Introducing Bayes blogging: a review of Richard Carrier’s Proving History, part 1

I mean to be doing a lot of thinking about probability theory, Bayesianism, etc. over the course of the next few months, perhaps longer. And I’ll be blogging it. Thus begins a (probably irregular) series of posts that I’m going to dub “Bayes blogging,” because it sounds better than “probability blogging.”

To start off with, I’m going to review Richard Carrier’s Proving History. This review is long overdue. I first read the book near the beginning of last year, but am only now reviewing it as part of a commitment to blogging more about these issues.

As you may know, Proving History is the first book in a two-book series arguing that Jesus never existed. This first book, though, is primarily about methodology, and that’s what I’m going to be caring about here. In fact, I read it precisely because I’d heard it strongly recommended on those grounds by Luke Muehlhauser.

I want to start off by saying that Carrier may be right about everything he really needs for the larger project on the historicity of Jesus, namely that Bayes’ theorem is potentially quite useful to historians and if an argument is clearly nonsensical on Bayesian terms it should be discarded. But I’m not going to be talking about those kind of Bayesian basics in this review. If you need that, I recommend Eliezer Yudkowsky’s “An Intuitive Explanation of Bayes’ Theorem.”

Instead, I’m going to be focusing on the philosophical side of the book. This is clearly an important part of the book–from an e-mail exchange I had with Carrier:

You say, “An example of resolving philosophical questions with Bayes’ Theorem is my chapter ‘Neither Life Nor the Universe Appear Intelligently Designed’,” what would be the equivalent statement about Proving History? Would you say it’s an example of how to resolve an analytical debate (about the historical method and epistemology)?

Oh, yes. Proving History is itself a work in the philosophy of history, covering the logic, epistemology and semantics of historical argument. And in that respect it is an example or model that might be useful to consult if anyone wants to expand the approach to other subject fields in philosophy.

Also, I recognize much of the material in Proving History, was focused on historical method, but chapter 6, “The Hard Stuff,” goes pretty far into general philosophical debates, how would you describe that chapter?

Yes, that’s true. Because I wanted the book to be as approachable to humanities majors as possible, I put all the most complex and difficult stuff I could in the last chapter so people who have no worries about those issues can skip it, while those who want those concerns resolved can dive in. It’s aim is to shore up issues that needed to be resolved for the rest of the book, and thus most of its examples still come from and deal with history. But of course many of those issues would similarly be raised if Bayes’ Theorem were applied to any other subject field in philosophy, so it does have a broader utility. Although again I’d say mostly indirectly, as my concern throughout was with answering historians and philosophers of history specifically. Other subjects were not as much on my radar. But inevitably the questions and applications dealt with there are often the same in any field.

So, let’s look at the philosophy: A key part of Carrier’s thesis is that even when it may not seem like we’re talking about probability, we actually are. For example, on p. 24, he says:

when we say something is “probable,” we usually mean it has an epistemic probability much greater than 50%, and if we say it’s “improbable,” we usually mean it has an epistemic probability much less than 50%.

Similarly, on p. 50:

Every time we say something is “implausible” or “unlikely,” for example, we are covertly making a mathematical statement of probability (and if this is not already obvious, I will prove it in the next chapter, beginning on page 110).

For those who would say we sometimes have no idea what probability to assign to a claim, Carrier endorses the principle of indifference. On p. 81:
All probability must be conditional on what you know at the time, so if you know nothing as to whether it would be higher or lower, then so far as you honestly know, it’s 0.5.”
Another key part of Carrier’s thesis is that all valid historical reasoning can be captured in Bayesian terms. He writes (on p. 104 and 106):

In fact, any valid form of hypothetico-deductive method is described by BT…. Bayes’s Theorem models and describes all valid historical methods… All become logically valid only insofar as they conform to BT.

Even if these claims are plausible, the way Carrier argues for them often ends up being problematic. He talks of “proving” his claims, and says things like, “I shall establish this conclusion by formal logic” (again on p. 106), as if he could derive them through the mathematical certainty of Bayes’ theorem itself. The trouble is that he seems not to clearly distinguish between Bayes’ theorem on the one hand, and Bayesian methodologies and philosophical claims on the other.

For example, on p. 81 Carrier writes:

The objection most frequently voiced against BT [Bayes' Theorem] is the fact that it depends on subjectively assigned prior probabilities and therefore fails to represent objective reasoning.

This makes the views of the critics sound silly, as Bayes is a piece of mathematics, which you shouldn’t reject for those kinds of reasons. You might, however, reject certain philosophical claims about Bayes for those reasons. This suggests not just a failure to distinguish between different kinds of claims, but also that Carrier doesn’t understand the views of people who might object to his project.

Similarly, starting on p. 106, Carrier spends a fair amount of breath saying how no valid argument or method can contradict Bayes’ theorem. This may be technically true, but it’s unclear whether this gets Carrier as much as he would like. Bayes’ theorem, again, just tells you what to do once you have certain probabilities (which it’s not clear we can always get), while historical methods and arguments often don’t explicitly deal with probability.

Of course, Carrier would claim that the outcome of any historical argument is always implicitly probabilistic. And he promised to “prove” it starting on p. 110. It turns out that a key part of the argument here is to refer the reader back to his previous discussion of a fortiori reasoning. From p. 85:

There are several tricks to ensure your use of BT is adequately objective. The most important is employing estimates of probability that are as far against your conclusion as you can reasonably believe them to be. This is called arguing a fortiori, which means “from the stronger,” as in “if my conclusion follows from even these premises, then my conclusion follows a fortiori,” because if you replaced those estimates with ones even more correct (estimates closer to what you think the values really are), your conclusion will be even more certain.

Carrier claims that this approach can solve the problem of not knowing exactly what numbers to plug in to Bayes’ theorem to get a result. In a way, it’s a very good point, which goes a lot way to making Bayes’ theorem practically useful to historians. But I’m skeptical that it goes all the way to showing that all valid historical reasoning can be captured in Bayesian terms.

To see why I hesitate, here, it can be worth reflecting on just how difficult it is to assign probabilities to historical claims sometimes. Consider an example Carrier an I have both spent a lot of time talking about, the claims of Christian apologists about Jesus’ resurrection.

When I need to explain why the apologists’ arguments are no good, the first thing I’m going to do is go to examples like the story of Joseph Smith and the Book of Mormon, or if I want the analogy to be exact as possible use a hypothetical example like Carrier’s Hero Savior of Vietnam example.

If it turns out that they’re the type of person that thinks maybe Joseph Smith was demonically influenced, I can lecture them a bit about the history of paranormal investigation, the troubles the Society for Psychical Research ran into, and so on. But assuming they agree that they acknowledge that they don’t/wouldn’t accept those other claims, I can then argue that the evidence for Jesus’ alleged resurrection is no better so they should accept that claim either.

Now you can put a Bayesian gloss on all this, and say that the prior improbability of a miracle is too low for the meager evidence presented by the apologists to overcome it. But it’s hard to know where to begin saying where those probabilities are exactly, even a fortiori.

It’s wildly inadequate to merely say they’re “low,” because as William Lane Craig points out, we have no trouble believing reports of lottery numbers being picked when a given number only had a one in a million chance of being picked. So, if you accept the argument from the analogy of Joseph Smith and Hero Savior, we know the probability needs to be really low. But how low is that? It’s hard to say.

The most I really think we can do here is give the argument from analogy, and then argue that what we’ve done can be made sense of in Bayesian terms. But that doesn’t necessarily mean we could get the right conclusion using Bayes alone, with no sanity checks from things like the argument by analogy to make sure the numbers we were plugging in to Bayes’ theorem made sense.

Or maybe we can go purely Bayesian if we’re careful enough… but what Carrier says in the sections of the book where he purports to “prove” his philosophical claims don’t actually seem to prove this.

This post is already quite long, and I haven’t even touched the chapter on “The Hard Stuff,” the most philosophically dense chapter of the book. So let me end this post here, and I’ll come back to that chapter in part 2.

  • Randall

    Chris, do you have any background in probability theory?

    • Chris Hallquist

      Limited. A stats course, and what I’ve picked up studying science and philosophy. Improving my knowledge of this area is one of the reasons for trying to blog about it.

  • Jacob Henderson

    William Lane Craig’s lottery analogy is a really poor example. If there are a million lottery numbers (one of them being the winning number), and if 500,000 people buy lottery tickets at all once, then there’s an instant 50% chance of someone winning the lottery. And if you figure in the buying of tickets over a period of time, then fewer players could actually lead to chances of winning greater than 50%.

    In other words, using the lottery as an example of a “believable one-in-a-million shot” is silly and probably shouldn’t have been mentioned.

    • Andrew G.

      To a Bayesian, the lottery paradox isn’t an issue, because your prior probability for the lottery being rigged has to be divided between all the participants, since you have no advance basis to pick out just one of them.

      Then, when you plug in the numbers, the posterior probability for the lottery having been rigged in favour of the eventual winner is virtually unchanged from the prior probability.

      So if there are a million tickets sold of which exactly one wins (which happens to belong to Fred Bloggs), and you initially believe that there’s one chance in a thousand that someone could rig the result, then your prior odds for “the lottery was rigged for Fred Bloggs to win” are not 1:999 but 1:999,999,999. Having observed that Fred won (P(E|H) =1, P(E|~H) = 0.000001), your posterior odds are back to 1:999 (actually 1:999.999999 but that’s a negligible difference), so you have no more (actually very slightly less) reason to believe it was rigged.

  • Patrick

    If there’s a one in a million chance that someone can perform real magic, then there ought to be three hundred wizards in this country right now.

    • MNb

      That’s a typical non-sequitur coming from people who don’t understand statistics. Statistics doesn’t say what’s ought to be; statistics is about probability.
      If there’s a one in a million chance that someone can perform real magic, than it’s still possible that nobody in the USA is a wizard – or all 300 million inhabitants are. These outcomes are just unlikely, ie have a low probability.

      • Steven Carr

        And the probability of zero wizards would be e^(-300). That is such a small number that it won’t even register on your calculator.

        ‘It’s wildly inadequate to merely say they’re “low,” because as William Lane Craig points out, we have no trouble believing reports of lottery numbers being picked when a given number only had a one in a million chance of being picked. ‘

        But the probability of the correct lottery numbers being reported in newspapers is so close to 1 that you would be dismissed as a lunatic if you tried to challenge the reported result in the courts.

        So William Lane Craig’s analogy is correct, but misleading. (I never thought I would ever type the words that Craig’s analogies were misleading, but there you go)

        • MNb

          Unless you are going to pull off the creationist’s version of Borel’s Law it’s still not zero, like Patrick seemed to imply. Like I wrote: low probability.

          • Steven Carr

            Are you claiming e^(-300) is not essentially different from zero? If there is a 1 in a million chance of being a magician, and there are 300 million Americans, we would expect 300 magicians, and be surprised if that number varied by more than 36 either way, and very surprised if it varied by more than 54.

            As for a 17 Sigma deviation down to zero magicians, even CERN scientists regard 6-Sigma deviations as being pretty certain….

      • Andrew G.

        Saying something has a “low probability” without quantifying how low is often very silly.

        To put some orders of magnitude on it (these are all very rough):

        “1 in a million” is your chance of being killed in a half-hour car drive, or being dealt a royal flush in a 5-card poker deal, or winning the UK national lottery jackpot by buying a dozen or so tickets. Your chance of spontaneously recovering from cancer diagnosed as terminal, without treatment, is usually somewhat better than this.

        “1 in a million million” is the chance of rolling snake-eyes on fair dice 8 times in a row, or being dealt a one-suit bridge hand from a properly shuffled deck.

        “1 in a million million million” (i.e. 10^-18) is about the chance of a specific Florida resident being killed in the next hour by a cocaine bale falling out of an aircraft. (see what-if.xkcd.com)

        So all of these are “low” probabilities, but you can see that we’ve covered a range from things which we know happen (people do die in car accidents, get dealt royal flushes, or recover from cancer) through to things which are frankly preposterous; but all of these probabilities are huge compared to e^(-300), which is about 10^(-130).

  • Patrick

    My apologies for not including the word “around.” I am aware of probability distributions.

  • http://theotherweirdo.wordpress.com The Other Weirdo

    I don’t understand the comparison between miracles and winning lottery numbers reports. We can see on TV the people who choose the winning lottery numbers. It’s a well-known and understood process. Winning lottery numbers don’t miraculously appear in the paper the next morning for no sufficiently explainable reason. There is nothing mysterious about them and deities are rarely invoked, except when one loses.

    Miracles, on the other hand, require complete suspension of natural law as we understand it, and a deity must perform them outside of any testable environment or evidence-gathering capabilities. With miracles, things do happen miraculously, as if by magic.

    What am I missing? How can the tiny chance of winning the well-understood process of lottery be compared against completely not understood creation by magical fiat?

    • Andrew G.

      Depends on who is doing the comparison.

      Probability theory in general and Bayesian updating in particular can be used honestly to evaluate miracle claims. One example I’ve actually worked through in the past is in the area of miraculous healing claims: if an apologist presents, say, a list of people who had all been diagnosed with terminal cancer but had “miraculously” gone into remission, you can fairly easily estimate the probability that this evidence would exist even if there were no miracles, and at least some sort of range of probability for the existence of the evidence assuming that miracles were possible, and get an odds ratio with which to update the strength of your belief in the existence or nonexistence of miracles.

      Used dishonestly, it’s just another tool in the apologists toolbox of lies.

      • AndrewR

        There’s a great bit of dialogue from “The Exorcist” that I always think of when miracle claims are brought up:

        The bureau drawer flies open on it’s own.
        KARRAS: Did you do that?
        REGAN/DEMON: Uh Huh.
        Karras pushes the drawer back in.
        KARRAS: Do it again.
        REGAN/DEMON: In time.
        KARRAS: No now.

        Working out the probabilities of miraculous faith healing is valuable, but we should also be like Fr Karras, “Do it again”.


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