Should We Use Bayes's Theorem to Evaluate God's Existence ( Pt.1 )
Against Bayesian number mashing. On misuse of formalism, misguidedness and ersatz rigour in Philosophy of Religion, and Philosophy more broadly
tldr; maybe, but if we do, being honest with ourselves and our audience about the severe limitations with what we’re doing.
In contemporary analytic philosophy of religion, some people elect to rationally go about evaluating what they thing about God’s existence using Bayes’s theorem. This approach has its merits and drawbacks. Some people will however present this Bayesian approach as if it’s the gold standard for reasoning, and as though those who aren’t participating in Numberwang are in some way failing in their intellectual duties. My intention here is to lay out some brief considerations that should at least temper the confidence we put in these methods in Philosophy of Religion. Nothing here is going to be decisive, but it should be a good start for laying out the terrain.
Bayesianisms
There are different types of Bayesians. Different Bayesians claim to be doing different things when they use Bayes theorem, and those different uses have different normative implications. To mention just two of these positions…
Bayesian approaches, or more appropriately Bayesian-Laplacian approaches1 to probabilities and statistics (purportedly) “differ” from frequentist approaches2 where Bayesians are more able to operate with imperfect information, uncertainties and emphasise the relationship of credences to agential action, frequentists depend more on empirical data that represents long run frequencies and the application of statistical “Golems”3 (dumb tools for inference and testing). That’s all a massive oversimplification and insufficient to really understand the motivations behind the different approaches, the history or if there is even really a difference between the two, or if that’s a sort of false binary. It will have to do, because this post is not about that!
Bayesian approaches typically apply Bayes’s rule in some form or other in order to ascertain the “inverse probabilities” i.e. rather than answering a question like “what’s the probability I will win this next lottery”, or “what’s the probability my next dice roll comes up a 6”, Bayes’s theorem helps to answer questions like “given I have a positive test for this disease, what’s the probability I actually have (had at the time of testing) this disease”, or “given some encypted U-boat messages, what’s the probability that X is the correct decryption sequence.”4
Bayes’s rule, in plain English, is that the probability of some hypothesis after taking into account some new piece of evidence is the probability of that evidence conditional on that hypothesis (i.e. how strongly that hypothesis statistically predicts that piece of evidence), multiplied by the probability of that hypothesis being true, all divided by the probability of observing that evidence (regardless of the hypothesis).
This is also a bit of an oversimplification. When we apply Bayes’s theorem we fix information “in the background” to make our judgements. We can express this as ‘k’ for background knowledge. This is important, because it can help (but I don’t believe is sufficient) to explain why different agents make different judgements. For example, if Richard Dawkins (atheist) and Thomas Aquinas (Christian) were asked to make assessments about whether or not the existence of life on earth is more expected conditional on atheism or Theism, then both would “fix” a lot of different facts in the background of their assessment. For example, Aquinas’s view of the world was geocentric, before the invention of the microscope, the cell, knowledge of DNA or natural selection or organic chemistry. Aquinas’s judgements about the relevant considerations given atheism or theism would be very different to Dawkins and their (I imagine) radically different evaluations of what the likelihoods are would be due to that different background information. As we can see, the background information affects every part of the equation, the likelihood of the evidence conditional on the hypothesis, the hypothesis (because we are evaluating that given what we fix) and the evidence, because we are evaluating the absolute probability of observing the evidence given our stock of knowledge.
One further piece to note here is that I mentioned not thinking that fixing background knowledge is sufficient to explain divergent assessments. This is to briefly engage with some things Bayesians will say. There is a “problem of priors” i.e. what a ‘pure’ prior would be in a complete state of ignorance. One response to this is to say it kind of doesn’t matter, because over time, updating on evidence, posterior probabilities will converge ( I will talk about that some other time ). Another response is to try to provide some theoretical apparatus for judging priors. Sometimes people appeal to “intrinsic probability” in reference to some kind of “pure” probability of a hypothesis like Theism or atheism sans anything else. One thought is that our description of the theory “contains” an amount of information. The problem is, descriptions vary in different languages and have different informational measurements in different languages. The move to get out of this is to say that you’re not talking about the informational content in any actual langauge, but the hypothetical ideal language of propositions (the thing that natural language is supposed to be an expression of); we don’t mean syntax we mean semantics (haha don’t you even know this real and true distinction that’s definitely real; haven’t you read Tarski’s deep and profound disquotational schema that disambiguates the two!) I think that this line of reasoning is mistaken, why even suppose an ideal language exists and underpins our actual meanings. As long as we can’t produce descriptions in this “ideal” language we can’t assess the informational content in the way we do with other languages (i.e. bit transmission length — yes yes questions about encoding etc. these aren’t my problems anyway as I think this approach is mistaken). For independent reasons in linguistics and philosophy of language and mind, I don’t think that there is a propositional content as philosophers mean it that words express. I don’t think there are entities, mental or otherwise called propositions. I intend to write various pieces, produce arguments, evidence and make videos and interview expert linguists articulating, explaining and defending this view as I believe it is very important for understanding philosophical questions. But suffice to say, I think that this move is based on a non-existent entity, people are imagining themselves to be measuring the informational content of a ghost!
--Suppose everyone had a box with something in it: we call it a "beetle". No one can look into anyone else's box, and everyone says he knows what a beetle is only by looking at his beetle. --Here it would be quite possible for everyone to have something different in his box. One might even imagine such a thing constantly changing. --But suppose the word "beetle" had a use in these people's language? --If so it would not be used as the name of a thing. The thing in the box has no place in the language-game at all; not even as a something: for the box might even be empty. --No, one can 'divide through' by the thing in the box; it cancels out, whatever it is.
That is to say: if we construe the grammar of the expression of sensation on the model of 'object and designation' the object drops out of consideration as irrelevant.
Wittgenstein, Philosophical Investigations, §293
Back to Bayes. The idea then is that after engaging in this updating procedure, we can arrive at a place where our confidence in various hypotheses is proportional to the evidence that we have and that’s Numberwang!
There are different things people claim to be doing when they offer up Bayesian arguments and depending on your views, these kinds of “updates” will have different consequences.
Strong Bayesianism
The Bayesian machinery is frequently used in statistics and machine learning, and some people in these fields believe it is very frequently the right tool for the job. I’ll call this position “weak Bayesianism.” There is a more extreme and more philosophical position, which I’ll call “strong Bayesianism,” that says that the Bayesian machinery is the single correct way to do not only statistics, but science and inductive inference in general – that it’s the “aspirin in willow bark” that makes science, and perhaps all speculative thought, work insofar as it does work.5
Strong Bayesians are those who think that this way of thinking about things and doing reasoning is how we should reason about everything. Of course we might not always do that, that would be an intellectual failing, but we should aim to be, this is the correct procedure for rationality. Of course, this independently has to be argued for ( and without begging the question ) — I think that can be done, though obviously I disagree. The relevant point is that in order for Bayesian assessments about hypotheses like Gods existence to be normatively binding, something like this will have to be true — a person considering a Bayesians “case for God” will have to believe that they are normatively compelled (or perhaps weakly in this case they are compelled) to update based on the evidence being offered. Of course that doesn’t “get the job done” because that person will also have to agree with the background knowledge and probability assignments of the Bayesian number masher in order to be compelled to believe in the conclusion of the Bayesian argument.
Likelihoodism
“Bayesians differ among themselves, but they tend to agree that computing the posterior probabilities of hypotheses is always an attainable goal. Likelihoodists claim that this goal is often impossible to achieve. They hold that when Bayesianism fails, discovering which of several specific hypotheses the evidence favors is often an attainable goal. Likelihoodism’s goal is more modest than Bayesianism’s. In a sense, likelihoodism is an attenuated Bayesianism; likelihoodism is what remains of Bayesianism when some of the latter is stripped away.”6
Sometimes, people offer Bayesian arguments where they only really focus on the likelihood comparison. All of the problems that Im going to bring up under the likelihood section will still apply to this approach. However, this approach attempts to skirt around some of the issues related to priors and actually providing any precise numbers — all you have to do is vaguely sort of intuit an inequality between two hypotheses that is “large enough” ( for your mother).
When people do this it’s not clear what the import is supposed to be for anyone else. Suppose they do assign likelihoods the way you do; so what? That doesn’t tell us anything about what we should believe, what we believe in total and so on. For example there are many atheists who think arguments like The Fine Tuning argument or Psychophysical Harmony argument are evidence for Theism. They’re still atheists, why? One explanation might be because they’re completely irrational, can you think of any others?
If you’re fuming at this and thinking “ah but any reasonable person will assign priors in such away that a likelihoodist argument is useful, I address this in a future post in this series… stay tuned.
Other Considerations
There be dragons. All of these areas of thought pertaining to rationality and probabilities are fraught with controversy and contention. I have highlighted two of the main ways one might be taking Bayesianism, however there are other types, and there are many other considerations which could have significant implications for the evaluations we given and the normative force of our considerations when presented as arguments. For example, considerations such as whether we are using a subjective or objective interpretation of probability, or if that is a false binary, whether we are going with something else and how that propagates through our reasoning. Im not going to go into all of the details here, but just keep in mind that anyone who tries to sell you a used car based on a Bayesian argument without laying all these cards down on the table is selling you up the river.
My reasons for highlighting these differences between things like Likelihoodists, Strong Bayesians etc. is that in offering Bayesian style arguments, Bayesian number mashers frequently brush over all of the details that pertain to their views on these matters and how what they’re doing, whilst impressing you with their command of mathematical vocabulary, is supposed to relate to your intellectual and rational life in any way. I am not saying there can’t be a connection, that these things can’t be persuasive in an appropriate rational way, but if they are to be and we’re not just having the wool pulled over our eyes —because we were intimidated by mathematics, or perhaps we liked the idea of presenting these Bayesian arguments ourselves, wouldn’t people think we are so smart and intelligent— then all of these details must be spelt out and made transparent. If not it’s utterly ambiguous what we’re supposed to be doing on the receiving end of some madman ejaculating numbers in our faces — it’s like someone is trying to sell us a Rockwell Retro Encabulator!
The appropriate response in such cases is given by Mark in Peep Show
Pull the other one Jeremy, it’s got bells!
In the discussion that follows Im going to mostly be talking about issues pertaining to Philosophy of Religion. However, many of these problems are generalisable to other Philosophical (or otherwise) applications of Bayes’s theorem.
Follow up Posts on Priors, Likelihoods, Evidence and summarising the usefulness and limitation of this approach coming over the next month. I thought I would split it up a bit to make things more digestible ( I also went over the length for en email newsletter…)
McGrayne S.B.,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; 2012
The Biggest Beef in Statistics, Very Normal (YouTube)
McElreath, R. Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd ed.). Chapman and Hall/CRC. ; 2020 [https://doi.org/10.1201/9780429029608]
King B., Lee M., Bayes’ Theorem: the maths tool we probably use every day, but what is it?; 2017 [ https://theconversation.com/bayes-theorem-the-maths-tool-we-probably-use-every-day-but-what-is-it-76140 ]
Sober E. The Design Argument. Cambridge University Press; 2018.
Really great piece Nathan! Thank you for sharing. Its too bad though that my prior probability in my own bayesian methods is .999 and my credence in your writing is .000001 so clearly and intuitively is just obvious that bayesian analytic philosophy is the best!!
Nice essay. I appreciate the Numberwang references, and the Rockwell Retro Encabulator video made me chuckle. Looking forward to the follow-up posts.