Thursday, December 08, 2016

Miracles and sample bias

I'm going to quote and comment on a discussion of miracles by Christian philosopher George Mavrodes:
An organization that sponsors a very large prize lottery in the United States recently informed potential entrants that the chance of winning the grand prize was approximately one in 100 million. I suppose that this is based on an estimate of the number of entries that will be received, or something like that. So if I were to submit an entry for this lottery the probability of my winning the grand prize would be approximately 0.00000001. That is, of course, a very low probability, and I would be very surprised if I won. Assuming that the lottery is fairly drawn, every other entrant would have that same probability of winning. Suppose now that the drawing has actually been held, and that we read a short news story about it. The newspaper reports that a certain man, Henry Plushbottom of Topeka, Kansas, is the winner of the grand prize. The antecedent probability—antecedent, that is, to the news story—of Henry’s being the winner is fantastically low. But what is now the consequent probability—consequent to the news story—that Henry really is the winner? My own inclination is to say that the news story makes the probability that Henry really is the winner quite high.
If my response is rational, however, then it seems to be the case that a single testimony, a testimony given in many cases by someone whom we do not know at all, is sufficient to produce an enormous change in probability. Something whose initial probability is so small as to be almost unimaginable is converted by a single testimony into something that is substantially more probable than not. I call this the Lottery Surprise. How could a single testimony have such an enormous effect on probability? And how does this fact bear on our assessment of the probability of miracles when there is some testimony at hand?
The news story about a lottery winner, therefore, involves two items, and each of them has a very low antecedent probability. It was antecedently improbable that Henry would win, for he was only one entrant among 100 million. It was also antecedently improbable that he would be named in the story as the winner, for his was only one name among 100 million different names that could have appeared in that story. 
The fact is that the news story involves two events, each of which, taken separately, is immensely improbable. In fact, they have the very same immense improbability. But taken together they support each other in such a way as to generate a substantial positive probability. If Henry is actually the winner, then it is probable that he will be named as such in the story, and if he is not the real winner, then it is fantastically improbable that he would be the one mistakenly identified in the paper. Therefore, his being identified as the winner makes it probable that he is really the winner. 
Hume, therefore, seems to believe the following proposition: (N) No one has ever risen from the dead.
Of course, Hume may have had a negative experience about resurrections, an experience that might be reported in this way: (E) Hume never observed any resurrection from the dead, he never met anyone who had been restored to life after dying, etc.
I have no reason to doubt (E), and I have no inclination to doubt it. I think it is very likely that Hume never came across a genuine resurrection in his whole life. And the same is true of me. I also have never observed a resurrection. But although (E) is true, and the corresponding proposition about me is also true, these propositions have no real relevance with respect to the probability of (N). It is not the negative nature of propositions such as (E) that makes them irrelevant. It is, rather, the fact that Hume’s sample and my sample are far too small relative to the scope of (N). (N) is a general proposition whose scope includes millions upon millions of particular cases, all the human deaths that belong to the history of the world. Hume, we might suppose, had some direct experience of a few human deaths and of what happened soon thereafter. Perhaps a dozen or so family members and friends. But even fifty or one hundred would be far too small to have a significant bearing on the probability of (N).
Of course, Hume’s negative experience is just what we should expect if (N) is true. If there simply are no resurrections, then Hume would not run into one. But Hume’s negative experience is also just what we would expect if there are real resurrections but they are quite rare. If there are, say, only half a dozen genuine resurrections among the many millions of deaths there have been in human history, then it is extremely unlikely that Hume’s tiny sample would have caught one of them. So that sample is entirely unreliable in distinguishing between a world in which there are no resurrections—that is, the world as described by (N)—and a world in which there are only a few resurrections. But that distinction is crucial to this case. For there is probably no aficionado of resurrections, or of miracles in general, who thinks that they are as thick in the world as fleas on a stray dog.
We can construe the probability of Jesus’ resurrection as being very low in the same way as we construe the probability of Henry’s winning the grand prize as being very low. If we take Jesus to be just a randomly selected person among the many millions of human beings who have lived in the world, and if we assume that resurrections are at best very rare in the world, then the antecedent probability of Jesus’ being resurrected is very low. But this is just the sort of case to which the Lottery Surprise applies. That is, it is just the sort of case in which a single testimony generates an enormous change in the subsequent probability. George Mavrodes, “Miracles”. W. Wainwright, ed., The Oxford Handbook of Philosophy of Religion (Oxford, 2005), pp. 304-22.

Mavrodes makes a number of good points:

i) How a single report can dramatically upgrade our assessment regarding the probability that something happened.

ii) Sometimes, the number of improbable incidents in a story makes the story increasingly unlikely. We think an alibi is fishy if it has improbable incidents. And the more improbable incidents, the more implausible the explanation.

But as Mavrodes explains, that can be simplistic. There are situations in which two or more independently improbable events reinforce each other rather than multiplying the overall improbability. There are situations in which we'd expect that if one improbable incident occurs, a related improbable incident will occur. 

iii) If miracles, or at least evident miracles, are rare, then the fact that many or most people don't observe them is consistent with their occurrence. That doesn't cast doubt on their occurrence. Negative experience doesn't render them suspect. Rather, that's to be expected if they are rare. That's a consequence of sample selection bias. 

iv) He says: "If we take Jesus to be just a randomly selected person among the many millions of human beings who have lived in the world, and if we assume that resurrections are at best very rare in the world, then the antecedent probability of Jesus’ being resurrected is very low."

But, of course, Jesus isn't a randomly selected person among millions (or billions).

By the same token, he says: "Assuming that the lottery is fairly drawn, every other entrant would have that same probability of winning."

But in analogy to the Resurrection, the lottery is rigged so that Jesus was bound to win. Therefore, if the Resurrection is probable even if Jesus were a randomly selected individual among millions (or billions), and if the Resurrection is probable even if the lottery is fairly drawn, then a fortiori, it is exponentially more probable–indeed, a dead certainty–if the lottery was designed to select for Jesus. 

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