Logical v's Causal Dependence

At the end of a previous post, I promised to discuss the difference between logical dependence, the substance of probability theory, and causal dependence, which is often assumed to be the thing that probability is directly concerned with. Lets get the ball rolling with a simple example:

A box contains 10 balls: 4 black, 3 white, and 3 red. A man extracts exactly one ball, ‘randomly.’ The extracted ball is never replaced in the box. Consider the following 2 situations:

a) You know that the extracted ball was red. What is the probability that another ball extracted in the same way, will also be red?

b) A second ball has been extracted in the same manner as the first, and is known to be black. The colour of the first ball is not known to you. What is the probability that it was white?

You should try to verify that he answer in the first case is 2/9, and in the second case is 1/3.

Nearly everybody will agree with my answer to situation (a), but some may hesitate about the answer for situation (b). This hesitation seems to result from the feeling that when we write P(A|B) ≠ P(A), then B is, at least partially, the cause of A. (P(A|B) means ‘the probability for A given the assumption that B is true.’) If  true, then there would be no possibility for knowledge of B to influence the probability for A, because the colour of the second ball can have had no causal influence on the colour of the first ball.

In fact, it makes no difference at all in what order the balls are drawn, in such cases. The labels ‘first,’ ‘second,’ ‘n^th,’ are really just arbitrary labels, and we can exchange them as we please, without affecting the outcome of the calculation.

In case there is still a doubt, consider a simplified version of our thought experiment:

The box had exactly 2 balls, 1 black and 1 white. Both balls were drawn, ‘at random.’ The second ball drawn was black. What is the probability that the first was black?

The product rule can be written P(AB) = P(A|B)P(B). With this formulation, can we account for cases where A depends on B. When thinking about this dependence, however, it is often tempting to think in terms of causal dependence. But probability theory is concerned with calculations of plausibility with incomplete knowledge, and so what we really need to consider is not causal dependence, but logical dependence. We can verify that P(A|B) does not imply that B is the cause of A, since, thanks to the commutativity of Boolean algebra, AB = BA, and we could just as easily have written the product rule as P(AB) = P(B|A)P(A).

What is the probability that X committed a crime yesterday, given that he confessed to it today? Surely it is altered by our knowledge of the confession, indicating that the propositions are not independent in the sense we need for probability calculations. But it is also clear that a crime committed yesterday was not caused by a confession today. 

Edwin Jaynes in ‘Probability Theory: The logic of Science,’ gave the following technical example of the errors that can occur by focusing on causal dependence, rather than logical dependence. Consider multiple hypothesis testing with a set of n hypotheses, H₁, H₂, …, H_n, being examined in the light of m datasets, D₁, D₂, …., D_m. When the data sets are logically independent, the direct probability for the totality of the data given any one of the hypotheses, H_i, satisfies a factorization condition,

P(D1...Dm | Hi, I)  =   ∏ j P(Dj | Hi, I)

(1)

(The capital 'pi' means multiply for all 'j'.) It can be shown, however, that the corresponding condition for the alternate hypothesis, H_i'

P(D1...Dm | Hi', I)  =   ∏ j P(Dj | Hi', I)

(2)

does not hold except in highly trivial cases, though some authors have assumed it to be generally true, based on the fact that no D_i has any causal effect on any other D_j. (Equation (2) requires that P(D_j|D_i) = P(D_j).) The datasets maintain their causal independence, as they must, but they are no longer logically independent. This is because the amount that equivalent units of new information change the relative plausibilities of multiple hypotheses depends on the data that has gone before: the effect of new data on a hypothesis depends on which other hypothesis it competes with most directly.

In Jaynes’ example, he imagined a machine producing some component in large quantities and an effort to determine the fraction of components fabricated that are faulty by randomly sampling 1 component at a time and examining it for faults. The prior information is supposed specific enough to narrow the number of possible hypotheses to 3:

A ≡ ‘The fraction of components that are faulty is 1/3.’

B ≡ ‘The fraction of components that are faulty is 1/6.’

C ≡ ‘The fraction of components that are faulty is 99/100.’

The prior probabilities for these hypotheses are as shown at the extreme left of the graph below. The graph is the calculation of the evolution of the probabilities for each hypothesis as the number of tested components increases. Recall that, from Bayes’ theorem, each P(H_i|D, I) depends on both P(D|H_i, I) and P(D|H_i', I). Each tested component is found to be faulty, so the information added is identical with each sample, but the rates of change of the 3 curves (plotted logarithmically) are not constant.

Evolution of the probabilities of 3 hypotheses as constant new data are added.

Taken from E.T. Jaynes, ‘Probability theory: the logic of science,’ chapter 4. 

The ‘Evidence,’ plotted on the vertical axis, is perhaps an unfamiliar expression of probability information. It is the log odds, given by

E  =   10 log10   P( H ) P( H' )

(3)

with the factor of 10 because we choose to measure evidence in decibels. The base 10 is used because of a perceived psychological advantage (our brains seem to be good at thinking in terms of factors of 10). Because we have used a logarithmic scale, the products expressed above in equations (1) and (2) becomes sums, and for constant pieces of new information, we expect to add a constant amount to the evidence, if both factorization conditions hold. The slopes of the curves are not constant, however, indicating that this is not the case, and consecutive items of data are not independent: ΔE depends on what data have preceded that point. Specifically, wherever a pair of hypotheses cross on the graph, there is a change of slope of the remaining hypothesis.

When we calculate P(D|H_i, I) we are supposing for the purposes of calculation that H_i is true, and so the result we get is independent of P(H_i), which is why P(D|H_i, I) factorizes. But P(D|H_i', I)  is different, because when the total number of hypotheses is greater then 2, then H_i' is composite and decomposes into at least 2 hypotheses, so P(D|H_i', I) relies upon the relative probabilities for those component propositions.