Bayes Theorem And Probability of God: No Dice!

  photo probability-theory-3224.jpg

It is understandable that naturalistic thinkers are uneasy with the concept of miracles. So should we all be watchful not to believe too quickly because its easy to get caught up in private reasons and ignore reason itself. Thus has more than one intelligent person been taken by both scams and honest mistakes. By the the same token it is equally a  danger that one will remain too long in the skeptical place and become overly committed to doubting everything. From that position the circular reasoning of the naturalist seems so reasonable. There’s never been any proof of miracles before so we can’t accept that there is any now. But that’s only because we keep making the same assumption and thus have always dismissed the evidence that was valid.
            At this point most atheists will interject the ECREE issue (or ECREP—extraordinary claims require extraordinary evidence, or “proof”). That would justify the notion of remaining skeptical about miracle evidence even when its good. There are many refutations of this phrase, which was popularized by Karl Sagan. One of the major problems with this idea is that atheists rarely get around to defining “extraordinary” either in terms of the claim (why would belief in God be extraordinary? 90% of humanity believe in some form of God) [1] The slogan ECREE is usually said to be based upon the Bayes completeness theorem.  Sagan popularized the slogan ECREE but the mathematical formula that it is often linked to (but not identical to) was invented by the man whose name it bears, working in the  seventeen forties but then he abandoned it, perhaps because mathematicians didn’t like it. It was picked up by the great scientist and atheist Laplace and improved upon.[2] This method affords new atheism the claim of a “scientific/mathematical” procedure that disproves God by demonstrating that God is totally improbable. It is also used to supposedly disprove supernatural effects as well as they are rendered totally improbable.[3]
            It is often assumed that the theorem was developed to back up Hume’s argument against miracles. Bayes was trying to argue against Hume and to find a mathematical way to prove that there must be a first cause to the universe.[4] Mathematicians have disapproved of the theorem for most of its existence. It has been rejected on the grounds that it’s based upon guesswork. It was regarded as a parlor trick until World War II then it was regarded as a useful parlor trick. This explains why it was strangely absent from my younger days and early education as a student of the existence of God. I used to pour through philosophy anthologies with God articles in them and never came across it. It was just part of the discussion on the existence of God until about the year 2000 suddenly it’s all over the net. It’s resurgence is primarily due to it’s use by skeptics in trying to argue that God is improbable. It was not taught in math from the end fo the war to the early 90s.[5]
            Bayes’ theorem was introduced first as an argument against Hume’s argument on miracles, that is to say, a proof of the probability of miracles. The theorem was learned by Richard Price from Bayes papers after the death of the latter, and was first communicated to the Royal society in 1763.[6] The major difference in the version Bayes and Price used and modern (especially skeptical versions) is that Laplace worked out how to introduce differentiation in prior distributions. The original version gave 50-50 probability to the prior distribution.[7] The problem with using principles such as Bayes theorem is that they can’t tell us what we need to know to make the calculations of probability accurate in dealing with issues where our knowledge is fragmentary and sparse. The theorem is good for dealing with concrete things like tests for cancer, developing spam filters, and military applications but not for determining the answer to questions about reality that are philosophical by nature and that would require an understanding of realms beyond, realms of which we know nothing. Bayes conquered the problem of what level of chance or probability to assign the prior estimate by guessing. This worked because the precept was that future information would come in that would tell him if his guesses were in the ball park or not. Then he could correct them and guess again. As new information came in he would narrow the field to the point where eventually he’s not just in the park but rounding the right base so to speak.
            The problem is that doesn’t work as well when no new information comes in, which is what happens when dealing with things beyond human understanding. We don’t have an incoming flood of empirical evidence clarifying the situation with God because God is not the subject of empirical observation. Where we set the prior, which is crucial to the outcome of the whole thing, is always going to be a matter of ideological assumption. For example we could put the prior at 50-50 (either God exists or not) and that would yield a high probability of God.[8] Or the atheist can argue that the odds of God are low because God is not given in the sense data, which is in itself is an ideological assumption. It assumes that the only valid form of knowledge is empirical data. It also ignores several sources of empirical data that can be argued as evidence for God (such as the universal nature of mystical experience).[9] It assumes that God can’t be understood as reality based upon other means of deciding such as personal experience or logic, and it assumes the probability of God is low based upon unbelief because the it could just as easily be assumed as high based upon it’s properly basic nature or some form of elegance (parsimony). In other words this is all a matter of how e chooses to see things. Perspective matters. There is no fortress of facts giving the day to atheism, there is only the prior assumptions one chooses to make and the paradigm under which one chooses to operate; that means the perception one chooses to filter the data through.
            Stephen Unwin tries to produce a simple analysis that would prove the ultimate truth of God using Bayes. The calculations he gives for the priors are as such:
Recognition of goodness (D = 10)
Existence of moral evil (D = 0.5)
Existence of natural evil (D = 0.1)
Intra-natural miracles (e.g., a friend recovers from an illness after you have prayed for him) (D = 2)
Extra-natural miracles (e.g., someone who is dead is brought back to life) (D = 1)
Religious experiences (D = 2)[10]
This is admittedly subjective, and all one need do is examine it to see this. Why give recognition of moral evil 0.5? If you read C.S. Lewis its obvious if you read B.F. Skinner there’s no such thing. That’s not scientific fact but opinon. When NASA does analysis of gas pockets on moons of Jupiter they don’t start out by saying “now let’s discuss the value system that would allow us to posit the existence of gas.” They are dealing with observable things that must be proved regardless of one’s value system. These questions (setting the prior for God) are matters for theology. The existence of moral evil for example this is not a done deal. This is not a proof or disproof of God. It’s a job for a theologian, not a scientist, to decide why God allows moral evil, or in fact if moral evil exists. These issues are all too touchy to just blithely plug in the conclusions in assessing the prior probability of God. That makes the process of obtaining a probability of God fairly presumptive.

[1] find,
[2] Sharon Berstch McGrayne, The Theory that would not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy. New Haven: Yale University Press, 2011, 3.
[3] As seen with chapter (? Disprove) by Stenger and Unwin.
[4] McGrayne op cit
[5] ibid, 61-81
[6] Geoffrey Poitras, Richard Price, Miracles and the Origin of Bayesian Decision Theory pdf$$$.pdf
Faculty of Business AdministrationSimon Fraser UniversityBurnaby, BCCANADA V5A 1S6. Geoffrey Poitras is a Professor of Finance in the Faculty of Business Administration at Simon Fraser University. Lisited 12/22/12.
[7] ibid
[8] Joe Carter, “The Probability of God” First Thoughts. Blog of publication of First Things. (August 18, 2010) URL:  visited (1/10/13). Carter points out that when Unwin (an atheist discussed in previous chapter) puts in 50% prior he gets 67% probability for God. When Cater himself does so he get’s 99%.Cater’s caveat: “Let me clarify that this argument is not intended to be used as a proof of God’s existence. The sole intention is to put in quantifiable terms the probabilities that we should form a belief about such a Being’s existence. In other words, this is not an ontological proof but a means of justifying a particular epistemic stance toward the idea of the existence or non-existence of a deity.The argument is that starting from an epistemically neutral point (50 percent/50 percent), we can factor in specific evidence for the existence or non-existence of a deity. After evaluating each line of evidence, we can determine if it is more or less likely that it would entail the existence of God.”
[9] Metacrock, "The Scale and The universal Nature of Mystical Experience," The religious a priroi blog URL: see also the major argument I sue for documentation in that article,  In P, McNamar (Ed.), Where God and science meet, Vol. 3, pp. 119-138. Westport, CT: Praeger. linked in Google preview.
[10] Stephen D. Unwin, The probability of God a Simple Calculation That Proves the Ultimate Truth. New York New York: Three Rivers Press, Random House. 2003, appendix 238


Joe Hinman said…
I debated Jeff Lowder on Bayes(yes he knows more about it obviously) Still I think I have some good points, This was his attack on the Clarice I posted here just now,the "no dice" article.

part 1

part 2
Jason Pratt said…
1.) Not that it matters much but the charges of atheism against Laplace were evidently not true, but rather were extrapolations from his rejection of Christianity (in its religious claims; he still thought it morally beautiful). Insofar as his own printed work goes, which has confirmation from people who knew him, he was a nominal deist, perhaps a minimal deist in that he believed God created Nature but created it perfectly so that God had no need for further intervention within Nature (and so never did farther intervene, especially by miracles). This is the context of his famous discussion with Napoleon, which has been often misrepresented: Laplace was correcting Newton's calculations which, so far as Newton had gone, would have ended in the destruction of the Solar System long ago, so that Newton hypothesized that God occasionally interjected adjustments. Laplace's position was that, in his farther calculations, he had no need for that hypothesis, and when asked about it later expressed that had Newton lived to see the result he would have had his belief in God more properly affirmed (i.e. only an imperfect creator, on Laplace's theology, would have needed to make adjustments.) Laplace was given last rights as a cradle Catholic, but doesn't seem to have come back to Catholicism or any Christianity per se before death.
Jason Pratt said…
2.) 2000 was around the time Swineburne started using (what I'm calling a greatly mis-applied form of) Bayesian Theory to argue that the Resurrection of Jesus should be accepted as well over 90% probable. I do not personally recall an equally or more prevalent atheistic usage dating back that far, although admittedly this is anecdotal. I mention this to add experience confirmation to what you said, that BT was pretty much unknown in theology disputes until around that time (I'd put it a little farther back perhaps to the late 90s, but I know you meant the date generally) -- but from what I recall atheists were inspired by Swineburne's usage.

3.) According to Bayes' own notes (and the introduction of Robert Price when presenting Bayes' paper posthumously to the Royal Society, which can be found in various places for example the main appendix of Swineburne's edited handbook, Bayes's Theorem), his goal (implicitly in dispute with Hume) was to logically describe the process of inductive (also known as analogical) reasoning. This was also from the beginning connected to a mathematical predictor formula, verrryyyyyyy much different to the expression of BT which was not itself a mathematical predictor of probabilities. The actual algebraic formula can be found in the paper, and is far too complex to represent in a Blogger comment; the practical purpose of the formula is to calculate the odds of a ball positionally coming to rest after bouncing from a drop onto a sheet of paper (i.e. a plane). I don't know whether the actual math formula is being used in, for example, cracking the Enigma code and other famous uses dating from WW2, or some application of the non-math formula. But it is the non-math formula which was borrowed by Swineburne and then converted (wrongly in my estimation) to a mathematic operation.

4.) Also, according to Price (not mentioned by Bayes so far as I can recall from a fast rescan of his paper, but no doubt a point he agreed with), the philosophical/theological implication of his work would be that it thereby shows what reason we have to believe that in the constitution of things there are fixed laws according to which things happen, and therefore "the frame of the world" must be the product of "an intelligent cause". Thus success of the argument is taken as confirmation of theistic argument from final (i.e. ultimate, or first) causes. This is probably why BT hasn't been very important in theological debates pro or con since then; and I doubt it would be now except that Swineburne (if someone else didn't try it first at around the time he did) attempted to apply the inductive description as a math argument for calculating "probabilities" of this or that religious topic.

Jason Pratt said…
As usual for reference sake, here's my eyebleedingly detailed article from back in 2012 discussing BT, as a preliminary (which I never got around to) of discussing Jeffrey Jay Lowder's evidential argument against theism.

im-skeptical said…
On this issue, I agree with Joe. Bayes' Theorem is useful when we have enough background information to calculate the probabilities with some degree of accuracy. If you have to make a guess for prior probability, or just assign a 50-50 probability for something in lieu of having real probability values, Bayes' Theorem is of no value, and in fact, can easily be used to skew the numbers in favor of any position you like.
Joe Hinman said…
We agree, and you don't believe in miracles! ;-)
Jason Pratt said…
I should clarify that when I mean "implicitly disputing with Hume", I mean there is no direct evidence Bayes and Hume were reading and replying to each other. Hume and Price (Bayes' successor) did personally know each other and were great friendly opponents. It was Price who deployed Bayes against Hume's argument against miracles; according to some notes from Hume afterward, Hume actually thought Price's argument might have refuted him -- but this was never followed up on. The two men admired each other a lot.

If Bayes was responding to something from Hume, which is both tenable and (in a Bayesian induction sense {wry g}) likely, he would have been trying to answer Hume's earlier critique of inductive reasoning per se.

Cale said…
So, point of order:

Bayes's theorem is not contentious among mathematicians or statisticians. It never has been. It's trivially provable from the axioms of probability (both as formalized by Kolmogorov and from within the less rigorous theories of probability that preceded him).

No-one doubts that Bayes's theorem is correct.

What has changed, over the last...twenty years or so, roughly, is that methods of parameter selection, model updating, and hypothesis testing based around Bayes's theorem (rather than more traditional statistical methods) have become popular, and they have been accompanied by a shift in our views about what probabilities actually represent--away from the strict "frequentist" interpretation (on which a probability is an estimate of a frequency within a population) towards a Coxian interpretation (in which a probability maps to some credence or expectation).

There isn't really a serious argument against using Bayes to evaluate things like the probability that God exists, here, so I am not going to spend a whole lot of time on this post, but I figured that I would point out that the characterization of Bayes's role in probability theory and mathematicians' opinions of it presented here is just entirely wrong.
Joe Hinman said…
"Bayes's theorem is not contentious among mathematicians or statisticians. It never has been. It's trivially provable from the axioms of probability (both as formalized by Kolmogorov and from within the less rigorous theories of probability that preceded him). "

Not looking good for your scholarship dude, I footnoted that allusion it comes from:

Sharon Berstch McGrayne, The Theory that would not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy. New Haven: Yale University Press, 2011, 61-81 (seefn 5)

you take that up with her. That is a valid point made by a published author writing on the subject you need to address her work,I quoted her and you have no documentation.

"Sharon Bertsch McGrayne is a science writer and award-winning journalist. She has been a reporter for Scripps-Howard, Crain's, Gannett, and other newspapers covering education, politics, science, and health issues. She is a former science editor and writer for Encyclopaedia Britannica and the author"

valid documentation especially given you have none, it looks to me like you are juding the hsitory based upon the last few years she went back to before WWII,
Joe Hinman said…
There isn't really a serious argument against using Bayes to evaluate things like the probability that God exists, here, so I am not going to spend a whole lot of time on this post, but I figured that I would point out that the characterization of Bayes's role in probability theory and mathematicians' opinions of it presented here is just entirely wrong.

Obviously the consequences of my argument being true would be devastating for your project,to totally defeats your attempt to disprove God. so clearly it's serous, you try to pretend it's no big deal because you know you have no answer.

simple test, you have not answered my question about what you use as a basis for judging God's existnece. I quoted Sdikennaker saying you have to have accurate info for probability to work you tried to insist you do't so what's the basis for your analysis if it's not factual knowledge?

That guy I quoted he is an atheist who was arguing Agilent Unwin's use of Bayes to prove God.
Joe Hinman said…
Your assertion that on one donuts is also fallacious, Vic try Repert does and some of those I quoted do. the only things you can prove it works to do are things I've admitted to they are empirical. For stuff like God not given in sense data obviously another story.You can pretend like I haven;/t said anything you just lost the argumet,

In debate you don't answer an argument you lose it. you did not answer my argument preteen it's not an argument is not an answer. It's an an argument it beats your argument.
Joe Hinman said…
here's another source that confirms the fact that mathematicians have not always liked Bayes,


again Bayes works well for things like finding submarines but not fro things where we have no empirical grasp of the issue.

Jason Pratt said…
Cole: {{What has changed, over the last...twenty years or so, roughly, is that methods of parameter selection, model updating, and hypothesis testing based around Bayes's theorem (rather than more traditional statistical methods) have become popular}}

All of which bear on how new evidence weighs compared to current evidence in adjusting expectations of the truth or falsity of the hypothesis, which as I have consistently said is the whole point of BT.

What you're saying about the paradigm shift in application in the last 20ish years is true, but it doesn't bear on the popularity of trying to use BT on religious topics which started in the same period. When Swineburne introduced it to evaluating the Resurrection, he didn't appeal to new methods of model updating and hypothesis testing etc.; he just treated the inductive-logic expression of BT (which I shall have to repeat is _not_ the math formula connected to BT) as a math formula for multiplying and dividing a set of fractions, for which purpose he haxored the inductive logic expressions in such a way that they would look, on the face of it, like a very normal math-proportion argument. Which he knew better not to do, but did anyway. He, and the people he kind of shell-gamed into being impressed with it, have made _that_ 'version' of it so popular that this is what theists and atheists have been tussling about pro and con, instead of the actual uses of BT which you're referencing in the quote. Including in your Facebook article linked by Don.

Actually using BT, P(h|e&k) = (P(e|h&k) P(h|k)) / P(e|k), for evaluating topics like God's existence, which as I affirmed in the comments to the other thread is perfectly acceptable (because inductive analysis on any topic is perfectly acceptable), would be a refreshing change of pace! -- but also kind of boring in a way because it would just be the normal inductive arguments that all humans use every day on all topics (when we aren't using deduction and abduction instead).
Jason Pratt said…
What has been contentious, and revised with updates in the past few decades, is trying to convert that inductive process to a mathematical mode that machines (in effect) can work at. Which gets back to Bayes' own original project: expressing inductive reasoning in formal logic and then connecting that to a math formula for cross-verification. You could say that his own related math formula (about the ball bouncing to rest on a plane) is a strict "frequentist" interpretation on which a probability is an estimate of a frequency within a population; whereas the logical expression of P(h|e&k) etc. is more like a Coxian interpretation in which a probability maps to some credence or expectation. But it isn't a proportional math problem where a fraction P(h|k) gets multiplied by a fraction P(e|h&k) and then divided by a fraction P(e|k).

If that was true than P(e|h&k) would always necessarily reduce P(h|e&k) (because multiplying fractions together always produces a result closer to zero than closer to 1); and P(e|k) would always necessary increase P(h|e&k) (because dividing a fraction by a fraction not only always produces a result greater than either prior fraction but quickly exceeds 1.0 and ultimately goes to infinity!)

Which is not how induction works. If P(e|k) happens to be false, which mathematically would amount to 0, then the hypothesis is not therefore superconfirmed to be infinitely more probable than one hundred percent certain. (If P(e|k) happens to be true, mathematically amounting to 100%, that should mean the new evidence doesn't pertain to the hypothesis at all and so is neutral in affecting an estimation of (h)'s likelihood to be true -- which dividing by an integer of 1 would do, but the point is that "dividing by" P(e|k) ends up making (e) ludicrously and impossibly hyper-relevant in principle.) And P(e|h&k) ought to be able to raise or lower (or if you prefer count for or against) the estimation of (h) being true; but instead it always necessarily lowers (h) if used as a multiplied fraction.

For that matter, P(h|k) should also be able to count for or against (h) being true, but instead it always necessarily lowers an expectation of (h) being true if used as a multiplied fraction!

Using the inductive expression of BT as a math formula, at least in the sense of multiplying a pair of fractional probabilities (ranging from 0 to 100 percent) and then dividing by another fractional probability, would be grotesquely distorting and indeed invalid for evaluating the probability of any hypothesis being true, regardless of the topic. And yet this is exactly how you, and Don (representing theistic proponents of Swineburne's application), are using the (wrongly oversimplified) expression of P(T|E) = P(T) * P(E|T) / P(E) in disputing over the "probability" of "T" being true or false.

Jason Pratt said…
On a totally incidental side note, I'm a big fan of "Less Wrong"'s epic fan-fiction rewrite of the Harry Potter story, Harry Potter and the Methods of Rationality! So seeing Joe link to an article from him on the history of Bayesian Theory is highly amusing to me. {g} (One of his story arcs is named after and involves BT if I recall correctly.)

Jason Pratt said…
(Although I should add, looking over the article, that this isn't the LW author I'm thinking of but another contributor to his blog. Oh well.)


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