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Let A = the hypothesis that aliens are visiting Earth

Let E = evidence that aliens are visiting Earth

The posterior probability that aliens are visiting Earth, given some evidence, P(A|E), can be derived from the likelihood of the evidence P(E|A) and the prior P(A), using Bayes' theorem

P(A|E) = P(E|A) * P(A) / [ P(E|A) * P(A) + P(E|Not[A]) * P(Not[A]) ]. (1)

I wish to argue backwards from the assumption of a significant posterior probability in order to find an expression for the prior probability given the likelihoods

P(A|E) = 1/2. (2)

Substituting Eqn.2 into Eqn.1 we find

P(A) / P(Not[A]) = P(E|Not[A]) / P(E|A). (3)

Now let us assume that the evidence is made of cases that have natural explanations, N, and cases that don't have natural explanations Not[N].

P(E) = P(E) P(N) + P(E) P(Not[N]) (4)

On the hypothesis that aliens are visiting Earth both terms on the RHS of Eqn.4 are present so that we have

P(E|A) = P(E) (5)

On the hypothesis that aliens are not visiting Earth the second term on the RHS of Eqn.4 is zero (no non-natural explanations) so that we have

P(E|Not[A]) = P(E) P(N) (6)

Substituting Eqn.5 and Eqn.6 into Eqn.3 we obtain

P(A) / P(Not[A] = P(N) (7)

Let us assume that we have a report of a close encounter where the witnesses seem to be of good character, in a normal state of consciousness and were unlikely to be the victims of a hoax. For example see the following Winchester,UK encounter of 1976 with witnesses Joyce Bowles and Ted Pratt

https://youtu.be/x0Ec5NJLWOw

Let us assume that the probability of a natural explanation for the Winchester 1976 case (lying, hallucinating or victims of a hoax) is

P(N) = 1/10. (8)

Thus, if we take P(N) in Eqn.7 to refer to the probability that a group of close encounter cases can be explained by natural causes, we only need 10 good independent cases like the one above in order to deduce that even if the prior odds of alien visitation is only 1:10^10 we will still end up with significant posterior odds of 1:1.

Is this backwards argument from likelihood to prior valid?

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  • From a formal mathematical POV you can of course assume you know the values for the posterior probability and the conditional probabilities and use that to derive a value for the prior probability, but this would go against Bayesian inference as a methodology, which is supposed to be about updating your priors given evidence.
    – Hypnosifl
    May 6, 2022 at 19:16
  • This particular application of the Bayesian method is on shaky ground because (as I read it) it seeks to determine whether (a) Earth is being visited by aliens and the witnesses are correct, or (b) Earth is not being visited by aliens and the witnesses are mistaken, although in good faith. It doesn’t consider a third possibility that Earth is not being visited by aliens but some other visitors. A conspiracy-monger might point out that there’s an infinite variety of other possibilities and so the probability of at least one of these being true approaches unity.
    – Frog
    May 6, 2022 at 22:09
  • @Frog If the Earth is being visited at all then one can define those visitors as aliens for the purpose of the argument.
    – John Eastmond
    May 7, 2022 at 15:59
  • @John Eastmond true but there are explanations that don’t involve anything that could reasonably be considered a visitor. I don’t propose to go down that rabbit hole though
    – Frog
    May 7, 2022 at 20:57

1 Answer 1

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You have some dubious math here. The first issue is, you equivocate over what "E" is; is E a single piece of evidence, or is it all the evidence together?

A second issue is where you do:

Substituting Eqn.5 and Eqn.6 into Eqn.3

Equation 5 is only valid under the assumption that aliens are visiting Earth, and equation 6 is only valid under the assumption that aliens are not visiting Earth, and equation 3 is valid only under the assumption that P(A|E) = 1/2. You can't just use them all together as if you had derived them in the same context.

Leaving that aside, you end by saying that if one piece of evidence increases the odds ratio of aliens by a certain ratio, then ten pieces of evidence must increase it by that same ratio, ten times! That's not how evidence works. Imagine if you're investigating a crime and you find a hair from a suspect, which you decide makes it 10 times more likely the suspect committed the crime. If you find 10 hairs does that make it 10^10 times more likely? Of course not. The extra hairs don't really tell you much beyond what the first hair told you.

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  • Combining likelihoods of the data under different hypotheses in order to update one's prior belief given the data is how Bayes' theorem works. I should have explained more clearly that I generalize the argument to 10 independent close encounter cases so that I can say P(N)=10^-10.
    – John Eastmond
    May 7, 2022 at 12:30
  • @JohnEastmond No, that's not how Bayes' theorem works. You combine eqns 5, 6, and 3 to get P(A)/P(not A) = P(N) which is obviously false, since P(A)/P(not A) might be >1, and P(N) is <1. And the close encounter cases are not independent. The probability a tenth person says they saw aliens, given that nine people already said so, is different from the probability the first person says they saw aliens when no one had said that before.
    – causative
    May 7, 2022 at 14:25
  • @JohnEastmond Also, in practice, assuming by P(N) you mean the probability of a natural explanation for an individual alien report (without which assumption your equations 5 and 6 make no sense), P(N) is going to be close to 1 since most ufo reports do have other explanations. Whereas most would agree that the prior P(A)/P(not A) is close to 0 (this being prior to any evidence). So this should be another clue you have made a math error somewhere.
    – causative
    May 7, 2022 at 14:33
  • I assume P(A) + P(not A) = 1. When I consider a group of cases I assume that they are from different parts of the world and from different years. The only common aspect between the witnesses of the different cases is that everyone has been aware of the idea of alien visitation for the last 75 years. When I look at the Winchester case I find it hard to give any probability to hallucination or hoax. As for lying I think I am being very skeptical to give it 10%. The whole idea of the argument is one can have a low prior of 10^-10 and yet after 10 good cases end up a 50% believer.
    – John Eastmond
    May 7, 2022 at 15:55
  • @JohnEastmond Yes, that whole idea is incorrect. Generally speaking, in any context, there are diminishing returns to additional evidence of the same kind. If going from 0 reports to 1 report makes it 10x as likely, going from 10 reports to 100 reports does not make it 10^90 times as likely; in fact it barely makes it more likely at all. It does indicate there is probably some common cause for the 100 reports, but the common cause doesn't have to be aliens; it can be weather phenomena, hallucinations, or people lying for attention.
    – causative
    May 7, 2022 at 16:26

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