Bayesian argument to believe in aliens?

why is it that all of the UFO's we see are close up and never seen further out into space? there's all sorts of telescopes being pointed "Out There[tm]" but do we ever see unusual zig-zag motions in space or flying saucer shaped asteroids? no.

all the local phenomena in the world are likely to be either man-made or natural things that some perhaps have not been explained yet, but by no means are they exhibiting anything i'd consider alien.

i have, by number, many more billions of examples of people being nutty.
 
Now are you sure you really want to throw the "supernatural" into the mix?

I see you define it as "The supernatural cases could consist of any phenomena that can’t be explained by our current understanding of science." Would that mean that thunder was supernatural until the causes were discovered and suddenly became natural after that point? No, it was merely natural but not yet understood.

Instead of generalizing the argument by using the terms “supernatural” and “natural” I should have used the terms “unexplainable” (by current science) and “explainable” (by current science).
 
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Yes. Zeus throwing down lightning from the Olymp used to be a thing.
I should probably explain that a bit more.

In ancient Greece, lightning wasn't "unexplained". The explanation was that the gods caused this.

"natural" means that something occurs by itself, without human intervention.
"artificial" means human intent has a hand in it.

If you hear something funny that makes you laugh, it's a natural reaction. If your boss cracks a "joke", your laugh may be artificial. Food that grew by itself is "natural", food ingredients created by humans in a chemical factory are artificial.

Now if we have an explanation that involves intent, but no humans, it's neither natural nor artificial, but supernatural: gods, aliens or ghosts causing stuff to happen is supernatural.

A similar division is between the physical world and the spiritual word: thoughts are not things. The physical world is sometimes called the "natural world". Phenomena that go beyond the physical world are then supernatural, and that would include ghosts, gods, and psychic powers, but not extraterrestrials, unless these ETs used psychic powers to travel here and hide themselves.

I think that's where we can connect to the "what evidence" thread: evidence of aliens is not acceptable if it requires us to believe the aliens have supernatural powers; and "breaks the known laws of physics" is just code for that.

I have great confidence in the laws of physics. I couldn't rate any witness report as "90% certain" that involves violating a physical law. @johne1618 's bayesian argument fails on this, and arguments of the type "this fuzzy blob of pixels does impossible stuff and therefore is extraterrestrial" are destined to fail from the start.
 
A similar division is between the physical world and the spiritual word: thoughts are not things. The physical world is sometimes called the "natural world". Phenomena that go beyond the physical world are then supernatural
Thoughts are not objects, but are indeed physical actions. Thinking is, in that respect, analogous to walking, and neither can be accomplished without bodily actions. The fact that we cannot see electrical impulses is immaterial; they're still part of the natural world.
 
I have great confidence in the laws of physics. I couldn't rate any witness report as "90% certain" that involves violating a physical law.

The idea of my reverse Bayesian argument is to start by estimating the likelihood of a normal explanation for the evidence as a whole without any prior prejudices. Only later does one deduce the prior probability of the paranormal required to “balance” that likelihood of a normal explanation leaving a 50% posterior belief in the paranormal.

My problem with Carl Sagan’s “Extraordinary claims require extraordinary evidence” is that it squashes each individual paranormal claim without allowing a group of good cases to be combined to offer support for the paranormal.
 
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Sorry I'm late to this party.

Let A = Hypothesis that aliens are visiting Earth

Let Not[A] = Hypothesis that aliens are not visiting Earth

Let { CE_i } = A set of i independent Close Encounter events

Using Bayes' theorem we find...

Confusion of necessity (or 'affirming the consequent') is a fallacy in formal logic which the OP perpetrates in a number of ways.

Based on the valid inference rule modus tollens (or 'denying the consequent'):

(1) The proposition 'there has been a report of a close encounter event' is not a logical (necessary) consequence of either the proposition 'aliens have visited earth' nor the proposition 'aliens have not visited earth'. If the proposition 'aliens have not visited earth' is true, the proposition 'there has been a report of a close encounter event' could still be true (while referring to an honest report).

(2) If the proposition 'aliens have visited earth' is true, the proposition 'there has never been an actual close encounter event' could be just as true as the propositions 'there has been a report of a close encounter event' or 'there has been an actual close encounter event'.

(3) Thirdly and obviously, the proposition 'there has been a report of a close encounter' does not logically imply 'there has been a close encounter', nor does the proposition 'there have been hundreds of thousands of reports of a close encounter' logically imply 'at least one of them concerns an actual close encounter'.

The foregoing unwitting violations of modus tollens inherent in the initial conditions at the start of the OP renders all the ensuing discussion on the Bayes' theorem -- both in the OP and on the thread -- irrelevant and an entirely unnecessary derail.
 
sorry I'm late to this party
at least you bring the good stuff:
(3) Thirdly and obviously, the proposition 'there has been a report of a close encounter' does not logically imply 'there has been a close encounter', nor does the proposition 'there have been hundreds of thousands of reports of a close encounter' logically imply 'at least one of them concerns an actual close encounter'.

I believe the (root?) cause why so many fall for this trap can be found in publication bias.

People usually dont share their "no ufo" encounter (I would have loved to say "no hot dog" but fear many wouldnt understand...).

"Hey reddit, day #3625 of me seeing another bird in the sky"
 
at least you bring the good stuff:

I believe the (root?) cause why so many fall for this trap can be found in publication bias.

I'm sure that's one valid cause. The other major one being our natural tendency to believe convincing declarations, especially when they come in great numbers. An all-around sensible person who sincerely shares his observation and insists on it being the truth, reinforced by many 'similar' stories the world over, can be a powerful sell. It takes a rare detachment and independence of mind to not be affected at all. A project of intellectual empowerment that MB has indeed taken on.

People usually dont share their "no ufo" encounter (I would have loved to say "no hot dog" but fear many wouldnt understand...).

:)

"Hey reddit, day #3625 of me seeing another bird in the sky"

A windblown plastic bag #666 for me. :cool:
 
I believe the (root?) cause why so many fall for this trap can be found in publication bias.
Not really, the trap is logic reversal.

This is true:
• Close Encounters cause UFO reports (theoretically) with some chance.

The false logic reversal is:
• UFO reports prove Close Encounters, with some chance.

And that's just not true; it does NOT follow logically from the first statement at all.
And therefore it's completely irrelevant what number of reports (or, as you suggest, non-reports) we have, as it fails to give us valid information.

Only an actual verifiable UFO report could do that, and one (1) would suffice! but there isn't one. (Hence my use of "theoretically" above.)
 
fully agreed.

i first thought the amount of people falling for this trap could be increased by publication bias. but you are right, its not connected.

there could be 10000^x "non ufo" reports versus 10^x "ufo" reports and they would still form the same nonsensical suggestion like:

"Its insane to think that so many people misinterpret a balloon, but for the sake of the argument assume a super low probability of them being correct."

The issue remains that by doing this, they inherit that the accepted hypothesis is already set to "ufos are real" because they attribute a probability higher than 0 even if they fail to realize what they just did.

They basically say "Lets find out what the chances of a ufo report being correct is by assuming that ufos are real. And if only every 1:100000^x is real, then this means ufos are real".

which qualifies as a "circular argument" long before doing any calculations.
 
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PhD student in probability theory here, let me explain what is wrong with OPs argument.

The problem with all of OPs arguments is that he is simply https://en.wikipedia.org/wiki/Cherry_picking evidence. Let me explain.

Let me make OPs argument mathematically rigorous: Let "A" denote the hypothesis that aliens visit earth. Let x_1,\dots, x_n denote n people that have given testimonies of Alien encounters and let CE_1, ..., CE_n denote the events that "x_1 has given testimony of Aliens", "x_2 has given testimony of Aliens", ..., and so on up to CE_n.

OP assumes that the CE_1,...,CE_n are independent conditional on A, and that P(CE_k | A)=1 as well as P(CE_k | not A) = p (in the original argument, OP uses p=0.01). Even though all three assumptions are problematic, as discussed in the thread, we will admit them in order to highlight a much more important flaw in OP's argument.

Under these assumptions, OP correctly calculates that, since the x_k's were chosen so that CE_k happened, P(A | CE_1, ..., CE_n) = P(CE_1, ..., CE_n | A) / P(CE_1, ..., CE_n) * P(A) = P(A) * 1 / (P(A) + p^n P(not A)). If P(A) is chosen very small, i.e. P(A)<<1 (our prior belief in Alien visitations is very small), then this is approximately P(A) / p^n.

OP correctly states that for n large and p small, this gives a Bayes factor of about p^{-n}, which is huge. The example that OP gives is n=10 and p=0.01, so p^{-n}=10^20 is very big.

If the x_1, ..., x_n had been fixed BEFORE testimonials were made (i.e. if you had to predict which people will make testimonials), this would be a valid scientific argument and hypotheses of Alien encounters would enter scientific debate.

However, the x_1, ..., x_n were chosen AFTER the testimonials were made. In other words, by Definition, the C_1, ..., C_n are events of testimonials that we know already happened (if different people had given testimonials, then the C_1, ..., C_n would have been chosen differently). This is the Definition of cherry-picking evidence.

This is why scientific theories need to make correct PREdictions and not POSTdictions.

How large is the pool of evidence from which one can cherry pick? This is impossible to determine in general, because the mechanism by which you choose your testimonials is unknown. Let's be modest and say that out of the 8 billion people on earth, 100 million have a good enough access to internet/media in order to popularize Alien testimonials, about 1 million will actually reach you (the other ones may be in languages you don't speak etc. so you will not have access to them) and of these 1 million people about 1'000 actually would choose to make Alien testimonials, conditional on the event that there are no Aliens (incentives for these 0.1% of people include delusion, monetary gain, attention gain, ideological reasons, etc.). The actual numbers don't matter much and are chosen pretty arbitrarily to explain how you cherry-picked data. As you did, we assume that the testimonials are independent conditional on not A.

Assuming that you cherry-picked evidence from 1 million possible testimonials, as motivated by the above calculation, the actually correct event to condition on, in this case, is the event "out of 1 million people, at least 10 chose to make Alien testimonials". Let's call the number of people who make such testimonials T. Under the assumption "not A" with the above numbers, the distribution of T is Binomial(1 million, 0.001). So P(T>= 10 | not A) is a bit more than 1 - 10^{-412}. Also, P(T>=10 | A) is 1 by assumption. The Bayes factor is thus about 1/(1-10^{-412}), i.e. about 1+10^{-412}. This is 1.0000(408 more zeros)1. In other words, negligibly small. This is not surprising: Assume that you observe 1 million people, out of which you expect 1000 to make Alien testimonials even if there are no Aliens. Then you will be completely unimpressed by 10 people giving Alien testimonials. After all, you would expect 1000 of them even if there are no Aliens!!

However, it is even worse: It is not clear how many testimonials you would expect if there were aliens. You assumed that P(C_k | A) = 1. If this assumption is also assumed for all possible testimonials, instead of just the ones that we know happened, then one would expect 1 million testimonials to take place, if A is true. If you then observe only 10, the posterior probability of A given this observation is exactly 0, since the assumption P(testimonial | A)=1 does not allow anyone to not make a testimonial.

Your problems could be ameliorated if you could demonstrate that the mechanism by which you choose the testimonials is not based on cherry-picking. For instance, if you were to choose people at random à priori and see if they turn out to make alien testimonials (this is the whole idea of random polling); [if you choose people (at random) à priori, the probability of just by chance getting all the people that give alien testimonials conditional on there not being any Aliens is exponentially small in the number of people that made testimonials that were in excess of the expectation of the number of such testimonials conditional on there not being any Aliens on earth, by https://en.wikipedia.org/wiki/Large_deviations_theory]. However, the mechanism you use is, as mentioned above, based on choosing testimonials à posteriori. Therefore, this mechanism fails to generate Bayes factors in favor of A in excess of tiny factors such as 1+10^{-400}.


Let A = Hypothesis that aliens are visiting Earth

Let Not[A] = Hypothesis that aliens are not visiting Earth

Let { CE_i } = A set of i independent Close Encounter events

Using Bayes' theorem we find that the (posterior) probability that aliens are visiting Earth given a set of i independent close encounters, P( A | { CE_i } ), is given by

P( A | { CE_i } ) = P( { CE_i } | A ) P( A ) / [ P( { CE_i } | A ) P( A ) + P( { CE_i } | Not[A] ) P( Not[A] ) ]

Now I argue that the probability of a set of i independent close encounters given the hypothesis that aliens are visiting Earth, P( { CE_i } | A ), can be taken to be one by definition. A believer in the alien hypothesis will not be surprised to learn of a set of close encounters with aliens.

I wish to argue backwards to find an expression for the prior probability for aliens P( A ) that gives us an appreciable posterior probability for aliens given reports of i close encounters, P( A | { CE_i } ) = 1/2.

Substituting P( { CE_i } | A ) = 1 and P( A | { CE_i } ) = 1/2 into Bayes' theorem we get

P( A ) / P( Not[A] ) = P( { CE_i } | Not[A] )

P( A ) / P( Not[A] ) = P( CE_1 | Not[A] ) * P( CE_2 | Not[A] ) * P( CE_3 | Not[A] ) * ...

Let us assume that aliens are not visiting Earth and that the witness was awake during the encounter (no bed-time accounts) and that the encounter was faithfully recorded soon after the event. I would say that the witness was either lying, hallucinating or the victim of a hoax.

For the sake of argument let us assume that we have a compelling case so that P( CE | Not[A] ) = 1 / 100

Let us assume that we have 10 good cases of close encounters. In order to end up with an appreciable posterior probability of aliens given those cases, i.e. P( A | { CE_i } ) = 1/2, the prior ratio of our belief to unbelief in aliens, P( A ) / P( Not[A] ) is given by

P( A ) / P( Not[A] ) = (1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100) = 10^-20

Thus if we combine data from compelling close encounter cases even the most hardened skeptic should begin to believe in the alien hypothesis.
 
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My problem with Carl Sagan’s “Extraordinary claims require extraordinary evidence” is that it squashes each individual paranormal claim without allowing a group of good cases to be combined to offer support for the paranormal.
Your reasoning here and with the math in the OP would force you to believe in literally every religion ever, since they all have collections of experiencers. So do you believe every religion ever is correct, then?
 
This is not surprising: Assume that you observe 1 million people, out of which you expect 1000 to make Alien testimonials even if there are no Aliens. Then you will be completely unimpressed by 10 people giving Alien testimonials. After all, you would expect 1000 of them even if there are no Aliens!!
This reminds me of the textbook example for false positives:
If a test for a disease has a 1% false positive rate, that means out of 100 healthy people, that test is expected to flag 1 person as sick when they're not. The question is, if you had this test done and got a positive result, what is the chance you are actually sick?

The answer depends on the prevalence of the disease.

If 1 in 100 people are actually sick, then testing 100 people results in 1 false positive (healthy person tests as sick) and 1 true positive (sick person tests as sick). 2 people get positive results, one is sick, one is healthy, therefore the probability that you're sick when your test is positive is 50%.

If the prevalence is lower, say only 1 in 1000 people are actually sick, then testing 1000 people gives us 10 false positives and 1 true positive, so now a chance of a positive person being sick has dwindled to 1/11 or ~9%.

Now let's apply this to UFO reports. If a person reports a UFO but hasn't actually seen an extraterrestrial craft, that's a false positive. (We have plenty of evidence that these happen.) If a person reports a UFO that's actually extraterrestrial, that's a true positive.

The question is, if you see a UFO report, what is the chance that it's real? And as above with the disease, the answer depends entirely on the prevalence of exterrestrial encounters. And if you don't know that, you have no way of knowing whether there are true positives at all: the statistics are meaningless.

For example, you could run smallpox tests today, and get a number of positives, but they'd all be false positives because smallpox has been eradicated. If there are no alien visitors, all we're looking at is false UFO reports.

If you want to prove alien visits, statistics is not the way to go.
 
This reminds me of the textbook example for false positives:
If a test for a disease has a 1% false positive rate, that means out of 100 healthy people, that test is expected to flag 1 person as sick when they're not. The question is, if you had this test done and got a positive result, what is the chance you are actually sick?

The answer depends on the prevalence of the disease.

If 1 in 100 people are actually sick, then testing 100 people results in 1 false positive (healthy person tests as sick) and 1 true positive (sick person tests as sick). 2 people get positive results, one is sick, one is healthy, therefore the probability that you're sick when your test is positive is 50%.

If the prevalence is lower, say only 1 in 1000 people are actually sick, then testing 1000 people gives us 10 false positives and 1 true positive, so now a chance of a positive person being sick has dwindled to 1/11 or ~9%.

Now let's apply this to UFO reports. If a person reports a UFO but hasn't actually seen an extraterrestrial craft, that's a false positive. (We have plenty of evidence that these happen.) If a person reports a UFO that's actually extraterrestrial, that's a true positive.

The question is, if you see a UFO report, what is the chance that it's real? And as above with the disease, the answer depends entirely on the prevalence of exterrestrial encounters. And if you don't know that, you have no way of knowing whether there are true positives at all: the statistics are meaningless.

For example, you could run smallpox tests today, and get a number of positives, but they'd all be false positives because smallpox has been eradicated. If there are no alien visitors, all we're looking at is false UFO reports.

If you want to prove alien visits, statistics is not the way to go.

If you wanted to use population-wide statistics to collect evidence for aliens having visited earth, you would have to demonstrate that the rate of alien encounter testimonials is much higher than what is expected if there were no aliens visiting earth. (See also caveat below.)

Let me make an example: For the smallpox test, let us say that the test false positive rate is 1%. In a population with no smallpox, you therefore expect about 1% to test positive. If you run the test on 1000 people and 20% are positive, this is strong evidence that smallpox is not, in fact, eradicated. However, if indeed 1% test positive, then you do not need any smallpox infections to explain this result.

It is the same with the Alien encounters here: If suddendly everyone came screaming that they saw Aliens, this would be good evidence that Aliens visited earth. But the reports about Aliens we have are rare individual cases, and the number of such reports does not exceed what I would expect to see if there are indeed no Aliens. So we do not need Aliens to explain these reports (or in other words, the number of reports we do see is sufficiently low so that they are not evidence in favor of Aliens).

This argument can be made even stronger: If there were indeed Aliens visiting earth, you may expect way more reports of this than you currently find. (I.e. "if Aliens have visited us, why isn't everyone talking about it?") Coming back to the smallpox example: If there was an active epidemic of smallpox in some region, then you would expect (for example) much more than 20% of tests to be positive. If you then only get a positive rate of 1%, this is strong evidence against such an active smallpox epidemic, more precisely, if your false positive rate is 1%, it is evidence that, up to the statistical error of your tests (which depends on the number of people that you test, it is not precisely defined here but can be made more rigorous if you specify a concrete Bayesian model), very few or no people are infected.

Of course all of this depends also on how you choose your model. Bayesian models are quite flexible and finding models that are useful to describe reality is a challenge, I recommend for instance this book by Andrew Gelman, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Donald Rubin on how to tackle it: http://www.stat.columbia.edu/~gelman/book/

——

Caveat: In general it is a bad idea to try and determine objective scientific truths or facts by using opinion polls (except if one wants to determine truths about the preferences, views, needs and capabilities of a population, for which polls are great and why democracy works: it gives people what they want), lest one fall prey to the bandwagon fallacy: The opinions of different people need not be independent of one another (for example during the times of Nazi Germany, the large majority of the population believed that the NSDAP regime was good for their country, this is due to many different opinions being affected simultaneously by Propaganda), and secondly, public opinion is simply not evidence in and of itself, but rather a bad estimator of the publicly available evidence : Public opinion (unlike possibly expert opinion) is not influenced by the truth when no evidence is present in forming the public‘s opinion (which, for most specific issues, seems to me to be a majority of cases), and it is more generally independent of the truth when conditioning on the available evidence (for example: That smoking cigarettes increases the probability of lung cancer is a scientific fact. About 20% of people smoke nonetheless. But this is not evidence that smoking doesn’t cause lung cancer; even if 80% of people were smokers, this would still not be evidence that smoking causes lung cancer, because conditional on the scientific evidence, the number of people that smoke is independent of whether smoking causes lung cancer or not).
 
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It is the same with the Alien encounters here: If suddendly everyone came screaming that they saw Aliens, this would be good evidence that Aliens visited earth.

——

Caveat: In general it is a bad idea to try and determine objective scientific truths or facts by using opinion polls (except if one wants to determine truths about the preferences, views, needs and capabilities of a population, for which polls are great and why democracy works: it gives people what they want), lest one fall prey to the bandwagon fallacy: The opinions of different people need not be independent of one another (for example during the times of Nazi Germany, the large majority of the population believed that the NSDAP regime was good for their country, this is due to many different opinions being affected simultaneously by Propaganda), and secondly, public opinion is simply not evidence in and of itself, but rather a bad estimator of the publicly available evidence : Public opinion (unlike possibly expert opinion) is not influenced by the truth when no evidence is present in forming the public‘s opinion (which, for most specific issues, seems to me to be a majority of cases), and it is more generally independent of the truth when conditioning on the available evidence (for example: That smoking cigarettes increases the probability of lung cancer is a scientific fact. About 20% of people smoke nonetheless. But this is not evidence that smoking doesn’t cause lung cancer; even if 80% of people were smokers, this would still not be evidence that smoking causes lung cancer, because conditional on the scientific evidence, the number of people that smoke is independent of whether smoking causes lung cancer or not).

I'm glad you inserted the caveat to contextualize the bolded point.

The statistical argument of larger sample sizes reducing the margin of error is always tricky when it concerns subjective reports by humans. It comes too close to argumentum ad populum.

Even if the majority of humanity reports ghosts, aliens or weirdly whizzing objects (which, by the way, they do at least in terms of ghosts, I live in Asia where almost everyone has seen a ghost and if you haven't, you're the weird one), that doesn't give us much to work with in terms of rigorous scientific examination.

The communal, the cultural as well as the constructivist nature of human perception and memory whereby we fill in the blanks or inaccuracies with all sorts of pre-conceived peer-influenced ideas is a significant factor in rendering all anecdotal evidence, when unsupported by non-LIZ physical records, suspect.

That most of these ghost testimonies are honest and sincere has little to do with this critical factor.
 
If you wanted to use population-wide statistics to collect evidence for aliens having visited earth, you would have to demonstrate that the rate of alien encounter testimonials is much higher than what is expected if there were no aliens visiting earth.
Yes.

more precisely, if your false positive rate is 1%, it is evidence that, up to the statistical error of your tests (which depends on the number of people that you test, it is not precisely defined here but can be made more rigorous if you specify a concrete Bayesian model), very few or no people are infected.
An important caveat here is that false positive error rates tend to depend on external factors.

For example, a Covid quick test might have a higher false positive error rate for people who suffer from a cold, because the HCoV viruses causing the cold are similar to SARS-nCoV-2. If you're testing a random population for Covid with this test, then your false error rate may be lower than if you're only testing people with cold symptoms. This means you can't conclude there's a Covid epidemic based on that difference. And then cold prevalence varies by season, so if you test a random population in the winter, you'd see more false positives than if you do that test in the summer!

So we don't really know what happened until we run a better test (which is why quick tests should be followed up by more reliable laboratory tests).

The same goes for UFO reports: if the number of reports surges, there could be an alien invasion; or a new hallucinogenic drug hit the streets, or a popular UFO TV programme has viewers look to the sky more often than usual, or the DoD is now soliciting UAP reports from all service members.

We still need better evidence; throwing statistics at UFO reports is not the way to go.
 
Let A = Hypothesis that aliens are visiting Earth

Let Not[A] = Hypothesis that aliens are not visiting Earth

Let { CE_i } = A set of i independent Close Encounter events

Using Bayes' theorem we find that the (posterior) probability that aliens are visiting Earth given a set of i independent close encounters, P( A | { CE_i } ), is given by

P( A | { CE_i } ) = P( { CE_i } | A ) P( A ) / [ P( { CE_i } | A ) P( A ) + P( { CE_i } | Not[A] ) P( Not[A] ) ]

Now I argue that the probability of a set of i independent close encounters given the hypothesis that aliens are visiting Earth, P( { CE_i } | A ), can be taken to be one by definition. A believer in the alien hypothesis will not be surprised to learn of a set of close encounters with aliens.

I wish to argue backwards to find an expression for the prior probability for aliens P( A ) that gives us an appreciable posterior probability for aliens given reports of i close encounters, P( A | { CE_i } ) = 1/2.

Substituting P( { CE_i } | A ) = 1 and P( A | { CE_i } ) = 1/2 into Bayes' theorem we get

P( A ) / P( Not[A] ) = P( { CE_i } | Not[A] )

P( A ) / P( Not[A] ) = P( CE_1 | Not[A] ) * P( CE_2 | Not[A] ) * P( CE_3 | Not[A] ) * ...

Let us assume that aliens are not visiting Earth and that the witness was awake during the encounter (no bed-time accounts) and that the encounter was faithfully recorded soon after the event. I would say that the witness was either lying, hallucinating or the victim of a hoax.

For the sake of argument let us assume that we have a compelling case so that P( CE | Not[A] ) = 1 / 100

Let us assume that we have 10 good cases of close encounters. In order to end up with an appreciable posterior probability of aliens given those cases, i.e. P( A | { CE_i } ) = 1/2, the prior ratio of our belief to unbelief in aliens, P( A ) / P( Not[A] ) is given by

P( A ) / P( Not[A] ) = (1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100)*(1/100) = 10^-20

Thus if we combine data from compelling close encounter cases even the most hardened skeptic should begin to believe in the alien hypothesis.

Surely any Bayesian argument depends 100% on your priors....your 'given that..'

The problem lies in how you define 'compelling case'. One man's 'compelling' may be another man's ' nah...nothing to see here'.
 
Now I argue that the probability of a set of i independent close encounters given the hypothesis that aliens are visiting Earth, P( { CE_i } | A ), can be taken to be one by definition. A believer in the alien hypothesis will not be surprised to learn of a set of close encounters with aliens.
Please use the Reply button so we can see the post you are referring to.
 
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