Wednesday, December 5, 2012

A bit more clarity on Bayes's Theorem and Loftus's Outsider Test for Faith

A couple of days ago, I wrote a rather long piece detailing the effect that John Loftus's Outsider Test for Faith has by examining how it should be affecting a (usually implicit) Bayesian analysis. I stand by my original assessment still, but I would like to add something in order to clarify a point and to add a great deal of strength to what is presented without needing any complex or difficult-to-win arguments.

To make it quite clear, I want to quote a half a paragraph from Loftus's Why I Became an Atheist (original edition) where he introduces the Outsider Test for Faith. From pp. 66-67,
The outsider test is simply a challenge to test one's own religious faith with the presumption of skepticism, as an outsider. Test your beliefs as if you were an outsider to your faith. An outsider would begin her journey as a disinterested investigator who didn't think the religious faith in question is true since there are so many different religious faiths in the world. An outsider would be someone who was only interested in which, if any, religious faith is correct and would have no intellectual affiliation with any of them at all. She would have to assume that her culturally inherited religious faith is probably false. (emphasis added, see below)
This is where Loftus first lays out the OTF in Why I Became an Atheist, where he defines it (so one can see why I recommend reading his writing if they are not yet doing so). Much of the rest of the chapter this paragraph is pulled from justifies two legs upon which this can stand, while quelling some objections
  1. The OTF is a justified activity for any believer to engage in;
  2. The geographic component to the distributions of religious faiths are the predominant factor that such a disinterested investigator will have to face down, e.g. that being born in Saudi Arabia gives a better than 99% chance of being a Sunni Muslim, while being born to Christian parents in the United States gave better than a 99% chance of being a Christian (until recently, at least--lots of people are leaving religion now, after all).
While I haven't read the yet-to-be-released book entitled Outsider Test for Faith, also by Loftus (release: next spring), I actually think Loftus sets the bar pretty low here in terms of what the OTF is meant to accomplish. I want to finish this short post by illustrating how I mean that, speaking in the language of Bayes's Theorem (please see that post now, if you haven't yet, to ensure clarity).

The italicized text in the quote from Loftus above indicates that Loftus's main conception of the OTF at the time he wrote Why I Became an Atheist focused only on one of the three numbers: the prior probability (that the religion being tested is true). In the long post I made previously, I make the case that, indeed, by far the heaviest bit of argument to be made (and being made!) is in the consequent P(e|h,b) (which I called t for convenience there) that measures how likely it is that the evidence we see in the world matches what we would expect to see if the hypothesis that the religion under examination is true.

Here's why working only on the prior probability is a relatively weak approach. Suppose we are dealing with a serious believer who is willing to take the test. First, if this serious believer is a "faith-head," one who would estimate that the evidence-based probabilities completely in his favor (t=P(e|h,b)=1 and n=P(e|~h,b)=0), no matter what prior probability he assumes for his religion, Bayes's theorem tells him that the likelihood his religion is true is 100% (if you think about this, it makes sense trivially: this position assumes the evidence is perfectly explained by his religion). In other words, you could easily get this faith-head to concede that there are millions upon millions of other religions out there, from the outside all just as likely to be valid as the next, and yet a Bayesian analysis will return absolute certainty for him that his particular religion is correct. Thus, more has to be dented than the prior for the OTF to have any effect.

Now, suppose another case where we have a very sincere (or deluded) believer, but not one that is impervious to looking at the evidence from the outside. Imagine now instead a very serious believer who holds that there is a 99% chance the evidence we see is what we would have if his religion is true and a 1% chance that there could be any other explanation for the evidence we see, i.e. t=P(e|h,b)=0.99 and n=P(e|~h,b)=0.01. Loftus's original conception primarily has this believer assault the prior probability without looking at the weight of other evidences. Here's what the math says about this.
  • If the OTF gets him to admit a still-biased 50% chance that his religion is true, a very common error among people who do not understand probabilities, then Bayes's theorem allows him to conclude that it's 99% likely that his religion is true. In other words, the OTF will have no chance of convincing him.
  • If the OTF gets him to admit that there are 4 equally likely positions to take (say, Christianity, Islam, Judaism, and "other")--surprisingly common among Abrahamic believers--he will put his prior at 25% and Bayes's theorem will tell him that there is a 97% chance that his religion is true. The OTF will not convince this person.
  • If the OTF gets him to admit 10 equally likely religions, for a prior of 10%, Bayes's theorem will tell him that his own is 91.7% likely to be true. The OTF will not convince this person.
  • In order to achieve an outcome from Bayes's theorem of 50%, it will be incumbent upon the person to recognize and admit that there are 100 religions that are equally likely to be true. This will be a tall order since there are only 20 "major" religions in the world.
  • If the person can be convinced, however, by the OTF that there are 1000 religions to consider equally, historically speaking not ridiculous, then Bayes gives back a 9% chance that the religion is true, and for 10,000 religions, Bayes returns just under a 1% chance.
In short, relying solely on getting a believer to face the reality of many religions in the OTF (and thus reduce the prior) will not succeed in convincing him in most realistic cases. Certainly, there is ample room to wiggle (or apologize) out of this.

The easiest point to make to help this problem:

Now, in the previous post I noted that because the naturalist position has a 100% chance of explaining nature from the outsider perspective, that this number is unlikely to be low for believers taking the OTF. This may not be accurate, though, as many sincere believers think the world would simply be chaos if any other explanation but their God were behind it. So, I'll amend that to say that serious believers will underestimate this number (n=P(e|~h,b)) perhaps quite grossly. This opens the door very widely for the OTF to have success, though.

The simple argument that will add strength is to get believers to recognize that the same process that is reducing the prior (per above) is admitting that there are other viable explanations for the evidence (the universe).

In more technical terms, reducing the estimate on the prior and raising the estimate on the value of evidence being supported by other explanations, i.e. n=P(e|~h,b), is essentially the same. It would even be possible to claim on this argument alone that the complement of the prior (e.g. 95% if the prior is 5%--that which would add to 100% with the prior) is the lowest value that n=P(e|~h,b) can take.

Consider the example above to see the impact that this change has, we can use even our super-sure, faith-head with t=P(e|h,b)=1 here (even though that's inconsistent--see next section).
  • If the OTF gets him to admit "don't know," i.e. a prior of 50%, then Bayes returns only a 66% chance that his religion is true: two out of three. Whether this would be convincing or not is hard to say, but it's far from "my soul burns in hell if I'm wrong" certainty.
  • If the OTF gets him to admit a prior of 25% (four religions), then Bayes tells him he's only able to be 30% sure that he is right.
  • If the OTF gets a prior of 10% out of him, then he can only be 11% sure.
  • If the OTF gets a prior of 1%, then he can be 1% sure.
  • Likewise, within a narrow margin of error, for lower priors.
This is very significant because this situation still assumes a believer who is 100% sure that the evidence in the world is accurately explained by his religion.

Thus, the easiest argument for the OTF to make, in addition to the recognition that the prior should be reduced, is that the possibility that other explanations are correct is essentially no less than the complement of the prior.

Consistency is the next easiest point

People like to be consistent (because not being consistent causes cognitive dissonance, which is actually uncomfortable). Thus, it should be relatively easy for the OTF to make the point that one cannot be 100% sure that their religion predicts the evidence and that there is some chance that other explanations predict the evidence. That, of course, is nonsense. 

This problem isn't as obvious if the person doesn't believe that there is a 100% chance that their religion predicts the evidence, but it still shouldn't be hard for the OTF to make the next most straightforward assertion and have it taken seriously.

The probability that any particular religion predicts the evidence should be no higher than the probability that particular religion is true. In other words, the OTF should easily be able to make the case that the prior probability sets an upper boundary for t=P(e|h,b).

[EDIT (12/7/2012): A commenter (below) noted that a and t here are technically independent, and certainly that is generally true. I do not wish to mislead anyone here into thinking that they must be related to one another. The (exclusionist) religions (i.e. the big ones), however, contain within themselves the assertion that nothing else could explain the evidence, so this would add weight to the claim that the prior should serve as an upper bound since the very existence of other religions severely challenges the notion that "this is what we would see if there was indeed one true religion."]

Let's examine the effects of this assertion on the outcomes of Bayes's theorem if t=P(e|h,b)<(prior) and n=P(e|~h,b)>(1-prior):
  • If the OTF can get a believer to admit a "don't know" position of 50% for the prior, then this construction tells them exactly that: 50%, i.e. "you don't know."
  • (As a pointed aside: the infamous argumentum ad populum. "But so many people believe in my religion! It has to be right!" Well, let's use Christianity here, since it's the biggest religion going. We'll be generous and say that all brands of Christians are Christians, including the nominal ones that aren't even really believers at all any longer. In that case, 33% of the world's population is Christian (and if we look historically over time, this number would be smaller). This results in a argumentum ad populum prior of 33% from the OTF, if we want to concede it. Bayes's theorem, in the full analysis (admitting modification to all three relevant numbers) gives this person a 19.5% chance that their religion is true, a bit less than one in five. Probably not enough to be getting on with.)
  • If the OTF can get the believer to admit 4 religions, i.e. a prior of 25%, then Bayes's theorem tells him that there is only a 10% chance his religion is true. This is likely to crack the walls of many faiths.
  • If the OTF can get the believer to admit to 10 religions, Bayes's theorem tells him there's only a 1.2% chance his religion is true. This is, by definition, essentially a decimation.
  • If he admits 100 religions, there is a 0.01% chance his religion is true, and if he admits 1000 religions, there is a 0.0001% chance (1 in 100,000) that his religion is true.
  • This number gets small fast for more religions, say if one were to make the argument that every religion that does exist, has existed, will exist, or could exist must be considered, plus no religion at all. Bear in mind that there are over 40,000 denominations of Christianity alone here.
This is enormously significant, and a good reason to call the OTF a "silver bullet" because one should immediately note what didn't have to happen here (and thus why the two consequents are still heavily generous in the believer's favor). This construction hasn't taken into account a single argument for or against any theology; it merely analyzes against the existence of other religions. That means that science needs no appeal here, and apologetics needn't be bothered with--save the least respectable kind that argues for why a particular religion deserves a double-standard (which falls outside of the OTF, which presupposes a disinterested observer that gives no favor to any particular sect or creed).

Again, because this is hard to overstate--the OTF can be this effective on one, and only one point of debate: how many religions do you admit exist and have a possible claim at being true? There needs be no debating evolution, no discussing whether or not Hitler was Christian or atheist, no mention of how many people died under Stalin, no comments about the origin of the universe, and no nonsense about without God whether or not there can be morals. One question is all that's needed to seriously break open religious faith: how many religions do you admit exist and have a possible claim at being true?

It cannot be denied by anyone that wishes to get into the real argument, which the OTF may have shown to be essentially unnecessary in principle (but unfortunately not in practice, in all likelihood), that the degree to which the evidence is explained by any religion, especially the Abrahamic ones, is abysmal. Indeed, I argue that it is zero (almost surely), end of story, but anything larger than billions to one will be very, very difficult to defend against the Problem of Evil (or Problem of a Silent God, more generally) alone (and hence the PoE is the "rock of atheism"). Similarly, that naturalism isn't a completely effective explanation for everything (even if all of the explanations aren't had) is going to be very difficult to defend, and thus the other consequent should be within billionths of one (I argue it really should be one, almost surely, end of story).

That is why the vast majority of the heavy indictments of religion from Richard Dawkins, Sam Harris, Christopher Hitchens, and others (including myself!) have worked so hard on these points--if the evidence is considered honestly, Bayes's theorem will show no mercy on any prior, at least anything short of 100% certainty. As Sam Harris has pointed out, though, "If someone does not value evidence, what evidence could you provide to change their mind?" (Capturing that the 100% certainty prior probability--i.e. mind completely already made up--case requires more work.) Attacking the evaluation of the evidence simply doesn't work against devout faith until something cracks it open. The OTF is something of a silver bullet because it says, simply enough, that you don't have to (though the rest of Harris's statement still makes for a real problem--they have to value logic or this is still out the window).

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22 comments:

  1. You don't appear to have read or taken account of any rebuttals of the OTF. Certainly you don't take into account my arguments in Chapter Six of True Reason, "John Loftus and the Outsider-Inside Test for Faith."

    I have not come to expect most critics of Christianity to read much of what they are criticizing, so I can't say I'm shocked. But you're defending a battleship that has already sunk, and has a few years of coral growing in its bridge, with sting rays swimming in and out.

    The book is extremely cheap in its electronic version, and there are free blog posts (by me and by others) that dispute John's premises. Just ignoring contrary arguments might work in Gnuistan, but not if you plan to lug your argument out into the bright sunshine of the bigger world.

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    1. Hi David, thanks for the comment.
      I have not read your rebuttal to the OTF. I have done an analysis of what the OTF is trying to achieve as viewed through Bayes's theorem.

      You are also correct--I'm not particularly interested in reading a rebuttal to it. I might be moved to do so, but as you may have surmised, I fully expect the burden of proof is on you (and other religious folks) to establish that it's even worth our time to read your rebuttals. Last time I checked, no one has yet offered any credible evidence that any God exists, and I'm pretty sure it will make real news if that ever happens. The weight of authority and tradition (let's not call "revelation" an argument with any weight--that's just silly) simply do not stack up to "credible evidence."

      Of course, you might pin on me that the above comment about God is unfair since I've dedicated this post particularly to whether or not religions are true. Since the validity of a religion depends upon its centerpiece being real, I have a hard time accepting that challenge, but I'll concede it here (as in, in this paragraph only) anyway. Last time I checked, no one has offered any credible evidence that any of the religions are true either. The foundations of Christianity, in particular, are staggeringly poorly substantiated, even without having to appeal to a concept of a God (and prevarication on the definition of the word "exists"--perhaps you should read Chapter 4 of my own book on that).

      I guess what I'm saying with this is that I haven't lugged my arguments out into the bright sunshine if you consider the position you're arguing from (a theology that cannot establish its central premise) "bright sunshine." Establish that it is indeed sunshine, and I'll give that some consideration, but lo, you can't. Folks have been trying for thousands of years and have never really been able to do that.

      Let me clarify for you what I have done here (and perhaps what the OTF is), though, since you seem to have missed it. I have put numbers (and thus weight) behind the idea that if someone steps outside of their religious beliefs and considers them as someone outside of them would, say an educated atheist or member of another religion, they would find very little reason to accept those religious beliefs. It's really not very complicated.

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    2. James: I am increasingly amazed at this tendency by Gnus to defend their own ignorance as if not knowing what the other side is saying were some kind of a virtue, or as if an intellectual publicizing his or her views on a subject had no responsibility to read both sides and know what he or she is talking about.

      I'm not inclined to waste time, bombing rubble. You say "last time I checked," but if you haven't checked on this issue, yet are willing to write about it, why should random readers assume you've checked on that one, seriously and dispassionately?

      This is your blog, and you can write whatever you like. But if you're not going to grapple with opposing arguments, however bright you might be, however objective and well-meaning, it just won't be worthwhile grapplying with yours.

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    3. Sorry David, you've misunderstood me. I have checked for evidence of God. I checked very hard. I found none. To my knowledge, that hasn't changed, and it would be the biggest news story in the world if it was found.

      Do I need to put this out in Bayesian terms for you to understand them, or will that cause you to miss them more fully?

      For what it's worth, I greatly enjoy and am flattered by the fact that you're trying to get me to legitimize your position by examining it and giving it some consideration, but as I made very plain: I've examined the evidence over the last decade, found nothing but the opposite, and haven't heard anything significant to the contrary yet--and it would be huge news. Do email me, please, when yours makes international headlines. I'll be glad to look at it then and offer my criticisms or congratulations, depending on the circumstances.

      Some diddly philosophical argument isn't going to cut it (see Ch. 4 of my book to see why), which is why those never, ever make news (except maybe in Huffington Post or religious, read: biased, periodicals).

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    4. James: I don't really care about headlines. Well, if Huffington Post wanted to publicize my books (aside from a few shots from Victor Stenger), I'd be happy about that, but that's no test of truth.

      Looks like two ships just pass in the night, here. I'm no judge of your mathematics. And you don't seem to even want to consider my critique of the philosophical and empirical premises on which it seems to be based.

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  2. Hi James,

    Good start on applying BT to the OTF. There are a couple of things I noticed which are in error, though (these are actually relatively common misunderstandings of BT, by the way, so don't feel bad :-) ). Your analysis doesn't hinge upon either of them, but understanding them correctly is fairly important to getting a Bayesian argument right.

    The most major one is this statement: "The probability that any particular religion predicts the evidence should be no higher than the probability that particular religion is true. In other words, the OTF should easily be able to make the case that the prior probability sets an upper boundary for t=P(e|h,b)."

    This is misunderstanding the concept of the 'likelihood', which is the conditional probability of the evidence *given* that the hypothesis is true. This number can and does easily range from 0 to 1 depending on the hypothesis and the evidence, and is generally not at all related to the prior probability by any such constraints.

    I emphasized 'given' above, because it helps to separate out the idea of the hypothesis being true (prior) from the idea of the evidence being true. For example:

    Suppose my 'religion' predicts (through prophecy) that the next fair coin I flip will show Heads. That is, *given* the hypothetical scenario that my religion is 100% definitely true, no questions, *then* the probability of Heads is 1. Likewise, the probability of Tails would be 0.

    Now, the coin may or may not come up Heads, of course. And the prophecy will very likely have no real logical effect because it's extremely unlikely that my 'religion' in this example is actually true. In other words, the prior probability of my religion being true is close to 0. Notice how the likelihood (1) has basically no connection to the prior (0).

    What you are probably thinking of in the quoted statement is not the likelihood probability, but the *joint* probability that my religion is true *and also* that the coin turns up Heads. This probability comes from multiplying the prior times the likelihood, and so it will be strictly less than or equal to the prior.

    So, the prior, a, can range from 0 to 1, and the likelihood, t, can independently range from 0 to 1, whereas the joint probability, a*t, will indeed be strictly less than or equal to a.

    (cont'd)

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    1. Thanks, I really appreciate commentary like this!

      Let me defend my application of these two "errors," as I have committed them intentionally but apparently failed to communicate things clearly because I wanted to try to keep already exceedingly long posts (about math, no less) short. I'll address them per the comment they are attached to.

      The first error: I know the prior and this consequent are independent, or essentially so, depending on the situation. I'm actually using this as a way to create an argument to get someone to consider the OTF more seriously, and indeed, this particular application is almost entirely a priori since it ignores all evidence besides the number of religions out there. What I was attempting to do with this construction is make it clear that a very heavy first-order argument can be made using the OTF by applying Bayes's theorem. I do agree that it stretches beyond the usual application to cite that the prior should be an upper bound for the weight of evidence here, somewhat at least. It would depend upon how we want to interpret "likelihood the evidence is explained by these religious claims." Granted, I know it is not "likelihood it is explained by these and no others."

      Indeed, from the honest outsider's perspective, this number I've called t is going to be very, very small, and so to say that the prior serves as an upper bound for it is hardly far-fetched. Some more tightness needs to be applied to making that case, to be sure.

      Additionally, the (exclusionist) religions (i.e. the big ones) contain within themselves the assertion that nothing else could explain the evidence, so this would add weight to the claim that the prior should serve as an upper bound since the very existence of other religions severely challenges the notion that "this is what we would see if there was indeed one true religion."

      Thanks, though, for trying to keep me honest on this point. It's pretty consequential, and I really should have defended it better (the paragraph just above does that, I think, and may get added in an edit for clarity's sake). I figured it would be a vastly simpler and shorter argument to look at what happens if we just let those numbers define the others, but I can totally see how it would be misleading as I've written it.

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    2. Thanks for the clarification. I think also that I may have jumped too early to the conclusion that you were mistaking likelihood for joint probability, and that probably biased my reading from that point on, since I didn't originally grasp what you *were* saying.

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    3. I also suffer incessantly from having at least six thoughts in my head at almost all times, so my first draft with writing tends to be a bit of a shotgun blast of ideas, sometimes all mashed into one incomprehensible sentence. Because I make essentially $0 off this blog, though I do realize it is my face to the world to build my reputation as a writer in this field, I simply cannot justify committing the time to doing a full edit and clean rewrite of my ideas for this kind of publication. I'm much more careful when I feel it's more serious, but this is, after all, a blog.

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    4. Heh! I know exactly how you feel. In my case, it turns out to be related to my ADHD (http://en.wikipedia.org/wiki/Attention_deficit_hyperactivity_disorder) which had been previously undiagnosed until only a few years ago. Now that I know what I'm dealing with, my strange patterns of thinking suddenly make a lot more sense.

      No worries about blog vs. 'professional'. I'm mostly responding to try to provide information so you can fast-track to the next level (always good to have fellow skeptics who can use (and explain!) the heavy duty tools like Bayes' Theorem).

      Also, one of the best ways to *learn* something is to try to *teach* it. So, you're kinda my guinea pig as I try to explain what I think I know about BT. (Apologies! Probably should have asked first. lol) :-)

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  3. The second error is actually found in your previous post, where you go from the idea that 'the more religions you consider, the closer each religion's prior approaches 0', to the idea that 'therefore each religion's prior is actually 0'.

    This error comes from transitioning between considering Bayes' Theorem in terms of discrete hypotheses (a finite set of religions, each with their own prior), to BT in terms of continuous hypotheses (any particular religion being a point in a continuous space of possible religions). It is true that treating each religion as a point renders its probability as 0, but when you transition into looking at things continuously, you no longer consider individual points (there are too many of them; infinite, in fact), but instead consider intervals or regions of possible religions.

    E.g. consider the possible versions of Christianity, each with a specific instant of time when the Rapture occurs. Any one of them would have an infintesimally small (i.e. zero) probability, but when you consider the infinite interval from now to the end of eternity, then that entire interval of conditional probabilities must sum up to 1. In other words, given that a version of Christianity is true which sets a specific instant of time for the Rapture to occurs, the probability (likelihood) that that instant of time falls between now and the end of eternity is 1 (certain).

    When dealing with continuous probabilities, we no longer rely upon the discrete probabilities themselves. Instead, we use probability density functions (pdfs) and cumulative density functions (cdfs), which are more like the derivatives and integrals of continuous probabilities than they are like individual probabilities themselves.

    At this point, Bayesian reasoning gets a lot more complicated, and I would recommend reading ET Jaynes' book Probability Theory: The Logic of Science, which is a must-read for anyone serious about learning how to use Bayes Theorem correctly, IMO.

    I would recommend doing as Richard Carrier has done, which is to stick to discrete hypothesis testing, until the continuous case becomes more widely accessible to lay people (perhaps with the help of software). Keeping things discrete has the advantage of being much easier to explain and understand, and it avoids all sorts of seeming 'paradoxes' which require careful reasoning to avoid in the continuous case. See Jaynes' book for more details on these kinds of paradoxes and how they are successfully avoided by using BT very strictly and carefully.

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    1. Now, this one... I expect a lot of this kind of thing.

      Let me indicate for you, first of all, that infinite and discrete are *not* mutually exclusive. Indeed, consider the counting numbers: 1, 2, 3, and so on. Discrete, check. Infinite, check. Probability of pulling the number 5 out of all of them? Zero (almost surely). We're still in the discrete, though.

      We could easily imagine dealing with possible religions in exactly the same way without defaulting to something so trivial as "different resurrection dates/times." I will be the first to admit that such constructions are by far the easiest to come up with, but we needn't be limited so.

      Consider, for example, that we simply define each religion based upon a different conception of God. How many conceptions of God are there? Without getting silly, it becomes pretty difficult to make the case that the number is finite if we accept even one premise that is commonly included: "God is nonphysical." If God is nonphysical, what limitations are there on such a being? How many of them could there be? That's without even considering the number of "physical" Gods that could exist.None of this needs to be done continuously if we accept the premise of "exclusionist" Gods that wouldn't tolerate the notion of being worshipped incorrectly.

      Now, of course, I made this entire argument in the finite case in the first place and only noted that it can be made against an infinite sample space.

      Many thanks for the book recommendation. I'll compare it against my other probability theory textbooks.

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    2. Haven't read your responses yet, but want to quickly reply to clarify something:

      "Let me indicate for you, first of all, that infinite and discrete are *not* mutually exclusive."

      D'oh! I should have mentioned that I'm not actually a professional mathematician (unless you consider computer science a branch of math), and I will undoubtedly have made sloppy errors in my terminology such as the one quoted.

      I tried to be precise so that I would state the issue correctly, and I hope my errors don't detract from comprehension. I'm basing my commentary off of what I've been reading from Jaynes. When in doubt, I'll defer to his much clearer writing.

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    3. No worries, and please do. I don't have Jaynes's book (which is quite expensive! I need to sell a lot more of my own before I can justify the expense to my wife!), and as may surprise you, none of the probability theory textbooks I have do more than give Bayes's theorem a cursory mention in an appendix or homework problem! It's certainly not considered a central result within probability theory, despite its usefulness. Indeed, mathematically, it's very nearly trivial.

      The language that Jaynes uses will hopefully be good for offering me clarity on this. The case I'm pointing at here does not seem to be covered in many places, and were we to approach it via integrals, it would have to be abstract integration (Lebesgue integration) against the counting measure. I should note, though, that for the continuous case, it isn't Bayes's theorem that changes but rather the method of calculating the probabilities that go into Bayes's theorem.

      On the other hand, doing the entire thing simply with limits provides (what appears to be, on first analysis) a clear way to use Bayes's theorem in the context I am using it, and it seems not to violate absolutely any mathematical rules or laws that I'm aware of.

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    4. Jaynes refers to paradoxes of infinite sets, so he was indeed referring to finite vs. infinite, but I mistakenly used 'discrete' as a synonym of 'finite'.

      Jaynes' point is that unless the method of taking limits is clearly defined then there can be more than one 'correct' answer, leading to apparent paradoxes.

      He goes on to develop *continuous* hypothesis spaces, ending up with the usage of pdfs and cdfs rather than the simpler probability mass functions typically associated with basic probability theory. This was the distinction I was trying to point to.

      Though technically (I suppose) a continuous distribution involves infinite sets (the Reals), Jaynes makes the case that you can still keep the 'probability as logic' endeavour applying to finite propositions/hypotheses by using the cumulative distribution functions (cdfs) to cut up the hypothesis space into a finite number of mutually exclusive intervals. From there, I think he finally goes on to say that you can then take the limit as the number of hypotheses goes to infinity and use this limit as an approximation to the 'practical reality' that there can only ever be a finite number of tiny hypotheses in any practical application of Bayesian reasoning (e.g. any actual computer program has limited physical memory).

      His point (and mine, borrowed from him) in all of this is (I'm pretty sure, but not 100% sure) that we don't actually go from 'approaching zero' to 'actually zero'. Instead, we use pdfs and cdfs, do the calculations of BT using them, and then do a finite process of integration to get our concrete answers for any real-world problem.

      In other words, it may seem intuitive to 'go to zero' for any particular religion, but in practical terms, when a person talks about their specific religion, they are not really talking about a single point in the hypothesis space, but a region/interval of hypotheses, which *does* have a non-zero probability.

      Using the Rapture example, someone may claim their religion states that the Rapture will happen on Jan 1, 2013. This is not actually a religion-as-a-point-with-zero-probability, but a religion-as-an-interval-with-non-zero-probability. The interval would, practically speaking, be 'sometime on Jan 1, 2013', perhaps 'some time between 00:00:00.000 and 23:59:59.999 on Jan 1, 2013'

      The prior distributions, likelihoods, and posterior distributions would be handled as pdfs and cdfs, and you'd calculate the probability of 'Jan 1, 2013' as:

      P('Jan 1, 2013'|B) = cdf('23:59:59.999 on Jan 1, 2013') - cdf('00:00:00.000 and on Jan 1, 2013')

      Since the cdf is monotonic increasing (by definition), and the pdf is (depending on choice of prior distribution) non-zero everywhere from now to eternity, then this difference will be non-zero.

      Hope that clarifies what I was trying to get at.

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    5. Yes, that does clarify what you're getting at. I apologize for neglecting to answer that "approaches" jumps to "is" zero thing for you previously.

      I'm not trying to argue that the probability that God exists or a religion is true goes to zero. I'm trying to argue that it *is* zero, almost surely. Indeed, I'm making the argument (in the book, not here, fwiw) that it begs the question to state a priori that any number other than zero, almost surely, applies to that probability, i.e. that the sample space can be thought of meaningfully as being both infinite and discrete with the varying gods being distinct individual points within that space.

      Let me use something akin to something I've heard Richard Carrier say that now makes my eyebrows raise. He said (in his talk at Sk4) essentially that we have to look at the situation in terms of what we can argue for: is one in ten an overestimate? one in one hundred? one in a million? one in a billion? and then we should choose the smallest number that we can that we can be reasonably comfortable must be an overestimate.

      All of this terminology, though, suggests that there is an accurate probability--as does an application of Bayes's theorem. All of these numbers are literally pulled out of his ass, though, and it's completely clear that they are (which is not something we should be criticizing him for as it actually makes the right point anyway). My question turns that on its head, though: what is the largest probability that an apologist could argue for?

      Could an apologist make a meaningful case that there is a one in one hundred chance that God exists? If so, based on what? None of the philosophical arguments hold any water. Could (s)he make the case that there is a one in a million chance that God exists? If so, based on what? One in a billion? If so, based on what? It seems to me that the only thing it's based upon is the unwillingness of anyone to say that the number actually is zero, predicated on the belief that that number is disallowed.

      This led me to conclude that any number they throw out there, however small, might be question-begging. Then I realized the thing about "almost sureness" breaks their argument that "well, you can't say it's zero because that rules out all possibility." Frankly, no, sir or ma'am, actually, it doesn't. In conclusion, I have managed to convince myself that there has been no argument an apologist has put forth that gives any positive probability to the claim that God exists (esp. their special God), and any other number begs the question until such an argument can be given and validated.

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    6. As far as textbooks, I haven't got around to it yet, but when I was looking for good texts, "Information Theory, Inference, and Learning Algorithms" by David J.C. McKay got lots of recommendations, and it's a free download for onscreen viewing (http://www.cs.toronto.edu/~mackay/itila/book.html ).

      A hardcopy of Jaynes might be available in a university library. There is a fragmentary (but mostly complete) PDF version of Jaynes linked from Wikipedia in the References section of http://en.wikipedia.org/wiki/Bayesian_probability . The relevant chapters are 1 through 4, especially 4 (but the others are excellent groundwork leading up to 4). This is the original version I started with (it's a draft from 1996). It contains all the relevant material. Good for review.

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    7. "All of this terminology, though, suggests that there is an accurate probability--as does an application of Bayes's theorem. All of these numbers are literally pulled out of his ass, though, and it's completely clear that they are (which is not something we should be criticizing him for as it actually makes the right point anyway). My question turns that on its head, though: what is the largest probability that an apologist could argue for?"

      I highly recommend going to Jaynes for a satisfying answer to this question. Chapter 6 on Elementary Parameter Estimation is the chapter that finally convinced me that the Bayesian approach is far superior to any existing approach, precisely because it's the only approach that even comes close to answering your questions.

      To boil it down and crudely summarize: What you really need is a 'pre-prior' distribution that covers an extremely broad range of assumptions, so that basically, 'Whatever the true value of the parameter is, this pre-prior will allow us to estimate the true value to an ever-increasing level of accuracy, given enough background research.

      For example, what is the probability of someone being named 'Jesus' back in the day? Well, initially, you would have no clue, but it's somewhere between 0 and 1. And if you do some research, you'll find that certain names back then were more common than others, and using this data, you'll be able to develop an accurate 'prior' for the *real* problem you're trying to talk about, which is the probability of Christianity being true (which may indirectly depend on the likelihood that someone would be named Jesus). So the pre-prior helps you to estimate reasonable priors (and likelihoods), and Jaynes gives a thorough exploration of how to do this and the theoretical and philosophical (pragmatic) justification for the method he defends (maximum entropy).

      "This led me to conclude that any number they throw out there, however small, might be question-begging."

      Exactly. Jaynes' book shows how to 'unbeg' that question, to coin a phrase. :-)

      A very closely related question, which simplifies thinking about it: Suppose a coin is tossed, producing a particular sequence of heads and tails. Is the coin fair, or biased? If it's biased, how much is it biased? What is its 'true' probability of landing heads vs. tails? This is a prototypical parameter estimation problem.

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    8. Thanks for the suggestions, and no worries about stepping in. I do appreciate it and feel you've helped quite a bit!

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    9. Also, the question of 'which religion is true, if any', is very similar, if not identical to, 'which scientific theory is true, if any'. These questions would share the same problems. So if any particular religion has a zero prior probability then the same would have to be said of any particular scientific theory. I believe Kolmogorov Complexity may be significant to answering this, though I'm not that far yet in my understanding to be sure. Basically, if I'm not butchering it, the idea is that every 'theory' has a non-zero prior probability (more specifically, a non-zero pdf distribution) of being true based on the probability of a random string of information specifying that theory. (Think of the million monkeys typing Shakespeare kind of situation.) Something to that effect, anyway.

      So, by this reasoning, every scientific theory, and every religion, has some remote chance of being true a priori, and from there you look at the real-world evidence, which rather quickly will eliminate some theories from contention, while winnowing down the remaining theories toward the truth.

      Incidentally, this reasoning provides a justification for Occam's Razor, whereby simpler (less complex) theories start out with higher base prior probabilities (the simple theory "A" is more likely than the compound theory "A and B"). There are additional justifications for Occam's Razor beyond this simple one, but it's an interesting result in itself.

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    10. Oh, and one last thing (I knew I was forgetting something) which might be of interest to someone with a background in math: Jaynes pretty much completely avoids/rejects defining probability theory in terms of set theory, and so he avoids things like measure theory, which I noticed you mentioned. He gives a justification for this in the book. I think this is one of the reasons his book is so popular, because frankly I can barely make heads or tails of measure theory and basically have to take it on 'faith', so to speak. For those not intimately familiar with the reasoning behind measure theory, Jaynes' focus on more intuitive concepts like finite sets of logical propositions allows one to focus on the nuts and bolts of probability theory without getting tangled up in esoteric set/measure terminology and concepts.

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    11. Wonderist, thanks for the reference to the Jaynes pdf text! It is proving to be an awesome read and a useful tool.

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