One common objection is the many gods argument: While it’s true you might be punished eternally (or, if you like, for 3^^^^3 years) if and only if you don’t follow, say, Christianity, it’s possible to imagine other scenarios where you would be punished if and only if you do follow Christianity; thus, it’s claimed, the different possibilities cancel each other out. In responding to the Pascal’s-mugging post, Michael Vassar suggested that we should have Equal priors due to complexity, equal posteriors due to lack of entanglement between claims and facts. But are the priors really equal? Intuitively, the anti-Christian God should take more bits to describe, since that hypothesis requires stating the entire concept of Christianity and then a little extra. I don’t know if that’s the case, but my point is it’s not obvious that, bit for bit, the hypotheses are identical in Kolmogorov complexity. Moreover, the set of relevant hypotheses is bigger than these two: There are tons of hypotheses according to which whether you follow Christianity will make a difference as to whether you suffer for 3^^^^3 years, and I’m not convinced that they all exactly cancel out in prior probability.

Moreover, is there really no entanglement? Is the probability of observing the world we do exactly the same given Christianity as given anti-Christianity? Is the probability, given Christianity, that billions of people would be persuaded of the truth of the Christian God’s message exactly the same as the probability, given anti-Christianity, that billions of people would be fooled into believing the Christian God’s message? Or for that matter, are the probabilities that billions of people will follow non-Christian religions equal under the two scenarios? And so on. There seems to be just too much data in the world for our probabilities to remain symmetric.

This relates to a second complaint about the wager: With vast amounts of data to process and an enormous space of possible religious hypotheses to search, Pascal’s wager (which is just an optimization problem) is computationally infeasible, especially for human minds. This is true, but even if we can’t find the global optimum (if one exists?), I don’t see why we shouldn’t make what local improvements we can, given our limited knowledge, processing ability, and creativity in specifying hypotheses. Just by considering a few basic factual predictions that various religions make, for example, it ought to be possible to separate hypotheses of similar prior probability by many orders of magnitude in their posteriors. We could make some progress on these back-of-the-envelope calculations even without having a full Solomonoff-inducting AI (though the latter would indeed be extraordinarily helpful).

In view of the high uncertainty surrounding the question of which religion (possibly including atheism) to choose, maybe it would be best to avoid making a commitment now, since you might learn more as time goes on that would affect your choice. Moreover, there’s a small chance that in trying to adhere to the commands of a particular religion and in surrounding yourself with fellow believers, you might blunt your ability to think rationally. This argument is fine as far as it goes (though you should also consider your probability of dying before you finally do make up your mind), but then why not spend considerable effort doing further research on the question of which religion to follow? The expected value of additional information would seem to be extraordinarily high.

You might reply that the problem of which religion to follow is overly narrow: There are lots of other projects to work on, perhaps involving more probable scenarios than does Pascal’s wager. For instance, maybe you’re aiming for physical immortality via ordinary materialist means and intend to spend all your time researching how best to stay alive until significant anti-ageing technologies kick in. Fair enough, but what if — as is true in my case — you’re more concerned about avoiding eternal suffering, rather than achieving eternal blissful life? Are there secular scenarios that would require you not to consider Pascal’s wager in the religious case in order to prevent yourself from experiencing massive amounts of suffering?

Finally, some might object to using an unbounded utility function because it leads to mathematical difficulties. I admit that I don’t like the idea of bounding utility functions, but even if we do that, can we not take the bounds big enough that we still allow speculative Pascalian scenarios to dominate over more minor, worldly considerations?

Mindful of Occam’s razor, and careful to avoid the type of reasoning that leads to the conjunction fallacy, we might agree that A’s claim is, in itself, more probable, because it is less specific. However, we have to condition our probability assessment on the evidence that A or B actually made his claim. While B’s claim is less intrinsically likely, the hypothesis that B’s claim is true has strong explanatory power to account for why B made the specific statements he did. Thus, in the end it may not be so obvious whether we should believe A’s claim more in situation A than we believe B’s claim in situation B.

To be concrete, let A be the event that A’s claim is true, B be the event that B’s claim is true, C be the fact that A made the claims he did, and D be the fact that B made the claim’s he did. We can agree that P(B) < P(A); for definiteness, say P(B) = 0.001*P(A). However, the relevant comparison is between P(A|C) and P(B|D). Bayes’ theorem says

P(A|C) = P(A)*P(C|A) / [P(A)*P(C|A) + P(~A)*P(C|~A)];
P(B|D) = P(B)*P(D|B) / [P(B)*P(D|B) + P(~B)*P(D|~B)] = 0.001*P(A)*P(D|B) / [0.001*P(A)*P(D|B) + P(~B)*P(D|~B)].

If either story is true, it’s fairly likely that the person would tell it to us the way it happened; for convenience, assume P(C|A) = P(D|B) = 1. Also assume that P(A) and P(B) are small enough that we can make the approximation P(~A) = P(~B) = 1 in our formulas. We then have

P(A|C) = P(A) / [P(A) + P(C|~A)];
P(B|D) = 0.001*P(A) / [0.001*P(A) + P(D|~B)].

Now, we can probably agree that P(D|~B) < P(C|~A). This is because, if the person wasn’t abducted, it’s less likely that he would give the exact details that B gave than the more general account that A gave. (I don’t mean to suggest that people who claim to be abducted are likely to give short accounts. I mean, rather, that the probability of giving any particular highly detailed account is less than the probability of giving any particular less detailed account.) If we decide that P(D|~B) < 0.001*P(C|~A), then P(B|D) > P(A|C).

There are some cases in which it may be appropriate automatically to give lower probability to more specific claims, even in light of the above reasoning: e.g., in the case of futurists who predict elaborately detailed scenarios. In this case, unless we have reason to think they’ve come back in time from the future, the equivalent of what I called P(D|B) above is likely to be low. That is, even if the futurist’s claims were true, this would not make it very likely that the futurist would successfully predict all of them, since–unlike person A or B, who saw the aliens–the futurist hasn’t viewed firsthand exactly how things will turn out.