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I'm afraid I have to take severe exception to this statement.
You give the human species far too much credit if you think that our mere ability to dream up a hypothesis automatically raises its probability above some uniform lower bound.
I am aware of your disagreement, for example as expressed by the absurd claims here. Yes, my basic idea is, unlike you, to give some credit to the human species. I think there's a limit on how much you can disagree with other human beings-- unless you're claiming to be something superhuman.
Did you see the link to this comment thread? I would like to see your response to the discussion there.
At least for epistemic meanings of "superhuman", that's pretty much the whole purpose of LW, isn't it?
My immediate response is as follows: yes, dependency relations might concentrate most of the improbability of a religion to a relatively small subset of its claims. But the point is that those claims themselves possess enormous complexity (which may not necessarily be apparent on the surface; cf. the simple-sounding "the woman across the street is a witch; she did it").
Let's pick an example. How probable do you think it is that Islam is a true religion? (There are several ways to take care of logical contradictions here, so saying 0% is not an option.)
Suppose there were a machine--for the sake of tradition, we can call it Omega--that prints out a series of zeros and ones according to the following rule. If Islam is true, it prints out a 1 on each round, with 100% probability. If Islam is false, it prints out a 0 or a 1, each with 50% probability.
Let's run the machine... suppose on the first round, it prints out a 1. Then another. Then another. Then another... and so on... it's printed out 10 1's now. Of course, this isn't so improbable. After all, there was a 1/1024 chance of it doing this anyway, even if Islam is false. And presumably we think Islam is more likely than this to be false, so there's a good chance we'll see a 0 in the next round or two...
But it prints out another 1. Then another. Then another... and so on... It's printed out 20 of them. Incredible! But we're still holding out. After all, million to one chances happen every day...
Then it prints out another, and another... it just keeps going... It's printed out 30 1's now. Of course, it did have a chance of one in a billion of doing this, if Islam were false...
But for me, this is my lower bound. At this point, if not before, I become a Muslim. What about you?
You've been rather vague about the probabilities involved, but you speak of "double digit negative exponents" and so on, even saying that this is "conservative," which implies possibly three digit exponents. Let's suppose you think that the probability that Islam is true is 10^-20; this would seem to be very conservative, by your standards. According to this, to get an equivalent chance, the machine would have to print out 66 1's.
If the machine prints out 50 1's, and then someone runs in and smashes it beyond repair, before it has a chance to continue, will you walk away, saying, "There is a chance at most of 1 in 60,000 that Islam is true?"
If so, are you serious?
Thank you a lot for posting this scenario. It's instructive from the "heuristics and biases" point of view.
Imagine there are a trillion variants of Islam, differing by one paragraph in the holy book or something. At most one of them can be true. You pick one variant at random, test it with your machine and get 30 1's in a row. Now you should be damn convinced that you picked the true one, right? Wrong. Getting this result by a fluke is 1000x more likely than having picked the true variant in the first place. Probability is unintuitive and our brains are mush, that's all I'm sayin'.
I agree with this. But if the scenario happened in real life, you would not be picking a certain variant. You would be asking the vague question, "Is Islam true," to which the answer would be yes if any one of those trillion variants, or many others, were true.
Yes, there are trillions of possible religions that differ from one another as much as Islam differs from Judaism, or whatever. But only a few of these are believed by human beings. So I still think I would convert after 30 1's, and I think this would reasonable.
If a religion's popularity raises your prior for it so much, how do you avoid Pascal's Mugging with respect to the major religions of today? Eternity in hell is more than 2^30 times worse than anything you could experience here; why aren't you religious already?
It doesn't matter whether it raises your prior or not; eternity in hell is also more than 2^3000 times worse etc... so the same problem will apply in any case.
Elsewhere I've defended Pascal's Wager against the usual criticisms, and I still say it's valid given the premises. But there are two problematic premises:
1) It assumes that utility functions are unbounded. This is certainly false for all human beings in terms of revealed preference; it is likely false even in principle (e.g. the Lifespan Dilemma).
2) It assumes that humans are utility maximizers. This is false in fact, and even in theory most of us would not want to self-modify to become utility maximizers; it would be a lot like self-modifying to become a Babyeater or a Super-Happy.
Do you have an answer for how to avoid giving in to the mugger in Eliezer's original Pascal's Mugging scenario? If not, I don't think your question is a fair one (assuming it's meant to be rhetorical).
I don't have a conclusive answer, but many people say they have bounded utility functions (you see Unknowns pointed out that possibility too). The problem with assigning higher credence to popular religions is that it forces your utility bound to be lower if you want to reject the mugging. Imagining a billion lifetimes is way easier than imagining 3^^^^3 lifetimes. That was the reason for my question.
My answer (for why I don't believe in a popular religion as a form of giving in to a Pascal's Mugging) would be that I'm simultaneously faced with a number of different Pascal's Muggings, some of which are mutually exclusive, so I can't just choose to give in to all of them. And I'm also unsure of what decision theory/prior/utility function I should use to decide what to do in the face of such Muggings. Irreversibly accepting any particular Mugging in my current confused state is likely to be suboptimal, so the best way forward at this point seems to be to work on the relevant philosophical questions.
Oddly, I think you meant "Pascal's Wager".
Pascal's Mugging. Pascal's Wager with something breaking symmetry (in this case observed belief of others).
Yes, I suppose it is technically a Pascal's Mugging. I think Pascal thought he was playing Pascal's Mugging though.
Privileging the hypothesis! That they are believed by human beings doesn't lend them probability.
Well, it does to the extent that lack of believers would be evidence against them. I'd say that Allah is considerably more probable than a similarly complex and powerful god who also wants to be worshiped and is equally willing to interact with humans, but not believed in by anyone at all. Still considerably less probable than the prior of some god of that general sort existing, though.
Agreed, but then we have the original situation, if we only consider the set of possible gods that have the property of causing worshiping of themselves.
No. It doesn't lend probability, but it seems like it ought to lend something. What is this mysterious something? Lets call it respect.
Privileging the hypothesis is a fallacy. Respecting the hypothesis is a (relatively minor) method of rationality.
We respect the hypotheses that we find in a math text by investing the necessary mental resources toward the task of finding an analytic proof. We don't just accept the truth of the hypothesis on authority. But on the other hand, we don't try to prove (or disprove) just any old hypothesis. It has to be one that we respect.
We respect scientific hypotheses enough to invest physical resources toward performing experiments that might refute or confirm them. We don't expend those resources on just any scientific hypothesis. Only the ones we respect.
Does a religion deserve respect because it has believers? More respect if it has lots of believers? I think it does. Not privilege. Definitely not. But respect? Why not?
No, it's a method of anti-epistemic horror.
You can dispense with this particular concept of respect since in both your examples you are actually supplied with sufficient Bayesian evidence to justify evaluating the hypothesis, so it isn't privileged. Whether this is also the case for believed in religions is the very point contested.
Yes, this seems right.
A priori, with no other evidence one way or another, a belief held by human beings is more likely to be true than not. If Ann says she had a sandwich for lunch, then her words are evidence that she actually had a sandwich for lunch.
Of course, we have external reason to doubt lots of things that human beings claim and believe, including religions. And a religion does not become twice as credible if it has twice as many adherents. Right now I believe we have good reason to reject (at least some of) the tenets of all religious traditions.
But it does make some sense to give some marginal privilege or respect to an idea based on the fact that somebody believes it, and to give the idea more credit if it's very durable over time, or if particularly clever people believe it. If it were any subject but religion -- if it were science, for instance -- this would be an obvious point. Scientific beliefs have often been wrong, but you'll be best off giving higher priors to hypotheses believed by scientists than to other conceivable hypotheses.
Also... if you haven't been to Australia, is it privileging the hypothesis to accept the word of those who say that it exists? There are trillions of possible countries that could exist that people don't believe exist...
And don't tell me they say they've been there... religious people say they've experienced angels etc. too.
And so on. People's beliefs in religion may be weaker than their belief in Austrialia, but it certainly is not privileging a random hypothesis.
Your observations (of people claiming to having seen an angel, or a kangaroo) are distinct from hypotheses formed to explain those observations. If in a given case, you don't have reason to expect statements people make to be related to facts, then the statements people make taken verbatim have no special place as hypotheses.
"You don't have reason to expect statements people make to be related to facts" doesn't mean that you have 100% certainty that they are not, which you would need in order to invoke privileging the hypothesis.
Begging the question!
This whole discussion is about this very point. Downvoted for contradicting my position without making an argument.
Your position statement didn't include an argument either, and the problem with it seems rather straightforward, so I named it.
I've been arguing with Sewing Machine about it all along.
I disagree; given that most of the religions in question center on human worship of the divine, I have to think that Pr(religion X becomes known among humans | religion X is true) > Pr(religion X does not become known among humans | religion X is true). But I hate to spend time arguing about whether a likelihood ratio should be considered strictly equal to 1 or equal to 1 + epsilon when the prior probabilities of the hypotheses in question are themselves ridiculously small.
Of course I'm serious (and I hardly need to point out the inadequacy of the argument from the incredulous stare). If I'm not going to take my model of the world seriously, then it wasn't actually my model to begin with.
Sewing-Machine's comment below basically reflects my view, except for the doubts about numbers as a representation of beliefs. What this ultimately comes down to is that you are using a model of the universe according to which the beliefs of Muslims are entangled with reality to a vastly greater degree than on my model. Modulo the obvious issues about setting up an experiment like the one you describe in a universe that works the way I think it does, I really don't have a problem waiting for 66 or more 1's before converting to Islam. Honest. If I did, it would mean I had a different understanding of the causal structure of the universe than I do.
Further below you say this, which I find revealing:
As it happens, given my own particular personality, I'd probably be terrified. The voice in my head would be screaming. In fact, at that point I might even be tempted to conclude that expected utilities favor conversion, given the particular nature of Islam.
But from an epistemic point of view, this doesn't actually change anything. As I argued in Advancing Certainty, there is such a thing as epistemically shutting up and multiplying. Bayes' Theorem says the updated probability is one in a hundred billion, my emotions notwithstanding. This is precisely the kind of thing we have to learn to do in order to escape the low-Earth orbit of our primitive evolved epistemology -- our entire project here, mind you -- which, unlike you (it appears), I actually believe is possible.
Has anyone done a "shut up and multiply" for Islam (or Christianity)? I would be interested in seeing such a calculation. (I did a Google search and couldn't find anything directly relevant.) Here's my own attempt, which doesn't get very far.
Let H = "Islam is true" and E = everything we've observed about the universe so far. According to Bayes:
P(H | E) = P(E | H) P(H) / P(E)
Unfortunately I have no idea how to compute the terms above. Nor do I know how to argue that P(H|E) is as small as 10^-20 without explicitly calculating the terms. One argument might be that P(H) is very small because of the high complexity of Islam, but since E includes "23% of humanity believe in some form of Islam", the term for the complexity of Islam seems to be present in both the numerator and denominator and therefore cancel each other out.
If someone has done such a calculation/argument before, please post a link?
Actually it doesn't, human generated complexity is different from naturally generated complexity (for instance it fits into narratives, apparent holes are filled with the sort of justifications a human is likely to think of etc.). That's one of the ways you can tell stories from real events. Religious accounts contain much of what looks like human generated complexity.
Here's a somewhat rough way of estimating probabilities of unlikely events. Let's say that an event X with P(X) = about 1-in-10 is a "lucky break." Suppose that there are L(1) ways that Y could occur on account of a single lucky break, L(2) ways that Y could occur on account of a pair of independent lucky breaks, L(3) ways that Y could occur on account of 3 independent lucky breaks, and so on. Then P(Y) is approximately the sum over all n of L(n)/10^n. I have the feeling that arguments about whether P(Y) is small versus extremely small are arguments about the growth rate of L(n).
I discussed the problem of estimating P("23% of humanity believes...") here. I'd be grateful for thoughts or criticisms.
P(E) includes the convincingness of Islam to people on average, not the complexity of Islam. These things are very different because of the conjunction fallacy. So P(H) can be a lot smaller than P(E).
I don't understand how P(E) does not include a term for the complexity of Islam, given that E contains Islam, and E is not so large that it takes a huge number of bits to locate Islam inside E.
I don't think that's true; cousin_it had it right the first time. The complexity of Islam is the complexity of a reality that contains an omnipotent creator, his angels, Paradise, Hell, and so forth. Everything we've observed about the universe includes people believing in Islam, but not the beings and places that Islam says exist.
In other words, E contains Islam the religion, not Islam the reality.
The really big problem with such a reality is that it contains a fundamental, non-contingent mind (God's/Allah's, etc) - and we all know how much describing one of those takes - and the requirement that God is non-contingent means we can't use any simpler, underlying ideas like Darwinian evolution. Non-contingency, in theory selection terms, is a god killer: It forces God to incur a huge information penalty - unless the theist refuses even to play by these rules and thinks God is above all that - in which case they aren't even playing the theory selection game.
It doesn't take a lot of bits to locate "Islam is false" based on "Islam is true". Does it mean that all complex statements have about 50% probability?
I just wrote a post about that.
Whoops, you're right. Now I'm ashamed that my comment got upvoted.
I think the argument may still be made to work by fleshing out the nonstandard notion of "complexity" that I had in my head when writing it :-) Your prior for a given text being true shouldn't depend only on the text's K-complexity. For example, the text "A and B and C and D" has the same complexity as "A or B or C or D", but the former is way less probable. So P(E) and P(H) may have the same term for complexity, but P(H) also gets a "conjunction penalty" that P(E) doesn't get because people are prey to the conjunction fallacy.
EDIT: this was yet another mistake. Such an argument cannot work because P(E) is obviously much smaller than P(H), because E is a huge mountain of evidence and H is just a little text. When trying to reach the correct answer, we cannot afford to ignore P(E|H).
For simplicity we may assume P(E|H) to be near-certainty: if there is an attention-seeking god, we'd know about it. This leaves P(E) and P(H), and P(H|E) is tiny exactly for the reason you named: P(H) is much smaller than P(E), because H is optimized for meme-spreading to a great extent, which makes for a given complexity (that translates into P(H)) probability of gaining popularity P(E) comparatively much higher.
Thus, just arguing from complexity indeed misses the point, and the real reason for improbability of cultish claims is that they are highly optimized to be cultish claims.
For example, compare with tossing a coin 50 times: the actual observation, whatever that is, will be a highly improbable event, and theoretical prediction from the model of fair coin will be too. But if the observation is highly optimized to attract attention, for example it's all 50 tails, then theoretical model crumbles, and not because the event you've observed is too improbable according to it, but because other hypotheses win out.
There are some very crude sketches of shutting-up-and-multiplying, from one Christian and a couple of atheists, here (read the comments as well as the post itself), and I think there may be more with a similar flavour in other blog posts there (and their comments) from around the same time.
(The author of the blog has posted a little on LW. The two skeptics responsible for most of the comments on that post have both been quite active here. One of them still is, and is in fact posting this comment right now :-).)
Wei Dai, exactly. The point about about the complexity of the thing is included in the fact that people believe it was the point I have been making all along. Regardless of what you think the resulting probability is, most of the "evidence" for Islam consists in the very fact that some people think it is true-- and as you show in your calculation, this is very strong evidence.
It seems to me that komponisto and others are taking it to be known with 100% certainly that Islam and the like were generated by some random process, and then trying to determine what the probability would be.
Now I know that most likely Mohammed was insane and in effect the Koran was in fact generated by a random process. But I certainly don't know how you can say that the probability that it wasn't generated randomly is 1 in 10^20 or lower. And in fact if you're going to assign a probability like this you should have an actual calculation.
This is a small point but "E includes complex claim C" does not imply that the (for instance, Kolmogorov) complexity of E is as large as the Kolmogorov complexity of C. The complexity of the digits of square root of 2 is pretty small, but they contain strings of arbitrarily high complexity.
E includes C implies that K(C) <= K(E) + K(information needed to locate C within E). In this case K(information needed to locate C within E) seems small enough not to matter to the overall argument, which is why I left it out. (Since you said "this is a small point" I guess you probably understand and agree with this.)
Actually no I hadn't thought of that. But I wonder if the amount of information it takes to locate "lots of people are muslims" within E is as small as you say. My particular E does not even contain that much information about Islam, and how people came to believe it, but it does contain a model of how people come to believe weird things in general. Is that a misleading way of putting things? I can't tell.
I agree that your position is analogous to "shutting up and multiplying." But in fact, Eliezer may have been wrong about that in general -- see the Lifespan Dilemma -- because people's utility functions are likely not unbounded.
In your case, I agree with shutting up and multiplying when we have a way to calculate the probabilities. In this case, we don't, so we can't do it. If you had a known probability (see cousin_it's comment on the possible trillions of variants of Islam) of one in a trillion, then I would agree with walking away after seeing 30 1's, regardless of the emotional effect of this.
But in reality, we have no such known probability. The result is that you are going to have to use some base rate: "things that people believe" or more accurately, "strange things that people believe" or whatever. In any case, whatever base rate you use, it will not have a probability anywhere near 10^-20 (i.e. more than 1 in 10^20 strange beliefs is true etc.)
My real point about the fear is that your brain doesn't work the way your probabilities do-- even if you say you are that certain, your brain isn't. And if we had calculated the probabilities, you would be justified in ignoring your brain. But in fact, since we haven't, your brain is more right than you are in this case. It is less certain precisely because you are simply not justified in being that certain.
At this point, if not before, I doubt Omega's reliability, not mine.
It is a traditional feature of Omega that you have confidence 1 in its reliability and trustworthiness.
Traditions do not always make sense, neither are they necessarily passed down accurately. The original Omega, the one that appears in Newcomb's problem, does not have to be reliable with probability 1 for that problem to be a problem.
Of course, to the purist who says that 0 and 1 are not probabilities, you've just sinned by talking about confidence 1, but the problem can be restated to avoid that by asking for one's conditional probability P(Islam | Omega is and behaves as described).
In the present case, the supposition that one is faced with an overwhelming likelihood ratio raising the probability that Islam is true by an unlimited amount is just a blue tentacle scenario. Any number that anyone who agrees with the general anti-religious view common on LessWrong comes up with is going to be nonsense. Professing, say, 1 in a million for Islam on the grounds that 1 in a billion or 1 in a trillion is too small a probability for the human brain to cope with is the real cop-out, a piece of reversed stupidity with no justification of its own.
The scenario isn't going to happen. Forcing your brain to produce an answer to the question "but what if it did?" is not necessarily going to produce a meaningful answer.
Quite true. But if you want to dispute the usefulness of this tradition, you should address the broader and older tradition of which it is an instance: that thought experiments should abstract away real-world details irrelevant to the main point.
This is a pet peeve of mine, and I've wanted an excuse to post this rant for a while. Don't take it personally.
That "purist" is as completely wrong as the person who insists that there is no such thing as centrifugal force. They are ignoring the math in favor of a meme that enables them to feel smugly superior.
0 and 1 are valid probabilities in every mathematical sense: the equations of probability don't break down when passed p=0 or p=1 the way they do with genuine nonprobabilities like -1 or 2. A probability of 0 or 1 is like a perfect vacuum: it happens not to occur in the world that we happen to inhabit, but it is perfectly well-defined, we can do math with it without any difficulty, and it is extraordinarily useful in thought experiments.
When asked to consider a spherical black body of radius one meter resting on a frictionless plane, you don't respond "blue tentacles", you do the math.
I agree with the rant. 0 and 1 are indeed probabilities, and saying that they are not is a misleading way of enjoining people to never rule out anything. Mathematically, P(~A|A) is zero, not epsilon, and P(A|A) is 1, not 1-epsilon. Practically, 0 and 1 in subjective judgements mean as near to 0 and 1 as makes no practical difference. When I agree a rendezvous with someone, I don't say "there's a 99% chance I'll be there", I say "I'll be there".
Where we part ways is in our assessment of the value of this thought-experiment. To me it abstracts and assumes away so much that what is left does not illuminate anything. I can calculate 2^{-N}, but asked how large N would have to be to persuade me of some fantastic claim backed by this fantastic machine I simply cannot name any value. I have no confidence that whatever value I named would be the value I would actually use were this impossible scenario to come to pass.
Fair enough. But if we're doing that, I think the original question with the Omega machine abstracts too much away. Let's consider the kind of evidence that we would actually expect to see if Islam were true.
Let us stipulate that, on the 1st of Muḥarram, a prominent ayatollah claims to have suddenly become a prophet. They go on television and answer questions on all topics. All verifiable answers they give, including those to NP-complete questions submitted for experimental purposes, turn out to be true. The new prophet asserts the validity of the Qur'an as holy scripture and of Allah as the one God.
There is a website where you can suggest questions to put to the new prophet. Not all submitted questions get answered, due to time constraints, but interesting ones do get in reasonably often. Are there any questions you'd like to ask?
This is a copout.
You've asked us to take our very small number, and imagine it doubling 66 times. I agree that there is a punch to what you say -- no number, no matter how small, could remain small after being doubled 66 times! But in fact long ago Archimedes made a compelling case that there are such numbers.
Now, it's possible that Archimedes was wrong and something like ultrafinitism is true. I take ultrafinitist ideas quite seriously, and if they are correct then there are a lot things that we will have to rethink. But Islam is not close to the top of list of things we would should rethink first.
Maybe there's a kind of meta claim here: conditional on probability theory being a coherent way to discuss claims like "Islam is true," the probability that Islam is true really is that small.
I just want to know what you would actually do, in that situation, if it happened to you tomorrow. How many 1's would you wait for, before you became a Muslim?
Also, "there are such numbers" is very far from "we should use such numbers as probabilities when talking about claims that many people think are true." The latter is an extremely strong claim and would therefore need extremely strong evidence before being acceptable.
I think after somewhere between 30 and 300 coin flips, I would convert. With more thought and more details about what package of claims is meant by "Islam," I could give you a better estimate. Escape routes that I'm not taking: I would start to suspect Omega was pulling my leg, I would start to suspect that I was insane, I would start to suspect that everything I knew was wrong, including the tenets of Islam. If answers like these are copouts -- if Omega is so reliable, and I am so sane, and so on -- then it doesn't seem like much of a bullet to bite to say "yes, 2^-30 is very small but it is still larger than 2^-66; yes something very unlikely has happened but not as unlikely as Islam"
If you're expressing doubts about numbers being a good measure of beliefs, I'm totally with you! But we only need strong evidence for something to be acceptable if there are some alternatives -- sometimes you're stuck with a bad option. Somebody's handed us a mathematical formalism for talking about probabilities, and it works pretty well. But it has a funny aspect: we can take a handful of medium-sized probabilities, multiply them together, and the result is a tiny tiny probability. Can anything be as unlikely as the formalism says 66 heads in a row is? I'm not saying you should say "yes," but if your response is "well, whenever something that small comes up in practice, I'll just round up," that's a patch that is going to spring leaks.
Another point, regarding this:
Originally I didn't intend to bring up Pascal's Wager type considerations here because I thought it would just confuse the issue of the probability. But I've rethought this-- actually this issue could help to show just how strong your beliefs are in reality.
Suppose you had said in advance that the probability of Islam was 10^-20. Then you had this experience, but the machine was shut off after 30 1's ( a chance of one in a billion.) The chance that Islam is true is now one in a hundred billion, updated from your prior.
If this actually happened to you, and you walked away and did not convert, would you have some fear of being condemned to hell for seeing this and not converting? Even a little bit of fear? If you would, then your probability that Islam is true must be much higher than 10^-20, since we're not afraid of things that have a one in a hundred billion chance of happening.
This is false.
I must confess that I am sometimes afraid that ghosts will jump out of the shadows and attack me at night, and I would assign a much lower chance of that happening. I have also been afraid of velociraptors. Fear is frequently irrational.
You are technically correct. My actual point was that your brain does not accept that the probability is that low. And as I stated in one of the replies, you might in some cases have reasons to say your brain is wrong... just not in this case. No one here has given any reason to think that.
It's good you managed some sort of answer to this. However, 30 - 300 is quite a wide range; from 1 in 10^9 to 1 in 10^90. If you're going to hope for any sort of calibration at all in using numbers like this, you're going to have to much more precise...
I wasn't expressing doubts about numbers being a measure of beliefs (although you could certainly question this as well), but about extreme numbers being a measure of our beliefs, which do not seem able to be that extreme. Yes, if you have a large number of independent probabilities, the result can be extreme. And supposedly, the basis for saying that Islam (or reincarnation, or whatever) is very improbable would be the complexity of the claim. But who has really determined how much complexity it has? As I pointed out elsewhere (on the "Believable Bible" comment thread), a few statements, if we knew them to be true, would justify Islam or any other such thing. Which particular statements would we need, and how complex are those statements, really? No one has determined them to any degree of precision, and until they do, you have to use something like a base rate. Just as astronomers start out with fairly high probabilities for the collision of near-earth asteroids, and only end up with low probabilities after very careful calculation, you would have to start out with a fairly high prior for Islam, or reincarnation, or whatever, and you would only be justified in holding an extreme probability after careful calculation... which I don't believe you've done. Certainly I haven't.
Apart from the complexity, there is also the issue of evidence. We've been assuming all along that there is no evidence for Islam, or reincarnation, or whatever. Certainly it's true that there isn't much. But that there is literally no evidence for such things simply isn't so. The main thing is that we aren't motivated to look at the little evidence that there is. But if you intend to assign probabilities to that degree of precision, you are going to have to take into account every speck of evidence.
I thought the salient feature of Islam was that many people believed it, not that it has less complexity than I thought, or more evidence in its favor than I thought. That might be, but I'm not interested in discussing it.
I don't "feel" beliefs strongly or weakly. Sometimes probability calculations help me with fear and other emotions, sometimes they don't. Again, I'm not interested in discussing it.
So tell me something about how important it is that many people believe in Islam.
I'm not interested in discussing Islam either... those points apply to anything that people believe. But that's why it's relevant to the question of belief: if you take something that people don't believe, it can be arbitrarily complex, or 100% lacking in evidence (like Russell's teapot), but things that people believe do not have these properties.
It's not important how many people believe it. It could be just 50 people and the probability would not be much different (as long as the belief was logically consistent with the fact that just a few people believed it.)