Monday, December 16, 2013

Conditional probability and probability comparisons

Suppose we want to say that event B is more likely than another A. What does that mean? A natural thing to say is that P(B)>P(A). But that doesn't fit our intuitions. For all measure zero sets then end up being equally likely. We can get a sharper comparison if we instead of starting with unconditional probabilities, we work with primitive conditional probabilities. A natural way that I've considered in the past is to say that AB if and only if P(A|AB)≤P(B|AB). This lets you compare tiny sets, like a single point to two points. But this approach has the intuitive disadvantage that then if we use uniform measure on [0,1], then [0,1]≤[0,1).

But there is a better way to generate a comparison from a conditional probability. Say that AB if and only if the two sets are identical or P(AB|(AB)∪(BA))≤P(BA|(AB)∪(BA)). It's not that hard (unless one is as sleepy as I am this morning) to show that this relation is reflexive, transitive and total—i.e., a weak order. Moreover, this weak order has the property that if A is a proper subset of B, then we're guaranteed to have strict inequality: A<B.

Update: A. Paul Pedersen informs me that De Finetti gives a definition equivalent to this on page 367 of the second volume of his probability book.


Dr. Rizz said...

Why think that belief in theism is necessary in order to receive some sort of afterlife benefit? God must know that he is hidden. A just God would not remain hidden and make belief in theism the way to access eternal goods or avoid eternal harms. False belief would be easy to fix in the afterlife. Hence, I think a the key to refuting the wager is to point that the odds of a just and loving God demanding belief under conditions of uncertainty are zero. This should follow analytically from the concept of fairness. Hence, I don't think one could have a duty to promote an inscrutable or not terrifically plausible belief as anything more than a serious possibility.

Alexander R Pruss said...

"Why think that belief in theism is necessary in order to receive some sort of afterlife benefit?"

Pascal's Wager doesn't really need necessity. All it needs is that belief in God increases the chance of such a benefit.

Dr. Rizz said...

Good point, necessity is not required for the wager to proceed. Yet, why wouldn't it follow analytically from the notion of God's desiring the best for his creatures (which would include the good of having us make well-informed choices) that we would have the opportunity to make a choice of such consequence from a much stronger epistemic position than we are currently in?

Alexander R Pruss said...

Well, I think we're in a very strong epistemic position with respect to whether God exists, but the point of the Wager is to grant for the sake of argument that we're not.

By the way, I just noticed that these comments are attached to the wrong post