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Bayesian inference and religious belief

We’re speaking here not of Bayesianism as a religion but of the use of Bayesian inference to assess or validate the evidence regarding religious belief, in short, the probability that God !=0 or the probability that the Pope is Catholic or, as Tyler Cowen put it, the probability that Lutheranism is true.

As a statistician and social scientist I have a problem with these formulations in large part because the states in question do not seem at all clearly defined—for instance, Lutheranism is a set of practices as much as it is a set of doctrines, and what does it mean for a practice to be “true”; also the doctrines themselves are vague enough that it seems pretty much meaningless to declare them true or false.

Nonetheless, people have wrestled with these issues, and it may be that something can be learned from such explorations.

So the following email from political philosopher Kevin Vallier might be of interest, following up on my above-linked reply to Cowen. Here’s Vallier:

Philosophers of religion have been using Bayesian reasoning to determine the rationality of theistic belief in particular for many years. Richard Swinburne introduced Bayesian analysis in his many books defending the rationality of theistic and Christian belief. But many others, theist and atheist, use it as well. So there actually are folks out there who think about their religious commitments in Bayesian terms, just not very many of us. But I bet there are at least 1000 philosophers, theologians, and lay apologists who find thinking in Bayesian terms useful on religious matters and debates.

A nice, accessible use of Bayesian reasoning with regard to religious belief is Richard Swinburne’s The Existence of God, 2nd ed.

Wow—I had no idea there were 1000 philosophers in the world, period. Learn something new every day.

In all seriousness . . . As indicated in my post replying to Cowen, I’m interested in the topic not so much out of an interest in religion but rather because of the analogy to political and social affiliation. It seems to me that many affiliations are nominally about adherence to doctrine but actually are more about belonging. This idea is commonplace (for example, when people speak of political “tribes”), but complications start to arise when the doctrines have real-world implications, as in our current environment of political polarization.


  1. Justin Smith says:

    I find the use of Bayes theorem for arguing for evidence of God(s) existing or Jesus’ resurrection being true, etc., very amusing. When supplied with subjective priors, Bayes theorem can be a “Drake Equation”, where it is true in some logical sense but silly in that one can manipulate it to get whatever outcome they want. Not science and not probability even.

    Some think that it may have been Bayes’ or Price’s intention in the first place to use it to refute Hume’s argument against miracles. (Unwin’s book) (Swinburne’s book)

  2. It was OJM that first pointed out to me the problem of using Cox theorem as a foundation for Bayesian reasoning when there is no good sense in which a particular statement can be true. For example what is the true year of manufacture of a classic car? The device has had almost every part on it replaced at one point or another…

    What is the true height of a given oak tree in mm? As the wind blows it flexes, and what reference datum are we using on the ground, and what happens when a bird alights on a top branch?

    So I’ve got an alternative concept, and it has to do with measuring how well a model fits data relative to how well the model is supposed to fit given the best possible values of the parameters, but I haven’t spent enough time polishing it up for publication.

    • Olav says:

      What sense of “best” are you talking about when you say the “best possible values of the parameters”?

      • In my conception the model must itself provide a notion of best. The best possible value of the parameters would be the one that maximizes the joint predictive probability of all the data it will ever be used for when expressed in a sufficiently small but finite precision number system.

    • Clyde Schechter says:

      “For example what is the true year of manufacture of a classic car? The device has had almost every part on it replaced at one point or another…”

      Well, think about a human body. Certainly by mid-life, nearly every atom in your body has been turned over during exchanges with the environment. Nor can you point to a constant configuration of things having an onset, because you have grown and developed and begun to senesce (is that a word?) in various ways–the configuration of your parts at a sufficiently micro level changes radically over the life cycle. Yet other changes appear to take place at a meso- or macro- level but not at the micro level (senescence). So that’s even more complicated.

      Yet my driver’s license does list a date of birth.

      • The boolean statement “there exists an officially issued drivers license with the name Clyde Schechter on which the data of birth is listed as xx/xx/xxxx” does have a boolean truth value, and if the date on your drivers license is the question under investigation, bayesian inference using Cox as the foundation can provide a logical foundation.

        But if the question is “There is an organism called Clyde Schechter whose age since conception is xxxxxxxxx.xxxxxxxx seconds” then if conception is a process that takes a certain amount of time, longer than 0.00000001 second (and it does) there is no exact numerical value that can be put in this statement and make it have a true boolean value. We can fool around with the statement and soforth, but there are plenty of cases where the consequences of this kind of thing are logically problematic and even practically problematic. For example surveys of the earth to establish property lines in fault zones, where the fault slides a couple centimeters a year…

        In my paper I derive an alternative that doesn’t rely on a 0/1 underlying truth value. It relies on the mathematical model in question expressing its own expected precision.

  3. Justin Smith says:

    Some ‘proofs’ of God existing, resurrection of Jesus, and other miracles, rely on Bayes theorem.

    For some examples, see:
    -“The Probability of God: A Simple Calculation That Proves the Ultimate Truth” by Unwin
    -“The Existence of God” by Swinburne
    -“Bayesian evaluation for the likelihood of Christ’s resurrection” (

    These are silly IMO, but use priors correctly, arguably, especially in a subjective Bayesian paradigm. It is like Bayes theorem is a big “Drake Equation”, true in some sense of the word, but easily manipulated to get whatever output one wants.

    Some argue that it may have been Bayes’/Price’s intention to use the theorem to refute Hume’s argument against miracles.

    Note that this could not possibly occur with frequentism. Are these ‘proof of God’ examples a mark against Bayesian statistics as a whole? I don’t think so. Just like misuses or misunderstandings of frequentist statistics or hypothesis testing or p-values do not count against frequentism.

    (sorry, I may have double-posted this)

    • Erik says:

      Being lazy I have picked the one source with a link to read it (the third one). And the prior is the only thing that I do not find very troublesome in particular. It’s the updating that is clearly fallacious – just giving two points on this:

      a) the assumption that multiple witnesses are independent from each other is clearly wrong.
      b) it is not correct to assume that some acquaintance of the author claiming something unlikely to have happened to him/her is equal to finding a similar claim in a famous historical document – when the fame of said document is based on the fact that the claims are otherwise surprising.

      It is also amusing to consider that the same line of arguments could be used to prove a) all other deities exist, b) alien visitors to Earth exist and c) Elvis being alive.

      The prior is not an issue. It is the application of the Bayesian framework which is incorrect here. And yes, you could do the same in a frequentist framework. Here is an equally naive version:

      H0: Man did not rise from the dead
      H1: Man did rise from the dead

      Data: Book says man did rise from the dead.

      Assuming the man did not rise from the data (and ignoring all other facts) p for Book says man did rise from the dead is certainly much smaller than 5% (after all, how many men who did not rise from the dead have books saying they did?).

      In fact, the same probability is used in the linked Bayesian argument and given as ~1e-8^[NUMBER WITNESSES QUOTED IN BOOK].

      I don’t think this unsound line of reasoning is an argument against either statistical framework. But it is certainly wrong to say that it could not possibly occur with frequentism.

  4. Eric Rasmusen says:

    Nice Sunday morning topic!

    Hume criticized theological statements that were unconnected to sense impressions as vacuous, which is indeed a big problem. Other theological statements are simple statements of fact, though, true or false, if hard to verify, e.g. did Jesus of Nazareth rise from the dead after 3 days and appear bodily to the Apostles, or not?

    I Corinthians 14 says: 12 Now if Christ be preached that he rose from the dead, how say some among you that there is no resurrection of the dead? 13 But if there be no resurrection of the dead, then is Christ not risen: 14 And if Christ be not risen, then is our preaching vain, and your faith is also vain. 15 Yea, and we are found false witnesses of God; because we have testified of God that he raised up Christ: whom he raised not up, if so be that the dead rise not. 16 For if the dead rise not, then is not Christ raised: 17 And if Christ be not raised, your faith is vain; ye are yet in your sins. 18 Then they also which are fallen asleep in Christ are perished. 19 If in this life only we have hope in Christ, we are of all men most miserable.

    You can apply Bayesian thinking to a theory like Lutheranism or Evolution as well as to a fact like the Resurrection or the partial descent of modern humans from Neanderthals, but defining what the probability is over, and what the event space is, gets tricky. Also, equivocation becomes easy and will happen even by accident rather than by design, and “No true Scotchman…” gets to be a problem.

  5. Regarding “Wow—I had no idea there were 1000 philosophers in the world, period. Learn something new every day.” Really? How many philosophers are there at Columbia? Of course, this depends on how one defines, “philosopher.” Just like saying Physics professors are physicists, and a proper subset of the total number of physicists, we could take the number of philosophy professors as a starting point for one of several rough estimates. At the University of Oregon, there are 14 tenure-track philosophy professors. Let’s call this 1/3 of the philosophy professors in Oregon, and Oregon about 1/100 of the total population of the US. This gives 4200 philosophers in the US alone! Take that, Ancient Greece!

    By the way: I’m sure you’re tired of people pointing this out, but your blog doesn’t update properly anymore. The main page is often an old one, and the comment bar doesn’t show the most recent comments. Refreshing sometimes helps.

  6. Great topic Andrew,

    As an aside but related

    I thought that Susan Haack’s presentation on Legal Probabilism: An Epistemological Dissent aligned with some of my own misgivings in assigning probabilities to events and predictions.

    • Olav says:

      Susan Haack apparently thinks that probability theory can only be meaningfully applied to frequencies or propensities. In other words, she’s dismissing the entire standard Bayesian approach, not just legal probabilism.

      • Thanatos Savehn says:

        Here’s my problem with what I’m guessing (1:17:00 video???) is her take:

        When the plaintiff lawyers launch a new mass tort, whether toxic, pharmaceutical, medical device, etc., their evidence is always founded upon some analysis of observational data. Thus, their liability claim flows from probabilities translated as risks. Usually the observed statistically significant (patent still pending) effect size is very small and is IMHO invariably the result of expert HARK-ing.

        The judge, always bedazzled by low p-values, says they have good evidence nevertheless and that I must mount a defense. My solution was to turn their numbers back on them to demonstrate (1) that it is very unlikely that any given plaintiff was harmed if you accept the results of their own evidence; and, even more deadly to their cause, (2) the risk is so low that compared to other quantified risks that people happily accept and do not guard against this is not a risk of a magnitude that would justify the imposition of liability (see Palsgraf etc.).

        In reply plaintiffs say that evidence may be founded upon probabilities but liability cannot because liability in the American and British systems is a decision about which side has the better narrative (model fit), i.e. juries decide which lawyer is the better HARK-er. I retort that what the courts do is set the outer limits of liability, what juries do is abduction plus stating their communities’ tolerance for risk, and what risk does is tell us whether the risk imposed was reasonable (and thus not amenable to the imposition of liability) or potentially unreasonable, in which case it’s a matter for the jury to decide.

        P.S. Haack’s YouTube video is way too long and I gave up listening to it when she started telling stories about getting her passport renewed and then mangled the purpose of self-authenticating documents so if I’ve misstated her position (I have read it I suspect in pieces elsewhere) it was out of ignorance and frustration rather than malice.

  7. Lord says:

    You may not believe in it, but when asked about what you do believe in, have as little belief in that as anything else, with doubt reigning or certainty as banishment of it.

  8. Olav says:

    Philosophers who work on this stuff realize that religions are partly doctrine and partly practice. They’re interested in whether it’s epistemically rational to believe religious doctrines; many philosophers who think’s it’s epistemically irrational to believe religious doctrines are perfectly willing to grant that it may still be practically rational to engage in religious practices. Some religious doctrines may indeed be too vague to admit of a truth value (although I’ll note that there’s also a large philosophical literature on vagueness, and the fact that a statement is vague doesn’t necessarily mean it can’t be true or false!). In any case, we can restrict our attention to religious claims that are not vague and are clearly either true or false, and that’s usually what philosophers do when they discuss these types of arguments. E.g. there was a man who lived 2000 years ago who died and then rose from the dead three days later, the world was created by an intelligent designer with a particular plan in mind, etc.

  9. Anonymous says:

    I’m atheist. I read your material because it’s educational, and about good practice of statistics. I just enjoy it. I don’t gender identify for this hopeless sense of belonging you describe, nor do I identify as a statistician.

  10. Paul Alper says:

    Andrew put it neatly:

    ” It seems to me that many affiliations are nominally about adherence to doctrine but actually are more about belonging. This idea is commonplace (for example, when people speak of political ‘tribes’), but complications start to arise when the doctrines have real-world implications, as in our current environment of political polarization.”

    When it comes to religious beliefs and “real-world implications,” unlike most of the contributors to this blog, I speak with ecclesiastical authority becase I am a long-time bishop in Kirby J. Hensely’s Universal Life Church:

    “The church’s growth was affected in part by social movements; during the Vietnam War, a widely circulated rumor claimed that ordination would qualify one for a [religious] legal exemption from the draft. Ordination requests increased dramatically, but the rumor proved to be false. The ULC and its founder, Hensley, were also featured in several publications during this time, including Rolling Stone, which further increased public awareness of the church.

    By 1974 the church had ordained over 1 million ministers. Also in 1974, a federal judge declared that the ULC was qualified for a religious tax exemption.”

    Hard to imagine a more real-world implication, even if unfounded, than avoiding death in Vietnam. And if you want to read more about the Vietnam War, read Seymour Hersh’s “Reporter.”

  11. Roger says:

    “Bayesian inference to assess or validate the evidence regarding religious belief”

    …. A ridiculous premise.

    Religion is irrational, emotional, mysticism — and thus cannot be “validated” by any evidence or reason.

    However, the conventional concept of “God” is easily disproved by reason — because that concept encounters logical contradictions at every step of rational analysis (akin to a Square-Circle concept).

    • Ever hear of steelmanning? It’s the opposite of knocking down a straw-man argument. Instead of taking the easy way out and attacking the least defensible version of a position or argument, you fix it as best you can to produce the strongest possible version, and address that version.

      WRT to the existence of a god, that would involve saying, “OK, let’s put aside the logically incoherent concepts of god, and ask what’s the closest thing to the conventional concept that is at least logically possible.” I imagine that would be something like a very powerful and very knowledgeable being (not literally omnipotent and omniscient, but so far beyond us as to perhaps appear so) who either created the entire visible universe or at least our solar system. One could apply Bayesian reasoning to the question of whether such a being exists. I think you could make a good argument that leprechauns are far more likely to exist than even such a limited god. And that argument would, I think, be more persuasive to an honest theist than simply pointing out the logical contradictions of a naive theism.

      • Erik says:

        I think neither way is very persuasive. The problem with the “limited God” concept is that there is no way to get people to agree about the likelihood function – too many unknowns. Just consider that you basically need to put the Drake equation in there as part of the probability without God. With God for the strongest position possible you would need to assume that a God would create our universe as it is with probability 1.

        Without a likelihood function there is no way to make this argument work.

      • Eric Rasmusen says:

        I actually have a paper on the leprechaun problem:
        “The Concealment Argument: Why No Conclusive Evidence or Proof for God’s Existence Will Be Found.” Logic and Biblical evidence suggest that God wishes that some but not all humans become convinced of His existence and desires. If so, this suggests that attempts to either prove or disprove such things as God’s existence, past miracles, or present supernatural intervention are doomed to failure, because God could and would take care to evade any such efforts. In tex and pdf. (

  12. someguy says:

    Bayes Theorem was invented by Reverend Bayes specifically to argue about religious belief. You atheists are shameless appropriators

  13. Dzhaughn says:

    If I say there are more than 5000 students and professors at Columbia, do you infer that I believe there are 5000 professors?

    I am not going to give up my comfortable illusion that there are only hundreds of philosophers.

  14. steven t johnson says:

    The notion that doctrines don’t matter that much because religion is about belonging is I think kind of misleading. Sadly it seems to be a way of minimizing the role of superstition and bigotry taught by religion. Oh, I agree that most individual adherents vary widely in their understanding of doctrine, and how they do or do not use it in everyday life. But this is very much like saying that soldiers in the army have widely varying opinions about this, that and the other thing, even about whether they want to fight, or how they fight. Actual religions, as opposed to apologetic notions of generic “religion,” are more like armies: The institution runs things, and what the members think is not directly relevant. As for the notion of tribalism, I think it is largely a meaningless pejorative. In real life, tribes are social institutions, just like armies or churches, with rules and property of some sort.

    As to the Bayesianism…Being literal minded, I look at Bayes’ theorem and get terribly confused about how this formula for the probability of one event conditional on another event serves to guide revision of probability for a singular event, much less revision of a model. But it seems to me that Andrew Gelman’s point is easier to see if you use this version of Bayes’: P(H/E)=P(E/H)*P(H)/P(E) It seems to me maybe you can think meaningfully about the probability of Lutheranism given the evidence from history, psychology, sociology, natural science and I suppose scripture and personal testimony too. But what however you conceive that, no matter what trial values you assign to the prior, I can’t see how you can devise any coherent notion of P(E) of the same nature.

    • Forget the generalized logic of probability, what does it even mean for Lutheranism to have a logical 0/1 truth value?

      Until we get past that step there’s no point in trying to generalize.

      With scientific models at least we can have numerical predictions, and if we don’t go by truth of an underlying Boolean statement, we can go by consistently predicting values that are not too far from observed data. But there is not much in the way of this kind of thing for “Lutheranism”

    • “As to the Bayesianism…Being literal minded, I look at Bayes’ theorem and get terribly confused about how this formula for the probability of one event conditional on another event serves to guide revision of probability for a singular event, much less revision of a model.”

      That word “event” is the problem. It’s frequentist terminology that presupposes repeated experiments. Try replacing it with “proposition”.

      • steven t johnson says:

        Mario Bunge once observed that propositions don’t have probabilities, if I remember correctly. It is entirely unclear to me how this is wrong. Also, even if I do substitute “proposition” for event, I have even less idea what a “formula for the probability of one proposition conditional on another proposition” could mean.

      • Justin Smith says:

        “It’s frequentist terminology that presupposes repeated experiments.”

        Wouldn’t you want to replicate in science? Don’t we want repeated experiments?

        • Usually in science we want models that have wide applicability, but that doesn’t mean we want to just do the same thing over and over again. Predicting the weather for example. We want to use the models every day to predict tomorrows weather, but every day we input different measurements and make different predictions. Each day and each prediction is not independent and identically distributed.

          Similarly there are plenty of singular events we still need to have uncertainty about: that Malaysian airlines crash from a few years ago. Where did it occur? We certainly don’t want to repeat that crash a few thousand times and see the frequency distribution of locations.

      • Corey says:

        steven t johnson, you can read Kevin’s own explanation of Cox’s theorem to discover what “formula for the probability of one proposition conditional on another proposition” could mean.

        Justin Smith, since the topic is probability as applied to religion and not science your question is a bit of a non-sequitur, but it’s worth addressing anyway. Yes, we want to find stable properties of the world that are repeatedly observable, but if you insist on repeated experiments as a demarcation criterion for science you have to exclude, e.g., cosmology. It’s also worth saying that Bayes works fine for repeated experiments — it’s just that it is also directly useful when there’s no notion of repetition (whereas frequentists need to choose a way to bridge the gap between a single event and an ensemble of them if they want to talk probability in situations where repetition is not really a thing).

  15. Jens Åström says:

    A defining aspect of religious belief, to me, is that it exists in absence of, or even in contradiction to evidence. Leap of faith anyone? Faith is borderline to a type I error, rejecting the null of a no God in absence of evidence of his existence. And it seems to be the whole point of faith. Something to be commended. So it seems the truly faithful will be immune to any argument of this sort. Then again, perhaps it sows the seeds of doubt that would encourage someone to leap back.

    • Mikhail says:

      So if God does exists, worshiping Satan is a type S error. Sound legit.

      • Jens Åström says:

        Yeah, and the Theodicy problem stems from an M error. If God exists, we have overestimated his godliness. And Pascal’s wager optimizes for a low Type II error, given the consequences of not rejecting the null. See, we can fit anything into this framework…

    • I’m not sure that we can generalize to that extent, i.e ‘ A defining aspecti of relgious belief, to me, is that it exists in the absence of, or even in contradiction to evidence.

      I prefer to use the terms ‘absolutist’ and ‘relativist’ as a means of distinguishing individual’s degrees of belief. [along a continuum]. I mean to suggest that people do change their beliefs over the course of their lives. Sometimes on the weight of the evidence or in the face of some experience. What I have found is that to get into an adversarial debate with either theists or atheists has drawbacks of its own. I’ve read some of the online debates. Generally ends poorly.

      • Jens Åström says:

        Oh, online debates about religion sounds horrible, what have I gotten myself into? Well I won’t claim to know what another person’s reasons for believing something is. But it still seems to me that religious “reasoning” is another beast altogether than logical reasoning based for example on Bayesian stats. Belief in the face of doubt seems to be a key value at least in Christianity. I guess that was my point.

        • Jens,

          I am not sure that all Christians are absolutists re; ‘belief in the face of doubt’. My father was a professor of comparative religions up through the 90’s. So I was able to hear many discussions about religion. I think the academics of religion were once guardians of dissent and new interpretations of Christianity at least. But if one chooses to, one can possibly find some semblance of Bayesian reasoning in some discussions and scholarship. I haven’t explored it, myself.

        • Eric Rasmusen says:

          No: standard Christian doctrine is that belief is not a sufficient condition to be “saved”. The Devil believes in God much more strongly than any person, for example. “Faith” is meant to include not just belief in God, but also love of God.

          • Eric,

            To clarify, I was suggesting that ‘what constitutes Christian Doctrine’ has been open to interpretation throughout. Admittedly, in going over the list of 20th Century Christian theologians, according to Wikipedia, I’ve read not even a tenth of them. Within each denomination, however, there have been considerable expansion of what Christian doctrine constitutes, including the importance of reason, which entail ‘degrees’ of belief. And that is a subjective exegetical interpretative process and one may be able use Bayesian inference as well. I haven’t thought much about the subject. So if I sound vague, apologies. But for the reasons that are laid out in this topic, it’s interesting query.

  16. Sorry for the typos. ‘aspect’ and ‘religious’.

  17. TwoHorns says:

    John Earman “Bayes, Hume, Price, and Miracles”

    “While there is no evidence of a direct connection between Bayes and Hume, the indirect connection that goes through Price is solid. Although we know little of the relationship between Bayes and Price, it must have been reasonably close since Bayes’s wiIll eft £200 to be divided between Price and one John Boyl (see Barnard, 1958) and since it was PIice who arranged for the posthumous publication of Bayes’s essay. In the other direction, Price was a persistent critic of Hume – not just on induction but on matters of religion and ethics as well. Despite the sharp differences in opinion the two men remained on remarkably cordial terms, dining together in London and in Price’s home in Newington Green (see Thomas, 1924). In the second edition of Four Dissertations Hume is lauded by Price as ‘a writer whose genius and abilities are so distinguished as to be above any of my commendations’ (1768, p. 382). And Hume in turn praised PIice for the ‘civility which you have treated me’ (Klibansky and Mossner, 1954,233–4). More intriguingly, Hume goes on to say that ‘I own to you, that the Light in which you have put this Controversy [about miracles] is new and plausible and ingenious, and perhaps solid’. Unfortunately, Hume left the matter hanging by adding that ‘I must have some more time to weigh it, before I can pronounce this Judgment with satisfaction to myself’ (ibid.). I will finish Hume’s unfinished task. I will claim that Price’s criticisms of Hume’s argument against miracles were largely solid. More generaIIy, I claim that when Hume’s ‘Of miracles’ is examined through the lens of Bayesianism, it is seen to be a shambles”

  18. TGGP says:

    When I was about 20 I went from being an ultracalvinist to atheist (or, technically, an agnotheist) when I learned about Bayesian probability and concluded the only intellectually honest thing to do was to ask myself what probability I assigned to God as described in the Bible, and concluded that such a fantastical story seemingly crafted for human psychology had only an epsilon chance of being true. But, as noted, I was coming from an extreme “Protestant” background fit for a nerd, unlike the orthopraxic & communitarian form of religion common for most of humanity.

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