may be obtaining solutions about
matters on which the human intellect has no right to any
definite quantitative opinion.
17. The nearest approach to any practical justification
of such judgments that I remember to have seen is afforded
by cases of which the following example is a specimen:--
"Of 10 cases treated by Lister's method, 7 did well and 3 suffered
from blood-poisoning: of 14 treated with ordinary
dressings, 9 did well and 5 had blood-poisoning; what are
the odds that the success of Lister's method was due to
chance?".[8] Or, to put it into other words, a short experience
has shown an actual superiority in one method over the
other: what are the chances that an indefinitely long experience,
under similar conditions, will confirm this superiority?
The proposer treated this as a 'bag and balls' problem,
analogous to the following: 10 balls from one bag gave
7 white and 3 black, 14 from another bag gave 9 white and
5 black: what is the chance that the actual ratio of white to
black balls was greater in the former than in the latter?--this
actual ratio being of course considered a true indication
of what would be the ultimate proportions of white and black
drawings. This seems to me to be the only reasonable way
of treating the problem, if it is to be considered capable of
numerical solution at all.
Of course the inevitable assumption has to be made here
about the equal prevalence of the different possible kinds of
bag,--or, as the supporters of the justice of the calculation
would put it, of the obligation to assume the equal _à priori_
likelihood of each kind,--but I think that in this particular
example the arbitrariness of the assumption is less than
usual. This is because the problem discusses simply a
balance between two extremely similar cases, and there is a
certain set-off against each other of the objectionable assumptions
on each side. Had _one_ set of experiments only been
proposed, and had we been asked to evaluate the probability
of continued repetition of them confirming their verdict, I
should have felt all the scruples I have already mentioned.
But here we have got two sets of experiments carried on
under almost exactly similar circumstances, and there is
therefore less arbitrariness in assuming that their unknown
conditions are tolerably equally prevalent.
18. Examples of the description commonly introduced
seem objectionable enough, but if we wish to realize to its
full extent the vagueness of some of the problems submitted
to this Inverse Probability, we have not far to seek. In
natural as in artificial examples, where statistics are unattainable
the enquiry becomes utterly hopeless, and all attempts
at laying down rules for calculation must be abandoned.
Take, for instance, the question which has given rise to some
discussion,[9] whether such and such groups of stars are or are
not to be regarded as the results of an accidental distribution;
or the still wider and vaguer question, whether such and
such things, or say the world itself, have been produced by
chance?
In cases of this kind the insuperable difficulty is in determining
what sense exactly is to be attached to the words
'accidental' and 'random' which enter into the discussion.
Some account was given, in the fourth chapter, of their
scientific and conventional meaning in Probability. There
seem to be the same objections to generalizing them out of
such relation, as there is in metaphysics to talking of the
Infinite or the Absolute. Infinite magnitude, or infinite
power, one can to some extent comprehend, or at least one
may understand what is being talked about, but '_the_ infinite'
seems to me a term devoid of meaning. So of anything
supposed to have been produced at random: tell us the
nature of the agency, the limits of its randomness and so on,
and we can venture upon the problem, but without such data
we know not what to do. The further consideration of such
a problem might, I think, without arrogance be relegated to
the Chapter on Fallacies. Accordingly any further remarks
which I have to make upon the subject will be found there,
and at the conclusion of the chapter on Causation and
Design.
1. It might be more accurate to speak of 'incompatible hypotheses with
respect to any individual case', or 'mutually exclusive classes of
events'.
2. The examples, of this kind, referring to human mortality are taken
from the Carlisle tables. These differ considerably, as is well
known, from other tables, but we have the high authority of De
Morgan for regarding them as the best representative of the average
mortality of the English middle classes at the present day.
3. I say, _almost_ any proportion, because, as may easily be seen,
arithmetic imposes certain restrictions upon the assumptions that
can be made. We could not, for instance, suppose that all the
black-haired men are short-sighted, for in any given batch of men
the former are more numerous. But the range of these restrictions is
limited, and their existence is not of importance in the above
discussion.
4. _Essay on Probabilities_, p. 53. I have been reminded that in his
article on Probability in the _Encyclopædia Metropolitana_ he has
stated that such rules involve no new principle.
5. This point will be fully discussed in a future chapter, after the
general stand-point of an objective system of logic has been
explained and illustrated.
6. Whitworth's _Choice and Chance_, Ed. II., p. 123. See also Boole's
_Laws of Thought_, p. 370.
7. Opinions differ about the defence of such suppositions, as they do
about the nature of them. Some writers, admitting the above
assumption to be doubtful, call it the most impartial
hypothesis. Others regard it as a sort of mean hypothesis.
8. _Educational Times_; Reprint, Vol. xxxvii. p. 40. The question was
proposed by Dr. Macalister and gave rise to considerable controversy.
As usual with problems of this inverse kind hardly any two of the
writers were in agreement as to the assumptions to be made, or
therefore as to the numerical estimate of the odds.
9. See Todhunter's _History_, pp. 333, 4.
There is an interesting discussion upon this question by the late
J. D. Forbes in a paper in the _Philosophical Magazine_ for
Dec. 1850. It was replied to in a subsequent number by Prof. Donkin.