Bayesian Atheism

James Lindsay has been doing some great blogging on how to apply Bayesian reasoning to model John Loftus’ Outsider Test for Faith (or OTF). A while ago I asked for recommendations of bloggers that often write about Bayes’ Theorem for a general audience (see Bayesian Blogging), and a few came up there. This is another.

Cover of The Christian DelusionFormulating and extensively defending the OTF is Loftus’ greatest contribution to the philosophy of religion and atheism. His best and most thorough treatment appears as chapter four in The Christian Delusion (a book I always recommend anyway as it contains lots of great chapters by great authors; and two by me). He is writing a whole book on it now. It should be out this year (I’ve seen advanced drafts and it’s good; I’ll blog it when you can buy it). The OTF is featured at Iron Chariots (which provides examples of looser expressions of the concept throughout history) and Loftus discusses it often at Debunking Christianity.

The basic idea is that you can only have a rational faith if you test it by the same standards you apply to all other competing faiths; yet when you do that, your religion tests as false as the others, and the same reasons you use to reject those become equally valid reasons to reject yours. Though this idea has been voiced before, Loftus is the first to name it, rigorize it, and give it an extensive philosophical defense; moreover, by doing so, he is the first to cause a concerted apologetic to arise attempting to dodge it, to which he could then respond. The end result is one of the most effective and powerful arguments for atheism there is. It is, in effect, a covering argument that subsumes all other arguments for atheism into a common framework.

Cover of God Doesn't; We DoLindsay, meanwhile, is an expert mathematician and author of God Doesn’t; We Do: Only Humans Can Solve Human Challenges (2012). His blog of the same title treats a number of issues in support of that book and its argument. I don’t always agree with him. But his blogging on Bayes’ Theorem is great. He started by talking about how Loftus’ OTF can be formulated using Bayes’ Theorem, to show why it can’t be dodged the way Christian apologists want. This led to further blogging on the subject, including a Bayesian analysis of “faith” in general. It’s worth checking out.

The first of these (on which the others build) is:

Here much of his argument is backed formally by my Bayesian models in The End of Christianity (edited by Loftus) for Christianity as a religion (chapter two) and for the design argument generally (chapter twelve); where most of the math is in the endnotes but the Bayesian logic is made explicit in each. These chapters especially explain why the evidence has a much higher consequent probability (a higher “likelihood” in sci-speak) on naturalism than on any kind of theism (much less Christian theism).

Combine those with Lindsay’s post and you should get a clear understanding why atheism is true and Bayesian reasoning proves it. Lindsay’s treatment will be especially helpful in understanding how atheists think like Bayesians all the time even when they don’t know it (and how Christians, in contrast, are really awful Bayesians). I give other examples of Bayesian atheism near the end of my talk Bayes’ Theorem: Lust for Glory (which is still my best intro to BT for beginners), which can supplement all this.

Lindsay continued blogging under the tag “Math” and what’s there so far is all Bayes’ Theorem stuff. Maybe that won’t always be the case, but keeping tabs on that tagged subject going forward might lead you to more gems about Bayesian reasoning. So far there are three other posts:

  • A Bit More Clarity on Bayes’s Theorem and Loftus’s Outsider Test for Faith (which shows how a BT-formulated OTF forces believers to confront facts that plain descriptions of the OTF might not; in short, it’s the probability of the evidence, and not just the prior probability, that’s the problem, although the OTF shows both are a problem for any honest believer)
  • Continuing My Bayesian Argument–The Role of Evidence (where he defends the OTF against accusations that it would lead to weird conclusions in other domains, which a BT analysis shows is actually not true; although he incorrectly applies the term a priori here: the prior probability in the OTF is not a priori, but based on background evidence regarding the number of observed religious faiths; a priori knowledge is by definition not based on any such evidence, and in particular neither are a priori probabilities; for an actual example of the latter, see my note 8, pp. 406-07, in TEC)
  • Defining Faith via Bayesian Reasoning (which builds a Bayesian definition of faith, when faith is used in any sense other than as a synonym of belief; this also provides an example of how many of Loftus’ rebuttals of critics of the OTF can be framed in Bayesian terms to show why he is right and they are not)

Good stuff so far. So I’m adding this to my list of Bayesian bloggers worth keeping an eye on. Another to add is Jeff Lowder.


  1. hjhornbeck says

    Bayesian Inference is wicked powerful when it comes to religion. I’ve been focusing on what I call “universal counter-proofs,” which argue against the existence of an arbitrary god, and most of them boil down to “apply Bayesian Inference to god X.” The most complete version of my arguments can be found here, if anyone wants to have a boo.

  2. says

    I arrived at the OTF looking at BT from a different angle. What it is basically trying to point out is that there’s no necessary relationship between being born into a Christian household (i.e. your parents, friends, etc. are all Christians) which is a pretty strong determining factor in most people being Christians, and Christianity actually being true. The evidence that we have is the geographical distribution of the various religions, in this case I thought that P(E) should be the probability, or percentage, of Christians who were raised as Christians; whose family, friends, etc. are also Christians. This would mean that P(E) is somewhere around 30%. 33% of the world is Christian and there are obviously some Christians who converted to Christianity from other religions.

    P(E | H), then, would be the probability of the evidence given that Christianity is true, and P(E | ~H) would be the probability of the evidence at hand given some other reason (e.g. Islam is true, atheism is true, etc.). Of course, P(E) = P(E | H)*P(H) + P(E | ~H)*P(~H), the denominator for BT. I even assumed for the sake of argument that P(H), the probability that Christianity is true, is some ridiculously high number like 97% just to prevent Christians from saying that I’m biasing the argument against Christianity. So if P(H) is 97% then P(~H) is 3%. Now we have P(E) = P(E | H)*P(H) + P(E | ~H)*P(~H), or 30% = P(E | H) * 97% + P(E | ~H) * 3%. If we assume that P(E | H) is a high number, again, to bias towards Christianity being true, we end up skewing things ridiculously. The equation would then become a simple enough algebraic one to solve in the manner of 3 = 97 + .3x, but with P(E | H) being 1, it forces P(E | ~H) to be a negative number, which makes no sense.

    Playing around with different values for both P(E | H) and P(E | ~H), it would seem to me that the only fair values would be to have P(E | H) = P(E | ~H), which is statistical independence. It literally means that whatever religion you were raised in has absolutely no relationship with that religion being true; it also means that it has no bearing on whether your religion is false. For both, you would need some other evidence to update your prior against. Of course, if a Christian appealed to something like “it just feels true” or “I had an experience I can’t explain, therefore Christianity is true” other religions use the same exact sort of argument, so we would end up in the same situation that P(E | H) = P(E | ~H) with E being the religious experience and H being the probability that the religion is true, again meaning that having a religious experience has no relationship with the religion being true. So again, you would need some other evidence to update against which is basically the fundamental premise behind the OTF.

    • says

      I don’t know what you are doing to get negative numbers. I’ll set that aside, since the paragraph is too confusing. But the rest seems to confuse priors with consequents.

      It cannot be 0.5 prior chance your religion is true, of course, since there are more than two religions, and all prior probabilities must sum to 1, and there is no prior certainty that any one religion is right more than any other. Thus, as Loftus says, the prior approaches zero (depending on how generically you define your religion: higher priors go to more generic faiths, because they subsume many specific ones), or at any rate is low (roughly equal to the number of options). Until you start introducing evidence. And that can happen at different stages.

      So if you assign prior just based on background evidence regarding options and how they are selected for you, it’s roughly 1 divided by number of options. But if you skip that and assign priors based on past experience with religion as an explanation of anything whatever, you get a very low prior (see how I do this for generic theism in The End of Christianity, pp. 280-84), and when you divide that further into specific religions, it even gets lower.

      It doesn’t matter which prior you start with, since all the evidence goes in eventually, and thus the outcome is the same. But the point of the OTF is that the plethora of religions and the evidence of how one got chosen for you makes the prior probability that you got the correct one very low. Therefore you need good evidence, much better than other religions have; and that means you can’t use any excuses or apologetics that would similarly rehabilitate any other religion, as all that does is negate the value of the evidence (any apologetics you use that they can use just makes e as likely on their religion as yours, so you get nowhere). This translates to: you need to be as hard on your religion as you are on everyone else’s. Only then can you claim to have a correct religion, if yours ends up with a high enough posterior probability to believe it. Loftus’ point is that no religion passes this test (not that no religion can, but that none do). Except, at best, atheistic naturalism.

      Your point that number of adherents has no evidential value is correct but only shows that we can’t set the prior by polling, BTW. So if 90% of the world is theist, that is not the prior probability of theism (nor is 33% the prior probability that Christianity is true when 33% of the population is Christian), due to the demonstrated fact that people are extremely unreliable at picking a true religion–as demonstrated by the fact that there are so many of them; and most are picked by happenstance rather than reasoning from evidence. Thus all selections are equally likely to be successful at picking the correct religion, therefore all options are equally probable, no matter how many relative adherents they have. Until you start comparing options on the evidence…but now we are past the prior and asking about likelihoods.

  3. Otis Graf says


    You wrote: “Combine those with Lindsay’s post and you should get a clear understanding why atheism is true and Bayesian reasoning proves it.”

    That statement is an exaggeration. Bayesian reasoning does not “prove” anything. It is a method of inference that takes into account evidence that has been learned. Your prior sentence about “higher probability” and “higher likelihood” is the proper way to phrase the argument. But you then slip into claiming “proof.”

    What is being sought by the method of Bayesian reasoning (and by the methods of science) are not proofs but the best explanations, and those explanations are allowed to change to make way for better explanations. That is how science works and how knowledge is gained. Claiming dogmatic “proofs” should be avoided.

    Otis Graf

    • says

      You should look up the different meanings and uses of the word “proof.”

      You are arguing from a single, very specific, comparatively rarely used definition of that word. Walk into any English-speaking courtroom and hear the first words out of the prosecutor’s mouth, “The prosecution will prove…” and you’ll realize your mistake. She is not misusing the word “prove.” You are just not matching the correct definition to the context.

      With that understood, it should be evident that you are just boxing at shadows here, preaching to the choir. Because I never mentioned anything about “dogmatic” proofs (much less formal logical-mathematical proofs).

    • rork says

      Wanna talk about math or science things, theories of learning, and have “baysian” in the title, then you might try to say things as carefully as we would. The context is statistical. A statistician would never have put it like you did. Maybe you have no business writing about this.

    • says

      Your attitude is precisely the elitism that cuts the public out of all access to learning and knowledge. If academics insist on speaking technolanguages unfamiliar to the general public, you may as well be speaking Klingon. Your knowledge simply won’t disseminate beyond the ivory tower.

      Communication to the public has to be in their language. Maybe you don’t value communicating to the public. But I do. And especially as my audience are people in the humanities (historians and history enthusiasts) I am especially obligated to speak in terms they will find familiar.

      Your obligation is to attend to context and read charitably. Not to ignore context and nitpick semantics in a pedantic and useless fashion.

      But then, maybe you like doing that. Most of the rest of us are rolling our eyes at it.

  4. says

    Very nice. I purchased his book, I should get it soon. Thanks for the recommendation. I just got Matt McCormick’s book in the mail. Your next one is next on my list, your last one definitely peaked my interest.

  5. says

    Thanks for the note that I’ve misused “a priori” in that one post, Richard. I’ll emend that shortly. Much appreciated! Thanks also for the commentary in general.

  6. says

    This was an excellent post, Dr. Carrier. It lead me down so many rabbit holes. I loved TCD (edited by Loftus), and regard his OTF with similar high-esteem. Though I left my faith in my early teens, I still nerd out over its landscape.

    Biblical historical criticism is the unsung hero of the ‘paths to atheism from theism’. People often divide into ‘science’ or ‘philosophy’ as the driving force behind their apostasy. Pretty much all the people that Loftus highlights in his books deserve so much more recognition!

  7. jimmo says

    Dr. Carrier,

    While I understand the need that “Communication to the public has to be in their language”, it seems to me that referring to “proof” in an historical context is doing “the public” a disservice in that it conflates the two definitions to the point where the “public” unnecessarily sees historical proof on the same level as scientific or mathematical proofs. As a member of the “public” I want to see terms that apply in the specific context as used by the experts and not something watered down or overly simplified. The people who are interested in this subject are typically not uneducated and are (hopefully) not dogmatic in their beliefs, so I see a necessity in making the terms clear and unambiguous . As I see it, the “public” involved in this discussion are interested in truly understanding the material and not simply looking to quote mine scholars.

    I have read several books on historiography (Tucker, Bentley, Iggers, to name a few) and to me a common thread is that history is not about proof but the “best explanation” or which explanation is most probable. Even your own book “Provong History” talks about Bayesian probability, which we all know does not prove anything, but only show which is most probable. I cannot recall ever reading a single book or article that says historians “prove” anything in a sense even close to “beyond a reasonable doubt” necessary in a criminal trial, but much closer to the “preponderance of evidence” found in civil trials.

    I personally do not find this issue either nitpicking sematics nor pedantic, but rather it is extremely useful to use clear and unambiguous terminology. At the very least, I think it necessary to take an approach like Dawkins in his books on evolution. He talks about DNA as being like a computer code, but is very clear about it not being the same thing in order to avoid christian apologists pouncing on “code” = “designer”.

    • says

      While I understand the need that “Communication to the public has to be in their language”, it seems to me that referring to “proof” in an historical context is doing “the public” a disservice in that it conflates the two definitions to the point where the “public” unnecessarily sees historical proof on the same level as scientific or mathematical proofs.

      First of all, there is already a huge difference between “scientific or mathematical proofs,” so you aren’t making sense here. A mathematical proof is what you meant by proof. A scientific proof is not anything like that, but a probabilistic demonstration. It is therefore the same use of “proof” that I am employing. The difference is only one of degree of certainty, as I explain in Proving History, pp. 45-49 (“From Science to History”).

      So you complain about an equivocal use of “proof,” then engage that equivocal use of “proof” yourself. Not helping your case.

      As far as historians using the word “prove” and “proof” you have got to be kidding. Those terms are everywhere in our literature. In the very sense I use them. Do I really have to drag out examples? I mean, want to wager money on this?

  8. says


    I have to admit it bugs me when you make sweeping statements like “Christians are awful Bayesians”. Many Christians are fully acquainted with Bayesianism and know plenty of the ins and outs of it. It’s especially difficult to hear given that a) there’s usually no reason given for these generalisations, and b) you seem to endorse quite significant butcheries of Bayesianism around the place. It’s hard to find anything in the literature which supports your methodology for coming to prior distributions, for example, and Lindsay’s attempted use of the principle of indifference in the OTF stuff is really quite difficult to find persuasive.

    • says

      That’s fair enough. I shouldn’t overgeneralize. Lindsay documents how some Christians (often well-educated ones) are awful Bayesians.

      But your contrary opinions simply don’t hold up in the face of what Lindsay and I have actually written.

  9. corwyn says

    I think Lindsay got it wrong when discussing the priors that fundamentalists might claim for Christianity being true. If he asked them to justify that 97% or whatever, they would start listing evidence (the bible, miracles, etc). Therefore it isn’t a prior at all (when the whole world is going to subsequently be used as evidence). The proper (IMHO) question for the fundamentalist is “I have written the name of a religion on this piece of paper, what do you think is the likelihood that it is the true religion?” Not only will this remove their bias, but it will get them asking the right questions, like “what do you mean by religion?” or, “How many were in that pool that you chose from?” From this even the fundamentalist will get the correct prior (given the assumptions), since they don’t want to give credence to the wrong religion.

    Thank you kindly.