# Does an improbable outcome under a known hypothesis increase the probability of an alternative hypothesis?

Suppose you observe a claimed psychic get 10 guesses right after he says he will guess a number between 1 and 10 that you’re thinking of.

After this happens, should your credence in psychism increase? What about after 200? Or 1000?

Intuitively, it seems that yes, your credence should increase. Yet when I think about it further, this all depends upon your prior probability of psychism. If you assume that it is possible apriori, then it seems that your credence should increase. If you don’t assume it is, it seems that you shouldn’t.

So what is the correct answer to this? If I had to bet on it, I don’t think it should increase your credence since I don’t see how the improbability of an outcome under chance increases the possibility of psychism. It would seem to only do this if we already knew psychism was possible apriori.

This seems to be an alternate phrasing of a question (see also this question and this question) that has been asked by the same user. The Bayesian answer to this one is pretty much the same as the answer for the old one, so I will just quote it.

Suppose for example that a person is standing on stage and says “God, if you exist, strike me with lightning right now” and a lightning strike occurs that barely misses him, is this evidence of God? Can it make it more likely that God exists?

A Bayesian analysis can be used to determine whether this evidence means we should increase our degree of belief in the existence of God. Let `M1` be a model of the universe where God exists, and `M2` be a model of the universe where God does not exist. The Bayes factor is the ratio of the marginal probability (integrating out an model parameters) of the observations `(D)` under both models:

``````K = P(D|M1)/P(D|M2)
``````

Where `P(D|M1)` is the probability of observing a lightning strike in these circumstances is M1 is true. Using Bayes rule, we can write that as

``````K = P(M1|D)P(M2)/P(M2|D)P(M1)
``````

where `P(M1)` represents our prior belief (before witnessing the lightning strike) that `M1` is the true model (the `P(D)` terms cancel). Equating and rearranging,

``````P(M1|D)/P(M2|D) = P(D|M1)/P(D|M2) x P(M1)/P(M2)
``````

So we can see that the Bayes factor tells us by how much the evidence changes our relative prior degree of belief in the two models to give us our posterior beliefs (the beliefs after seeing the evidence).

So if the probability of a lightning strike near the person is higher if God exists, `P(D|M1)`, than if god does not exist, `P(D|M2)`, then the observation will increase our rational Bayesian belief in the existence of God as `K > 1`.

The good thing about this is that we can steer clear of involving science, which is often used as a synonym for "rational" and gives a false impression of rationality when applied to subjects outside it's domain (e.g. anything non-falsifiable if you are a Popperian). This is not a criticism of science (I am a scientist of sorts myself), science gains value from its specificity.

The other good thing about this approach is that it separates out your prior beliefs and makes you state them explicitly (if you want to evaluate the posterior ratio) and it makes you explain how likely the observations are under your competing models.

This is how you should "...update your credence in a deity based on an event like this...". But there isn't enough information in the question to progress the analysis further at the current time.

Note if you are certain that God does not exist, the `P(M1)/P(M2)` is zero, and no amount of evidence will change your belief.

In short, yes, evidence that is more consistent with a hypothesis than with its negation will increase your belief (credence) in the truth of that hypothesis, unless your prior belief is zero (i.e. you consider it to be impossible a-priori, in which case it is a dogmatic belief and no amount of evidence can change your belief).

From a purely scientific perspective, the validity of a hypothesis is evaluated based on empirical evidence rather than a priori beliefs or assumptions. The scientific method involves formulating a hypothesis, making predictions based on that hypothesis, conducting experiments or observations to test those predictions, and then refining the hypothesis based on the results. This process may be repeated many times, with each cycle providing more evidence and making the hypothesis more robust.

In the case of the psychic phenomenon, the hypothesis that a person can consistently guess the number you're thinking of between 1 and 10 is testable. If the psychic gets it right significantly more often than would be expected by chance (about 10% of the time, if the numbers are chosen randomly), then that would be evidence in favor of the hypothesis.

However, it's important to be cautious about jumping to conclusions. Just because the psychic hypothesis fits the data does not automatically mean it's correct. There could be other explanations that also fit the data. Maybe the psychic is cheating, or perhaps there's a pattern to the way you choose numbers that the psychic has picked up on. In fact, these explanations are more consistent with our current scientific understanding than the notion of psychic abilities.

Can he do this with another person in another place? How do we know you are not a part of his team? There are lots of tricks like that done by so called "mentalists".

Now, regarding the a priori argument: The problem with this approach is that it's not based on empirical evidence. Instead, it's based on pre-existing beliefs or assumptions, which may or may not be valid. While it's true that we all have a priori beliefs that influence our interpretation of evidence (this is the basis of Bayesian inference), it's risky to rely too heavily on these beliefs when evaluating a hypothesis. Doing so may lead us to overlook important evidence, or make us too quick to accept a hypothesis that confirms our beliefs and too reluctant to accept one that challenges them.