Think: A Compelling Introduction to Philosophy (31 page)

or equally:

The first equation says that the probability of wearing jeans and
jacket = the probability of wearing the jeans x the probability of
wearing the jacket given that you are wearing the jeans. This last is
called a'conditional probability. The second says it is also equal to
the probability of wearing the jacket x the probability of wearing
the jeans given that you are wearing the jacket. These have to be
identical, by symmetry (since a & b is the same proposition as
b & a).

An English clergyman called Thomas Bayes (1702-61) looked
hard at this result. Since each of them is equal to Prob (a & b), each
of these is equal to each other:

So we can write down an expression for the probability of bgiven a:

This rather frightening-looking equation is a simple version of
what is known as Bayes's theorem.

The application of the result conies like this. Suppose now that
we have some hypothesis, and a piece of evidence for it. We are interested in the probability of the hypothesis h, given the evidence e.
We can write this as Prob (h/e). This is called the posterior prob ability of the hypothesis-its probability after the evidence comes
in. Then the theorem tells us that:

This directs us to three different things on which the posterior
probability depends.

Prob (h). This is known as the prior or antecedent probability of h.

Prob (e/h). This is the probability of evidence e, given It. It
is a measure of the fit between the hypothesis and the evidence.

Prob (e). This is the prior or antecedent probability of the
evidence itself.

Intuitively it can be thought of like this. There are three factors.
First, how likely is the hypothesis from the word go? Second, how
well does the evidence accord with the hypothesis? Third, how
likely is the evidence from the word go?

It is often useful to treat this last figure in terms of the different
ways the evidence might have come about. It is a figure that gets
larger the greater the number of quite probable alternative explanations of the evidence. And when it gets larger, the probability of
the given hypothesis on the evidence gets smaller. It has too many
competitors. So in practice the figure on the bottom measures how
many other ways there are in which that evidence could be explained, and how likely they are. We recognize the importance of
this intuitively. When the call-girl Mandy Rice-Davies was told
that some member of the aristocracy denied having had an affair with her, she replied, `Well, he would, wouldn't he?' She was in effect reminding people that the antecedent probability of this particular piece of testimony was high regardless of which hypothesis
is true, and this undermined its value as evidence. You could guess
in advance that whatever their relations, the aristocrat would have
said what he did. So his saying what he did was worthless as evidence.

The ideal would be: the hypothesis is antecedently quite likely.
The evidence is just what you would expect, given the hypothesis.
And there are not many or any other probable ways the evidence
could have come about.

In the case of the disease, Bayes's theorem puts the base rate up
front: it is the antecedent probability that you have the disease, of i
in i,ooo. The next figure, the fit between the test result and the hypothesis that you have the disease, is excellent: i, in fact, since the
test always says you have it if you do. But on the bottom line we
have the number of ways that evidence could have come about. Informally, there is the i in i,ooo chance of a true result plus the in in
i,ooo chance of a false positive. It is this that results in your overall
chance, given the evidence, being (approximately) i in Ii.

There is a nice way now of representing the impasse between
Straightie and Kinkie in the lottery for the Golden Harp. Suppose
S in the entire ninefold pattern-blue each time--on Straightie's
ticket. And suppose E is the part of it that is within our experience:
the five results of blue each time so far. Then

The antecedent or prior probability of S was I/6". The second figure
is good, however. If S indeed describes the way events fall out, then
the evidence E, i.e. the first five readings, is just what would be expected. Their probability is in fact i, given S. And the prior probability of E? That is just five readings of blue, which, given that blue
is one of six competing possibilities, is i/6'. Calculating out, we get
that Prob (S/E) is 1/61, which is just what we got intuitively before.

The trouble is that exactly the same formula gives exactly the
same result for Kinkie's ticket, K. You can easily see that the evidence has probability I, given K, and the prior probability of the
evidence is the same in either event.

Notice that the problem is not one of 'proving'that S will win, or
that K will not win. It is just one of finding some good reason to expect S rather than K. It is a question of comparing probabilities.
Hume's position is that even this cannot be done in S's favour. Reason remains entirely silent between them. And following Bayes's
analysis, he looks to be right. The debate between Straightie and
Kinkie is as stalemated as ever. In fact, if there was no reason for
preferring the ticket S to the ticket K in Heaven, a priori, then there
is no reason for preferring it after the evidence has come in. Or so
it seems.

We could now revisit a number of areas: the Zombie possibility,
the design argument, the likelihood of a good God creating or allowing evil, and especially the discussion of miracles, using Bayes's
theorem. It is a tool of immense importance. The fallacies it guards
against-ignoring the base rate, ignoring the chance of false positives-are dangerous, and crop up everywhere that people try to
think.

Of course, very often it is difficult or impossible to quantify the
prior' probabilities with any accuracy. It is important to realize
that this need not matter as much as it might seem. Two factors alleviate the problem. First, even if we assign a range to each figure, it
may be that all ways of calculating the upshot give a sufficiently
similar result. And second, it may be that in the face of enough evidence, difference of prior opinion gets swamped. Investigators
starting with very different antecedent attitudes to Prob (h) might
end up assigning similarly high values to Prop (lh/e), when (e) becomes impressive enough.

For interest, it is worth mentioning that there are quite orthodox methods of statistical inference that try to bypass Bayesian
ideas. Much scientific research contents itself with ascertaining
that some result would only occur by chance some small percentage of the time (less that 5 per cent, or less than i per cent, for example). But it then infers that probably the result is not due to
chance-that is, there is a significant causal factor or correlation of
some kind involved. This prevalent reasoning is actually highly
doubtful, and Bayes shows why. If the antecedent probability that a
result is due to anything else than chance is very, very low, then
even enormously improbable results will not overturn it. If I put
my hand in a shaken bag, throw seven Scrabble letters face down
on a table, shuffle them into a line, and turn them up, the actual result (PQAERTU, say) will be very improbable indeed. I night do
the same thing for a hundred years and not repeat it. But it was
chance, for all that. In this setup any result is going to be very improbable, and we should not be able to infer back to say that anything other than chance is responsible for it.That is the very kind of reasoning that fuels lunatic attempts to prove that the pattern of
occurrence of vowels in Shakespeare's plays is best explained by the
hypothesis that he was writing the Name of the Beast 666 times, or
whatever. In short, it is not just the fact that a result is improbable
that should prompt us to look for some special explanation. We
need some additional reason to think that the improbable result is
not just due to chance anyway. Chance is just as good at throwing
up improbabilities as design.

EXPLANATIONS AND PARADIGMS