Tuesday, October 1, 2013

A theistic frequentist theory of chances

An event E has divine-intention probability p if and only if E is part of a system S of events such that there is a probability function P applicable to events in S and God intends S to be such as to be described well by P. For instance, suppose over the history of the world a million coins are tossed and of these 50.1% land heads and other statistical properties of the coin tosses are described well by the assumption that the coin tosses are independent events with each outcome of equal probability. Then if God intended that the coin tosses be such as to be thus described, each coin toss has a divine-intention probability 1/2 of being heads.

In the case of finite sequences, standard frequentism suffers from the problem that it gives intuitively incorrect answers. In the above scenario, it would say that the probability of heads is 0.501. But surely it is possible for a million coins to be tossed, each with probability 1/2 of heads, and yet to get 50.1% of the coins landing heads.

Lewis's regularity theory of laws can get the same result as the theistic version, for in the case of probabilistic laws one trades accuracy against simplicity. But regularity theories of laws get the order of explanation mixed up: chances are supposed to be explanatory rather than descriptive (of course Humeans will say that this is a false dilemma, but they are wrong).

Heath White said...

Two questions. Caveat lector as I am an amateur at probability. Also, I have no axes to grind w.r.t. this topic.

1. How do you reconcile “God intends S to be such as to be described well by P” with “other statistical properties of the coin tosses are described well by the assumption that the coin tosses are independent events”? I would have thought that being intended as part of a system with an overall probability structure would make the coin tosses non-independent events. Or is ruling out this problem the function of “other”?

2. Suppose a million coins are tossed and they come up heads 50.1% of the time. What is the difference between having the probability of getting heads be .5, versus having the probability of heads be .501? That is, on what basis are we favoring the former description (explanation) over the latter? Simplicity? (Is .5 simpler than .501?)

Here are a couple of points I’m not sure what to do with. Surely it is POSSIBLE for a million coins to be tossed, each with probability 1/2 of landing heads, and to get heads 75% of the time. On your view, however (if I am understanding it right), it is not possible for this to happen and each coin toss to have divine-intention probability of landing heads be 1/2. Your view would say that each coin has DIP of landing heads = 3/4. That suggests that your view suffers from the same problem (if it is a problem) as frequentism.

Also, if we take this criticism of frequentism literally, it suggests that what is possible is a vague matter. For one might want to say that it is possible for a million coins with p=0.5 to land heads 50.1% of the time, but not possible for them to land heads 75% of the time. Where is the cutoff? 50.2%? 51%? 55%? And so on. Possibility is not normally thought of as vague, though perhaps we should revise that view.

Alexander R Pruss said...

Heath:

Ad 1: Well, they may not be causally independent, but they can still be statistically independent, in the sense that to within the tolerance of the statistical tests, there are no correlations between them.

Ad 2: Here's one answer: Distinguish between the epistemic and metaphysical questions. The epistemic question is a standard philosophy of science question--why do we think the 0.5 theory is true rather than the 0.501 theory which in some sense fits the data better? I am not addressing the epistemic question here. (Though I think simplicity is the answer. :-) ) The metaphysical question is settled by God's intentions. If God intended the rate of heads to be approximately 0.501, under that description, then that is the divine-intention probability (that's oversimplifying, because the coin flips are part of a larger system of probabilities). But if he intended it to be approximately 0.5, under that description, then that's the divine-intention probability.

As for the 75% option, there are a couple of moves one can make, though perhaps one should just bite the bullet.

a. Maybe God could intend approximately 0.5, and get 0.75. After all, 0.75 is approximately 0.5. :-) It depends on how much stress one puts on the divine intention part of the account versus the frequency part. Maybe one could even use God's antecedent intentions here. He has an antecedent intention to have frequency approximately 1/2, but the consequent intention of 75%.

b. I am attracted to pluralism about probabilities. Anything that satisfies the probability axioms is a probability. So I can say: yeah, you can get 75% heads even though each heads had independent probability 1/2, but in a different scenario, one where the probabilities aren't divine intention probabilities. But that won't satisfy. The intuition is, surely, that in THIS setup, with THESE probabilities, it's possible to get 75%.

c. One can bite the bullet. What the account preserves is the compatibility of probabilistic explanations with fine-grained providence, and the satisfaction of the axioms of probability. It would be nice if we could save the intuition that in this very scenario it was possible, with the probabilities being unchanged, to get 75%. But notice that this intuition is going to be hard to preserve on any view that has fine-grained providence, since any such view will threaten to swamp in-creation probabilities with divine propensities (if one can even talk of such).