In what follows, I’ll consider Carrier’s claims about the mathematical foundations of probability theory. What Carrier says about probability is at odds with every probability textbook (or lecture notes) I can find. He rejects the foundations of probability laid by frequentists (e.g. Kolmogorov’s axioms) and Bayesians (e.g. Cox’s theorem). He is neither, because we’re all wrong – only Carrier knows how to do probability correctly. That’s why he has consistently refused my repeated requests to provide scholarly references – they do not exist. As such, Carrier cannot borrow the results and standing of modern probability theory. Until he has completed his revolution and published a rigorous mathematical account of Carrierian probability theory, all of his claims about probability are meaningless.
Carrier’s version of Probability Theory
I intend to demonstrate these claims, so we’ll start by quoting Carrier at length. I won’t be relying on previous posts. In TEC, Carrier says:
Bayes’ theorem is an argument in formal logic that derives the probability that a claim is true from certain other probabilities about that theory and the evidence. It’s been formally proven, so no one who accepts its premises can rationally deny its conclusion. It has four premises … [namely P(h|b), P(~h|b), P(e|h.b), P(e|~h.b)]. … Once we have [those], the conclusion necessarily follows according to a fixed formula. That conclusion is then by definition the probability that our claim h is true given all our evidence e and our background knowledge b.
In OBR, he says:
[E]ver since the Principia Mathematica it has been an established fact that nearly all mathematics reduces to formal logic … The relevant probability theory can be deduced from Willard Arithmetic … anyone familiar with both Bayes’ Theorem (hereafter BT) and conditional logic (i.e. syllogisms constructed of if/then propositions) can see from what I show there [in Proving History] that BT indeed is reducible to a syllogism in conditional logic, where the statements of each probability-variable within the formula is a premise in formal logic, and the conclusion of the equation becomes the conclusion of the syllogism. In the simplest terms, “if P(h|b) is w and P(e|h.b) is x and P(e|~h.b) is y, then P(h|e.b) is z,” which is a logically necessary truth, becomes the concluding major premise, and “P(h|b) is w and P(e|h.b) is x and P(e|~h.b) is y” are the minor premises. And one can prove the major premise true by building syllogisms all the way down to the formal proof of BT, again by symbolic logic (which one can again replace with old-fashioned propositional logic if one were so inclined).
More specifically it is a form of argument, that is, a logical formula that describes a particular kind of argument. The form of this argument is logically valid. That is, its conclusion is necessarily true when its premises are true. Which means, if the three variables in BT are true (each representing a proposition about a probability, hence a premise in an argument), the epistemic probability that results is then a logically necessary truth. So, yes, Bayes’ Theorem is an argument.
He links to, and later shows, the following “Proof of Bayes Theorem … by symbolic logic”, saying that “the derivation of the theorem is this.”
For future reference, we’ll call this “The Proof”. Of his mathematical notation, Carrier says:
P(h|b) is symbolic notation for the proposition “the probability that a designated hypothesis is true given all available background knowledge but not the evidence to be examined is x,” where x is an assigned probability in the argument.
Like nothing we’ve ever seen
I have 13 probability textbooks/lecture notes open in front of me: Bain and Engelhardt; Jaynes (PDF); Wall and Jenkins; MacKay (PDF); Grinstead and Snell; Ash; Bertsekas and Tsitsiklis; Rosenthal; Bayer; Dembo; Sokol and Rønn-Nielsen; Venkatesh; Durrett; Tao. I recently stopped by Sydney University’s Library to pick up a book on nuclear reactions, and took the time to open another 15 textbooks. I’ve even checked some of the philosophy of probability literature, such as Antony Eagle’s collection of readings (highly recommended), Arnborg and Sjodin, Caticha, Colyvan, Hajek (who has a number of great papers on probability), and Maudlin.
When presenting the foundations of probability theory, these textbooks and articles roughly divide along Bayesian vs frequentist lines. The purely mathematical approach, typical of frequentist textbooks, begins by thinking about relative frequencies before introducing measure theory, explaining Kolmogorov’s axioms, motivating the definition of conditional probability, and then – in one line of algebra – giving “The Proof” of Bayes theorem. Says Mosteller, Rourke and Thomas: “At the mathematical level, there is hardly any disagreement about the foundations of probability … The foundation in set theory was laid in 1933 by the great Russian probabilitist, A. Kolmogorov.” With this mathematical apparatus in hand, we use it to analyse relative frequencies of data.
Bayesians take a different approach (e.g. Probability Theory by Ed Jaynes). We start by thinking about modelling degrees of plausibility. The frequentist, quite rightly, asks what the foundations of this approach are. In particular, why think that degrees of plausibility should be modelled by probabilities? Why think that “plausibilities” can be mathematised at all, and why use Kolmogorov’s particular mathematical apparatus? Bayesians respond by motivating certain “desiderata of rationality”, and use these to prove via Cox’s theorem (or perhaps via de Finetti’s “Dutch Book” arguments) that degrees of plausibility obey the usual rules of probability. In particular, the product rule is proven, p(A and B | C) = p(A|B and C) p(B|C), from which Bayes theorem follows via “The Proof”.
In precisely none of these textbooks and articles will you find anything like Carrier’s account. When presenting the foundations of probability theory in general and Bayes Theorem in particular, no one presents anything like Carrier’s version of probability theory. Do it yourself, if you have the time and resources. Get a textbook (some of the links above are to online PDFs), find the sections on the foundations of probability and Bayes Theorem, and compare to the quotes from Carrier above. In this company, Carrier’s version of probability theory is a total loner. We’ll see why. Continue Reading »