Wolfgang Schwarz

Conditional probabilities and Humphreys' Paradox

Expressions like 'P(A/B)', or 'the probability of A given B', seem to be used in various different ways. On one usage, P(A/B) equals P(AB)/P(B), at least if P(B) > 0. Call this the ratio usage. Simple versions of the ratio usage define P(A/B) as P(AB)/P(B), and so entail that P(A/B) is undefined whenever P(B)=0. But I would like to admit views into the family on which P(A/B) is taken as a primitive binary probability, governed by something like the Popper-Renyi conditions.

Another use of 'P(A/B)', more common in statistics than in philosophy, is where B gives the value of a parameter of the probability distribution P. Call this the parameter usage. For example, the probability of k heads in n tosses of a coin may be stated as

Here x stands for the probability of heads on an individual toss. P(k / x) is not to be identified with the ratio P(k & x)/P(x). In frequentist statistics, both numerator and denominator of this ratio are deemed nonsensical. Likewise for the inverted probability P(x / k). Unlike on the ratio usage, 'P(A/B)' on this usage therefore does not satisfy ordinary rules for conditional probabilities, such as Bayes's Theorem.

A third kind of use is one in which B in 'P(A/B)' denotes the setup, or condition under which A has the relevant probability. Call this the setup usage. Thus we might find 'P(heads/toss)', or 'P(future state is Y / present state is X). On many interpretations of probability, this too must be sharply distinguished from the ratio usage. The problem is best known for the propensity interpretation, where conflating the two readings leads to "Humphreys' Paradox". (But the problem also arises for other accounts such as that of von Mises.)

On the propensity interpretation, 'P(A/B)' measures the degree to which setup B is disposed to causally produce A. Again, this magnitude has little to do with the ratio of P(AB) over P(B), which may both be deemed nonsensical. And again, when P(A/B)=x, which means that there is a certain tendency x of B to produce A, then there is generally no tendency for A to produce B (nor even to have been produced by B); so P(B/A) is undefined, or at least has a value quite unlike what one would expect on the raio usage.

Humphreys, along with Fetzer and others, conclude from the fact that the setup usage does not satisfy the laws governing the ratio usage that propensity theorists should reject and revise the standard rules for conditional probabilities. Others, like Wesley Salmon, take this fact to be an argument against the propensity interpretation.

It would be better, I think, to not use the same notation for so many different things. On the propensity view, probabilities are only defined relative to a given setup. Each setup S brings with it a probability function P_S. These setup-relative probabilities obey the standard probability calculus, and nothing stops us from introducing standard conditional probabilities, governed by the standard ratio principles. Thus P_{toss}(2 / even) would denote the propensity of a die toss to produce the outcome 2 conditional on producing an outcome with an even number. This may be defined as P_{toss}(2 & even)/P_{toss}(even), or it could be taken as primitive to allow for conditional probabilities with zero-probability conditions.

Consider an example of Humphreys's. Our setup S consists of a photon source and a half-silvered mirror placed at some distance such that the chance of the photon hitting the mirror is 0.5. Let I be the event that the photon impinges upon the mirror, and T the event that the photon is transmitted through the mirror. Now the propensity of the photon hitting the mirror is causally independent of whether or not it afterwards gets transmitted: if we change the mirror's opacity, the probability of I remains unchanged. Thus

(1) P(I/TS) = P(I/~TS) = P(I/S).

On the other hand, the photon can only be transmitted if it hits the mirror: T entails I. So

(2) P(I/TS) = 1.

But now it follows that P(I/S) = 1, contradicting the assumption that the chance of I in S is 0.5.

What went wrong? Humphreys accepts (1) and rejects (2). McCurdy accepts (2) and rejects (1). What's really going on, I think, is that two completely different readings of the conditional stroke are being mixed together. (1) might better be written

(1') P_{TS}(I) = P_{~TS}(I) = P_{S}(I).

The claim is that I is just as probable to be produced under conditions S as under conditions ST or S~T. (These are somewhat odd "conditions" because they lie partly before and partly after the time of I. Assuming there is no backwards causation, only the part of the condition before that time is causally relevant to the production of I, which is why (1') holds. The point could also be made by instead reading S~T as a modification of S in which the mirror is opaque, and setting aside P_{TS}(I).) So here we have a straightforward setup usage.

(2) is meant to express a ratio-style conditional probability in setup S:

(2') P_S(I/T) = 1.

The claim is that under conditions S, the probability that the photon hits the mirror given that it later passes through it, is 1. This, in turn, might be understood as the ratio of the propensity of S to produce an outcome in which the photon impinges and transmits over the propensity of S to produce an outcome in which the photon transmits.

The general point is that the ratio notion of conditional probability, as well as perhaps the parameter notion, can be very useful even for propensity theorists. And there is no reason to choose between the three. If we stop equivocating, there is no paradox here. (1') and (2'), for example, do not entail anything that conflicts with the original description of the scenario. And the claim that P_S(I) may be defined while P_I(S) is undefined is as unproblematic as the claim that on time-relative notions of chance, P_t(A) can be defined while P_A(t) is nonsense.