The simple answer to today’s titular question is no, Bayesianism cannot quantify true belief.
This answer will disappoint the Effective Altruists, Rationalists, many professional statisticians, a score of others, and the gentleman who wrote the article of the same name in New Oxford Review. The gentleman is Daniel Sadasivan, a professor of physics at Ave Marie University in Florida.
Bayes is only a tool in probability, and a nifty one. It aids many calculations, which would be much more complicated without it, but it is no more important than, say, the Rule of Total Probability. But Total Probabilityism doesn’t have the same ring to it, does it, so it’s not likely to catch on.
What is of base or fundamental importance is not Bayes, but simply probability itself.
And this is probability itself: all must bow to the Universal Dominion of the King of All Formulae:
Pr(What I Want To Know | All The Evidence I Assume).
That reads “The probability of what I want to know given all the evidence I am assuming, including all tacit and implicit evidence.” That is it. That is all probability. This is The King.
I think it’s the simplicity of The King that throws people off. Probability must be complicated to be useful! But, no. If statistics were to jettison its old methods and embrace this unassuming powerful complete King of All Formula, all would be well. Alas, it is too easy, and things that are thought to be too easy are looked on with deep suspicion in academic science and philosophy.
I have told us many times that probability, the completion of logic, is nothing but the relation between propositions. That is all logic is, and since probability is logic, that is all probability is, too. The King is what this idea looks like in formula form. The vertical bar is the dividing line between the propositions. (In logic, because of old custom, we usually write the dividing line horizontally. See the Video.)
If what you want to know is, say, “The existence of God”, then you can use probability, but not until you have first amassed the evidence you assume is probative of the proposition of interest (the stuff to the right of the vertical bar, or on top of the horizontal one). Only then we can we discover if we can make numbers out of the relation. If we can, then Bayes may or may not be useful in that quantification. But it is the evidence that is of paramount importance. The formula is King: the evidence is Queen. The quantification itself, if possible, however complicated mathematically it turns out to be, is of secondary interest.
Bayesians, as they call themselves, have a different way of looking at it, and a bad habit. They say “Why don’t you announce, in numerical terms, what your prior probability of the proposition is, and then I’ll show you how to update that prior when new evidence arrives. Maybe think of this number as a bet, or how it makes you feel.” Suddenly the Bayes formula looms, and math happens. Which gives a satisfying scientific patina to the formula.
Yet if it were that easy, if all we had to do is assign priors to propositions, then update them with evidence, we’d be done with all the Big Questions by now, having had computer models (AI, if you like), solve the math for us.
You will have noticed I repeatedly emphasized if. Not all probability relations can be quantified. Not everything can be turned into a number. (We did this in Class recently.) Which is why Bayes is only of limited help. But even supposing every relation could be put to a number, would you be satisfied on hearing from some Expert. “The probability of God’s existence has been scientifically determined to be 93.8%”?
Put whatever number you like in there, it doesn’t matter (note to my comedic readers: between 0 and 100% inclusive). You will very quickly ask, if you care about the proposition, “What is the evidence?” Indeed, that is always the Queen of All Questions.
Suppose you like the evidence, which includes those premises which lead you to suppose the relation between the evidence and proposition can be quantified—you can even use Bayes’s formula!—then what? Let our Expert be correct in his assessment of 93.8%. What follows?
Do you believe? Do you disbelieve? What if the number is 100%? Or 0%?
Belief is not probability! Belief is an act, a decision. Probability is only a matter of the mind, an expression (stop me if you’ve heard this before) about the relation between propositions. What you do, what actions you take, because of those propositions and that relation is entirely something else. If you choose to believe that, say, “God exists”, that is an action that cannot be quantified (though as we saw in the Class, many do put numbers in attempts to quantify the unquantifiable).
Sometimes numbers can be put to actions. When beliefs have to do with money, quantification can be had. And are had, such as in bets. When beliefs are questions like “God exists”, they cannot. There is no putting a number on your belief. That is to say, many do put numbers on beliefs, but that is because we assume that if we want to be scientific, quantification is a must. Yet any numbers put to this propositions will be like those we covered in the class, crude, misleading, and incomplete.
Bayesianism thus makes the same genre of error P-values do: it mixes up decision with probability; it fails to separate the two acts of the mind. Bayes asks about your initial (“prior”) feelings. Logic (probability) asks you about evidence. When you give your “prior” in terms of a bet, you are bringing in an action and its consequences and conflating it with the relation between proposition of interest and evidence. There is then no separating the two after Bayes rears its head. See the Video for clarification.
Acts depend on consequences, and again sometimes these can be quantified, but often they cannot, even when the probability can. Simple gambles are an excellent example of that. You can quantify the probability of a roulette wheel coming up black, but what you do about a spin is in a different class from that probability.
From Sadasivan:
[F]ormerly Protestant YouTuber Cameron Bertuzzi applied a Bayesian analysis in his decision to convert to the Catholic faith. Using the described method [Bayes formula starting with a “prior”], he determined that he had a 93.8 percent certainty of the truth of the papacy. Due to this calculation, as well as other motivations, he accepted the truth of the papacy on faith.
If I told you that the gun you have pointed at your most hated enemy had a 93.8 percent certainty of the truth of its hitting its target, as determined by scrupulous attention to Bayes’s formula, would you open fire? Did Bertuzzi prove the “truth of the papacy”? He proved it to only have a 93.8 percent chance of truth. Where did his “faith” arise? By supplying the extra 6.2%?
Anyway, he decided to believe. Is that the right decision? For him it was. He was operating on acts and consequences which are closed to us, but meaningful to him.
And so it is, or so it ought to be, in all probability and statistics (which includes as a subset all AI).
Video
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