# Bayesian argument for combining extraordinary claims

This is an improved version of Backwards Bayesian argument for alien visitation?

It is said that extraordinary claims require extraordinary evidence and therefore this criterion is applied to every claim independently. But can extraordinary claims be legitimately combined before being weighed against our prior knowledge?

Let A = the proposition that aliens are visiting Earth

Let {C} = a set of witness testimonies of different close encounter events

I wish to use Bayes theorem backwards to deduce the prior odds P(A) / P(Not A) given that I end up with an evens posterior odds P(A | {C}) / P(Not A | {C}) = 1.

Bayes theorem can be written in terms of odds as:

P(A | {C}) / P(Not A | {C}) = P({C} | A) / P({C} | Not A) * P(A) / P(Not A)

If P(A | {C}) / P(Not A | {C}) = 1 we obtain the following expression for the prior odds given by

P(A) / P(Not A) = P({C} | Not A) / P({C} | A)

As the events are independent we can assume the witness testimonies are as well so that

P({C}) = P(C_1) * P(C_2) * P(C_3) * ...

Therefore we have

P(A) / P(Not A) = Product_i [ P(C_i | Not A) / P(C_i | A) ]

Let us assume there are 4 explanations for a typical close encounter witness testimony:

1. They are lying.
2. They were hallucinating.
3. They were the victims of a hoax.
4. It was aliens.

As we are assuming no prior knowledge at this stage we can assign equal probability to each alternative.

Therefore:

P(C_i | Not A) = P_1 + P_2 + P_3 = 3/4

P(C_i | A) = P_1 + P_2 + P_3 + P_4 = 1

Therefore the prior odds of alien visitation implied by even posterior odds is given by

P(A) / P(Not A) = (3/4)^N

where N is the number of close encounter testimonies.

Thus, given 100 testimonies of close encounters with aliens, even if our prior odds for alien visitation is only 1 in 10^13 we end up with a posterior 50% belief in alien visitation.

• This depends on the technical issue of whether one can treat an inferential statement like "aliens are visiting Earth" as if it was a single event when updating Bayesian probabilities based on eyewitness accounts. It was recently asked on Cross Validated, but there is no answer so far. Also, "assuming no prior knowledge at this stage we can assign equal probability" is problematic because one can easily manipulate the answer by manipulating the options this way, as Pascal's wager shows. You can easily get a fraction other than 3/4 here. Aug 27, 2022 at 17:37
• The 3/4 was only an illustration. For each witness testimony I would list all the realistic non-alien alternatives I could think of and then assign them admittedly subjective probabilities. I think the important point is that the prior odds required for an evens posterior belief in alien visitation decays exponentially with the number of close encounter cases. Aug 27, 2022 at 21:08
• "Extraordinary claims require extraordinary evidence" means that "it was aliens" should be marked as far less likely than ordinary alternatives. So maybe you can get your estimate to be valid (if the inference updating works as for single events) under the equiprobability assumption, but it would not mean much because the assumption is unrealistic. For that matter, "they were misinterpreting what they saw" seems most likely to me, but is not even listed. Aug 27, 2022 at 21:18