Monday, September 30, 2019

Classical probability theory is not enough

Here’s a quick argument that classical probability cannot capture all probabilistic phenomena even if we restrict our attention to phenomena where numbers should be assigned. Consider a nonmeasurable event E, maybe a dart hitting a nonmeasurable subset of the target, and consider a fair coin flip that is causally isolated from E. Let H and T be the heads and tails results of the flip. Then let A be this disjunctive event:

  • (E and H) or (not-E and T).

Intuitively, event A clearly has probability 1. If E happens, the probability of A is 1/2 (heads) and if E doesn’t happen, it’s also 1/2 (tails). (The argument uses finite conglomerability, but it is also highly intuitive.)

So a precise number should be assigned to A, namely 1/2. And ditto to H. But we cannot have these assignments in classical probability theory. For if we did that, then we would also have to assign a probability to the conjunction of H and A, which is equivalent to the conjunction of E and H. But we cannot assign a probability to the conjunction of E and H, because E and H are independent, and so we would have a precise probability for E, namely P(E)P(H)/P(H)=P(E&H)/P(H), contrary to the nonmeasurability of E.


Michael Gonzalez said...

What does intrinsic "nonmeasurability" mean in real life?

Philip Rand said...
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Alexander R Pruss said...

There is no way to assign probabilities while respecting the symmetry of the situation.

Wesley C. said...

Regarding nonmeasurability and brute facts - if the nonmeasurability of brute facts is a good reason to deny their actual possibility, then what about other nonmeasurable things like some mathematical constructs?

Does the PSR require that nonmeasurable mathematical constructs can't be actually instantiated or concretely realised?

Alexander R Pruss said...

I think there is some reason independent of the PSR to doubt the actual causal implementability of nonmeasurable mathematical constructs. Cf. the chapter on the Axiom of Choice in my book on the paradoxes of infinity.

I also think there is a difference between (a) the extreme nonmeasurability of uncaused contingent events and (b) the kind of nonmeasurability you would have in a concrete situation like throwing a dart at a nonmeasurable target. In (a), there are no probabilities at all attachable to the event. But in (b), there are probabilistic bounds, since the probability of hitting the nonmeasurable target is less than the probability of throwing the dart which in turn is less than one. Thus, (b) leads to less in the way of sceptical worries than (a).

Wesley C. said...

Could you explain more about how nonmeasurable mathematical constructs have probability bounds brute facts wouldn't have - since the fact the probabilities associated with them are nonmeasurable and so probability bounds don't seem applicable to limit that in itself? And would this apply to saturated nonmeasurable sets as well?

Alexander R Pruss said...

Imagine you're randomly throwing a dart at a round target C. Let A be a nonmeasurable subset of C. Suppose you also have a fair coin.

Now, let E be this event: you hit A with the dart AND your coin lands heads. Then the probability of E is less than or equal to the probability of heads, which is 1/2. So we have an upper bound on E. Yet E is nonmeasurable.

Similarly, we can get lower bounds. Let F be: you hit A with the dart OR your coin lands heads. Then the probability of F is at least 1/2.

Wesley C. said...

So from the fact the bounds are external to the circle and the subset, I presume this would also apply to saturated nonmeasurable sets, correct?