How can this counterexample to Bayesian reasoning be addressed?

Question

I'm having a hard time rationalizing how a particular theory is confirmed by Bayesianism in a particular example and yet seems to be completely unintuitive. As a reminder, in Bayesianism, an observation O confirms a theory if and only if the P(H|O) is > P(H) after the observation. This is usually done if the likelihood of P(O|H) is higher than P(O|~H). The higher it is, the better the confirmation.

Suppose now that we have 10^10 games in front of us. Each game has a 10^10 chance of winning a prize. Suppose we live in a world where the prize must be won by chance or occur through some supernatural, unknown force. To make this example more concrete, you can simply imagine some supposed psychic trying to guess a number between 1 and 10^10 among 10^10 different people where they all think of a number in that range. Assume that each person is thinking of a number in a uniform way, so don't worry about the exact mechanism here, since that's irrelevant to the example.

Suppose that after the nth try, the psychic gets his guess right. Technically, this confirms the theory that the psychic used mind powers to guess the number for that particular person. After all, P(O|H) = 1 but the probability of O|~H (in this case chance) is 1 in 10^10. Bayesians will argue that the prior for P(H) is so low (since we've never observed psychics) we still shouldn't believe that he is a psychic. But how low can this prior be? No matter what value you pick, you can simply change the example to an even larger number such that the likelihood difference makes up for that prior.

Now, of course, the response will simply be that he failed many other times. It seems unreasonable to only focus on the time he got it right. But we are not judging the hypothesis that he would guess all of them correctly. We are judging the hypothesis that he would guess that particular person's number correctly. A further response might be that the hypothesis is ad hoc. You are waiting for him to guess it correctly and then claiming that he guessed it right using mind powers. This seems ridiculous.

I agree that it seems ridiculous but there seems to be nothing in Bayesian terms that prevents the conception of ad hoc hypotheses. As mentioned before, the only way a hypothesis is confirmed is based on the previous formula. Ad hocness does not come into play. If you want to assign an ad hoc theory a lower prior, you can again change the example to a much higher number, such that the probability of getting that particular guess correct is lower than that prior.

How can Bayesians counter against this example? The only response to this I can think of is that for the n - 1 times the psychic fails, that should factor into the prior. But why? What if the person was just waiting for the nth try to get it correct? Why should the previous trials be factored in?

Answer 1

There are a few points you need to bear in mind.

1. Bayes' theorem is just that- a theorem, so it is mathematically correct.

2. Bayesian inference works with subjective probabilities.

3. As with any computational procedure, the 'garbage in, garbage out' rule applies.

4. Given 1) and 3), if the application of Bayes' theorem leads to nonsensical results, then it is not a fault with the theorem- there is something nonsensical about the way it has been applied.

5. Given 2), if your subjectives probabilities are off, then the output will be off too, so Bayesian inference is unlikely to be helpful if you are applying it to scenarios in which individual people can take widely differing views about the probabilities.

I suggest you adopt 4) and 5) as your starting point, and then examine your imagined scenario for nonsensical or inappropriate assumptions.

I will give you a clue to get you started.

What, exactly, is your hypothesis about the powers of the psychic? Suppose it were that he can predict for certain every guess of every person. Now suppose that he fails 9 times then guesses correctly. Clearly your hypothesis is false. Now, if you are prepared to accept his word when he says he wasn't trying on the first 9 guesses, then your hypothesis must be changed to another which is 'This psychic can predict for certain the guess of every person when he makes a special effort to do so, and he never lies about whether any failed guesses he makes are due to a lack of special effort.' How can you make reasoned assumptions about probabilities in connection with such a hypothesis?

In short, Bayesian inference is a tool that is well suited to certain applications and not to others. Your problem seems to how to decide where it works and where it doesn't, and for that I suggest you consider points 4) and 5) above.

Answer 2

This seems very similar to another question that you asked recently. In that question, you said that H was the hypothesis than an invisible monster causes a coin to always land on heads, and ~H was "chance", i.e., the hypothesis that the coin is fair. Those propositions would only be logical negations of each other if no other explanation of the coin's behavior were possible. You effectively assigned a prior of zero to reasonable hypotheses like the coin being loaded, which is smaller than the probability you assigned the invisible monster. If you had chosen a prior that reflected your true beliefs, you would have gotten a reasonable answer: the monster is always much less likely than the coin being loaded no matter how many times the coin comes up heads. Only evidence specifically favoring the monster (such as a failure of the cloaking device) could ever push it into your top-100 list of explanations.

Here, you are assigning zero probability to reasonable alternative explanations like cheating. The fact that you explicitly say

Suppose we live in a world where the prize must be won by chance or occur through some supernatural, unknown force.

doesn't make that supposition any less silly. You're setting your thought experiment in a world very different from ours, where none of the common sense you've developed from your life in the real world can be expected to apply. Why is it surprising that probabilities in that world seem strange? You assumed that they are strange. There is nothing ad hoc about the hypothesis of supernatural powers when no other hypotheses are possible, and you somehow know with absolute certainty that that is the case.

Answer 3

To make this example more concrete, you can simply imagine some supposed psychic trying to guess a number between 1 and 10^10 among 10^10 different people where they all think of a number in that range. Assume that each person is thinking of a number in a uniform way, so don't worry about the exact mechanism here, since that's irrelevant to the example.

Your example does not counter Bayesian reasoning:

If each of the 10^10 people is thinking of a different number, then any number the psychic guesses will have a corresponding person thinking about that number. The psychic has a 100% probability of choosing the "correct number" so it proves nothing.

The probability of the psychics success depends directly on what percentage of the available numbers have been selected from the 10^10 available including duplicate selections. So if the group selects out 70% of the available numbers the psychic has a 70% chance of selecting a number someone in the group selected. No proof of psychic behavior.

Answer 4

There's all the reason in the world to believe that the prior probability on the "psychic gets participant #492948378's number right" is less than 1 in $10^{10}$. That particular digit could have been replaced by any of the other participant's numbers, so the space of propositions relevant for modeling this situation is is H0, H1, H2...H999999999, so each of them will have a prior that scales inversely with the number of participants.

Note how the the likelihood factor and the priors scale in step in this case -- if psychic is just randomly guessing from a domain of 1 to N then the likelihood gain is about about a factor of N, but they'll need a population size P on the order of N in order to have a good chance of being successful. So the likelihood factor scales like N but the priors scale like 1/N.