Sunday, January 13, 2008

The Conditional Pascal Wager

For those of you not in the know pascals wager is an argument that runs something like this. Let us analyze belief in god as a gambling wager. The expected return of a wager is simply the sum of the possible returns multiplied by their probabilities. We have two wagers that we can place we can either believe in god or not believe in god. Say there is some probability P(G) that god exists. Lets say we place the wager of believing in god then if god does exist we go to heaven and enjoy eternal bliss, an infinite payoff, if we believe and god doesn't exist then we get nothing. Calculating our payoff we have our average payoff is P(G)*infinity + (1 - P(G)) * 0 = infinity regardless of what the probability of god existing is. On the other hand if we don't believe in god and god exists we go to hell and are eternally tortured which is a sort of negative infinity and if we don't believe in god and he doesn't exist then we again get squat. So according to the pascal argument the expected return of believing in god is infinite while the expected return of not believing in god is negative infinity. That was strong enough reason for pascal to decide to become devout and ditch mathematics. Here is a post on Booms blog expanding on pascal's wager. I disagree with the argument in general but it is the impetus for this post so I figured I should at least link to it.

The biggest problem in the reasoning of Pascal's wager is that it is assumed that the probability that god exists P(G) is independent of the event of believing in god. I have used some standard notation up to this point without declaring it but after this point I am going to need a lot more notation so I had better introduce it formally. An event is the set of all outcomes which share some property. For instance if we were to talk about the event that the third flip in a series of coin flips comes up heads then the event that corresponded to this would be all possible coin flips in which the third flip is heads. The function P(X) is the probability of the event X. The notation P(X | Y) denotes the probability of the event X given that Y occurs. This conditional probability is equivalent to the probability of the event X intersect Y divided by P(Y) So the conditional probability only makes sense when Y has a non zero probability. Let the event of believing in god be denoted B and let the event that god exists be denoted by G also let the complement of any event X be denoted by X*. The complement of an event is the set of all possible outcomes in which X does not occur. Finally let the reward function be R(X | Y) which is the return we get upon the event X occurring when we have made the wager Y.
If two events have no bearing on each other then the probability of the combination of their outcomes is equal to the product of their probabilities. If all this doesn't make perfect sense to you here is a wiki article that might help.

In pascal's wager we take the expected return of believing in god to be P(G)*R(G | B). This calculation implicitly assumes however that the probability of gods existence is independent of our choice to believe or not believe. In other words P(G) = P(G | B) otherwise the calculated return should be P(G | B) * R(G | B). Of course the argument would run something like that whether god exists or not is an aspect of physical reality and therefore it is not a probabilistic event and so our belief or non belief could not possibly alter the existence of god one way or another. Either god exists or he does not and believing one way or another cannot change that so the assumption of this independence of belief and actuality seems consistent. However Pascal's wager is a deterministic partial information game not a complete information probabilistic game but pascal's wager is a probabilistic analysis. Chess is a good example of a perfect information game meaning that there is nothing about the game that is hidden from the players. A partial information game is where each player has only part of the picture such as in a game of poker. Poker is not probabilistic because once the cards have been dealt there is no shuffling or random influence in the game you don't know what cards your opponent has or what is on top of the deck but it is not randomly determined during the game. A game of craps is an example of a complete information probabilistic game because at each stage of the game you know everything there is to know about the state of the game but the outcome is randomly determined.

Even though poker is not a random game but rather a partial information game we treat it as a random game because the laws of probability apply because one can use probability to see how likely any particular initial set up is. Thus because pascal is applying the rules of probability to this partial information game if we are to be consistent with our analysis we must treat the existence of god to be something that is truly random. Just like in poker we let anything that is not yet known to be something that is randomly determined during the course of the game not something that has already been determined otherwise our probabilistic analysis would be inconsistent or at least incomplete.

Now let us consider the probability P(G | B). If there is indeed no interaction between god and the universe meaning that the universe would be exactly the same if god existed as it would be if he did not exist. If the existence of god makes no difference in the universe then our belief must certainly be independent of his existence and therefore the original assumption of the wager holds that P(G | B) = P(G) So in essence the pascal wager considers the condition that gods existence has no effect on the world whatsoever. While this might work for some deists I think most people would like to think god has at least some effect on the universe you know a few miracles here and there and whatnot at the very least resurrect a Jesus or two.

If god does have an effect on the universe then things get a little bit more complicated for the argument. Lets say that our belief in god is a function of a number of independent factors at least one of which is the actual existence of god. Since the case where the existence of god and our belief are in no way correlated under the original pascal wager we will assume that the correlation between our belief and gods existence is greater than zero. Since not everyone believes in god we also know that the correlation is strictly less than 1. Since the existence of god does not change from time to time or from person to person that means that the probability of a person believing in god is affected in a constant way by the existence of god. In other words because the existence of god does not vary any particular persons belief in god can be treated entirely as a function of variables other than the existence of god.

So here we have an experiment, We keep the existence of god constant and vary everything else, race, class, gender, sexuality, intelligence, etc. When we factor in all of these conditions and then look at the belief of these people as a cross section then we can use a statistical technique whose name at the moment escapes me in order to see what the effective dimension of the distribution is or in other words how many things the belief is dependent on. This isn't at all where I was thinking I would end up with this but I like that so I am going to stop there for a bit and I will give you the rest of the tour of the original thought I had in a later blog post.