I want to say a little more about this calculation (in which I've corrected a misplaced decimal):
That's using Sam's numbers, the most important of which seems to be this:
Chance of Seeing This Evidence If H0 Is True: 0.0001 (0.01%—super low, because if it's all natural, it'd be weird to have so many matching, detailed reports without huge coincidences). — Sam26
You'll note that making this value really small is what makes the posterior probability so high, regardless of how low the prior probability was. It's a ratio: on top is the chance, however low, you assign to consciousness surviving death and people reporting that they experienced this; on the bottom is the total chance of people reporting that they have experienced survival, whether it happened or not (so we add the two cases to get a total).
One way to think of this is as an explanation of how quickly people can update. Consider the characters in a science fiction movie: maybe they don't believe in monsters or aliens, but when one is right in front of them, they might initially resist thinking it's real, but if it demonstrates that it is, they very quickly adjust. Similarly for rare events in real life. You may know for a fact that airplane crashes, church shootings, and tornadoes are rare, but when you're in one, you believe it not quite immediately, but quickly.
This is what Sam is asking of us here. The idea is that something you think unlikely has in fact happened: you never in a million years expected someone to tell you they had experienced the afterlife, but here they are. My prior credence was low; I've gotten the extremely unlikely evidence; now my posterior credence is high. The
more unlikely the evidence, the higher my credence will now be. (Hence, Sam above comparing these reports to "coincidences", which raises other issues not addressed here.)
Of course, that is not the view of the skeptic at all. There are two possibilities:
(1) Skeptics believe that these reports are not evidence of an afterlife, and therefore the likelihood of someone offering such a report, having had a near brush with death, just is whatever it is in real life. If five million people last year nearly died but survived, and five thousand of those reported experiencing the afterlife, then the odds of a survivor making such a report are 1000 : 1, and that's it. Whether there's an afterlife doesn't enter into it. Bayes's rule has no use here at all.
(2) Skeptics believe the reports do count, but not so much.
Let's look at how (2) works with an example.
Suppose the chances are 9 in 10 that people will comment favorably on a cute outfit. Suppose further that the chances are 3 in 10 they will comment favorably on an uncute outfit, out of politeness, etc.
How likely are people to say that your outfit is cute? We can't say, because we don't know the base rate ― we don't know how likely your outfit is to
actually be cute, so we can't do the calculation. Let's say half your outfits are cute. Out of 20 outfits you wear, 10 of them are cute and you get 9 comments, 10 of them are not cute and you get 3 comments; altogether you get 13 comments out of 20.
Now for the important question: what are the chances that your outfit is cute, given a favorable comment? 9 out of 10? 13 out of 20? Nope. The chances are given by the likelihood ratio of comments on cute outfits to comments on uncute outfits, scaled by the base rate. Given our 50-50 base rate, the chances that your outfit is cute, given a nice comment, are 3 in 4 (because genuine comments are three times more likely). But if only a quarter of your outfits are genuinely cute, a favorable comment makes it only even money that this is one of the cute ones. If only 1 out of 10 of your outfits are cute, the favorable comment gives you only a 1 in 4 chance that this is a cute one.
For our problem, let's say the skeptic considers the odds there's an afterlife a colloquial "million to one". That's the prior. To calculate the posterior odds, we need to know how much more likely we are to get reports of an afterlife, if there is one than if there isn't. It doesn't matter what the odds are, really ― both can be pretty likely or unlikely ― what matters is the ratio. Sam's estimate was that we are 2000 times more likely to get reports if there is an afterlife (0.2 : 0.0001).
Having gotten these reports, what would the skeptic say are the odds there's an afterlife? It's the likelihood ratio scaled by the base bate, in (rounded) odds form:
Still 500 to 1 against.
It's as if the skeptic says, out of a million and one universes, one of them is cute; reports of an afterlife are two thousand times more likely in that universe than in any of the other million; we have those reports, so what are the odds we're in that universe? Bigger than you might think, but still small because the base rate controls. Even if people in the cute universe are dramatically more likely to report an afterlife experience, our chances of being in such a universe ― according to the skeptic ― are so small that they remain small, even when we have those reports.
Sam's skeptic picked a colloquial prior of "a hundred to one", so instead the calculation was (rounding again):
or 20 chances out of 21, which is about 95%.
So it turns out ― as it almost always does with these kinds of problems ― that the most important estimate Sam gives is not (as I suggested above) the relative likelihood of reports, but
the base rate.
If you want to leave open the possibility that we live in a cute universe, you still have to consider:
(a) whether the reports that we do are acceptable as evidence at all;
(b) how much more likely that evidence, if accepted, is in cute universes rather than uncute ones; and
(c) how likely it is that we live in a cute universe.
What will determine whether this evidence controls is the difference between the likelihood ratio of the evidence, in (b), and the base rate of cuteness you give credence to in (c). Is one orders of magnitude bigger than the other? Which one? Sam gets the result he does by treating the evidence as twenty times more likely in the favorable case than the favorable case is unlikely.
(I'm not saying anything about how we might settle on one value or another here. It's just my understanding of the math, particularly for people who found that "95%" somewhat eye-popping. Ignore if you're better at probability than I am.)