Probability Is Not Frequentist, Nor Math, Nor Real, Nor Subjective

I once had a guy walk out on me at a scientific conference, full of my peers. He and I were disagreeing about the nature of probability.

The exact words have escaped my memory, but the sting that set him flying was, “Nah, you’re wrong.”

After my enemy fled, an older, stouter in mind, and braver, gentleman also disagreed with me. He insisted probability only applied to mathematical entities of what are known as the frequentist kind. I asked What about counterfactuals? What of one-offs? He said these are not probability. What are they then, since they are uncertain? Answer came there none.

David Siegel points us to that rationalist fellow Scott Alexander and his article “In Continued Defense Of Non-Frequentist Probabilities“. Judging by the enormous number of comments to the post, there is still great interest in the topic.

(Incidentally, does anybody else have trouble on the browser when on his site? It always bogs my whole system down.)

I have, of course, written a book on the subject: Uncertainty. I have strong opinions.

Easy ones, too. Because probability is easy. There is almost nothing to it. It’s so simple I can teach you all of probability in just three lines. Ready?

1. Pick a thing you are uncertain about; call this proposition Y;

2. Collect evidence probative of Y; call this evidence X;

3. Probability is how certain Y is given this X.

That’s it. There is nothing more to it. Not as far as the fundamentals go. Naturally, there is much that can be deduced from these three lines. Lots is a better word than much. Indeed, so much can be deduced that we haven’t come close to deducing all we can.

Let’s walk through some simple deductions, leaving any proofs to links or to the book.

A: Probability can be represented by numbers. The proof of this is not so hard, so do try your hand at it. Yet can be does not mean is, not always. Plenty of probabilities, and most of those we use in real life, are never quantified. Some examples below.

Incidentally, once we realize that probability can be represented by numbers, then all of mathematics opens to us. Still, probability is not math. It is what was said above.

B: Probability is not relative frequency. Let’s do an example drawn (a good joke) from real life. You have 10 murderers in a paddy wagon, seven black, two Hispanic and one white (the Asian slipped out to bring back takeout). You reach in and grab a murderer. Which is the probability the murderer is white?

The answer is, as you guessed, 1 out of 10, 10%, or 0.1; whichever way you want to write it. And this is not because upon “repeated” draws the ratio of times you pulled out a white to the total, a total which must approach Infinity itself, converges to some fixed number. The theory insists on this, which everybody forgets, even right after you remind them. This is the definition of relative frequency. The reason the probability is 1 in 10 is not because of any “frequency”. It because there is one white and nine non-white murderers.

But you will have noticed, I hope, that I cheated; or, rather, took a short cut. If you haven’t yet seen, then maybe you will in our next example.

C: Probability is not relative frequency. Yes, we’re doing this one twice. Rational Alexander gave this example: “What is the probability that Joe Biden will win the 2024 election?” This is equivalent, in essence, to this uncertain proposition: “Vladimir Putin would not have invaded Ukraine were it not for the US-sponsored color revolution.”

It’s obvious that both of these questions have propositions which are not certain. And that it makes sense to think of that uncertainty in terms of how true the propositions are with respect to the evidence you assume. The air is filled with talk about probative evidence about the election, for instance.

Even if you ignore the Infinity requirements of relative frequency, the reason we cannot use relative frequency is there is no fixed information or set that can be used to make a ratio for either proposition. Of what divided by what should the probabilities be calculated? For the election, all elections of Democrats versus only Republicans? Only recent ones? All elections everywhere? Just the USA? Of all men over seventy?

If you could pick, and justify, which set to use, it’s clear that this would be a limited set, and that you’d neglect a massive amount of evidence besides “frequencies” of past elections. Like Biden’s decrepitude and ineptitude, Trump’s bizarreries and his seething lunatic enemies, Regime vote fortification, and on and on.

It is because this set doe not exist that some say probability cannot be for “one offs” like our questions. Probability, to them, is defined circularly: probability is relative frequency because it is relative frequency. If the problem cannot be set up as relative frequency, it is not probability.

But for us, uncertainty is probability.

Now we can see how I cheated in B. The answer to the paddy wagon question was not 1 in 10; not unless the assumed evidence was the only evidence we decided to use, eschewing all else. That evidence had nothing to say about how the reaching in would be done, or which race would allow themselves to be grabbed, whether there was a window, and on and on. Which leads us to the next item.

D: Probability is objective. Once the “Y” and “X” are fixed, once the uncertain proposition and its probative information, that is, are set, probability is rigorously objective. It becomes a strict matter of logic, with just as much freedom in deriving the answer as a (pre-woke) middle school algebra problem. Which is to say, no freedom at all.

E: Probability is subjective. There is no guidance that says “Pick this Y” and “With this Y, we must pick this X”. When you have an uncertain proposition, the choice of which is subjective, the evidence you use with respect to it is also subjective. The coin-flipping machine is proof of this (blog, Substack). The probability of a head is either 1/2 or 1, depending on which evidence you pick. Naturally, custom and common sense will call some X wise or absurd—or inconsistent with other knowledge.

F: Probability is logic. From D and E we deduce probability is logic. In logic, once the premises are set, the conclusion follows. Logic is purely about the connection between propositions. Logic does not say where premises come from; they are subjective. Probability is exactly the same: it only gives the relation between propositions; it says nothing about where they arise.

G: Probability is not strength of evidence. Suppose, going back to the election question, you have investigated the matter thoroughly, looking, say, at county-level polling data, have incorporated voting and demographic trends, calculated the effects of “migrants”, that of local economies, and have even used AI (wow!), all evidence which gives you the probability of 1/2 that Biden wins.

In contrast, I use the evidence, “There are two men in the race, one of which must win”. Using that evidence, the probability I calculate is also 1/2.

Clearly, our evidence is not the same, though our probabilities are. Your evidence is strong, mine is pathetic. Probability doesn’t distinguish between them.

H: Probability is not explanation. The last example is proof enough that probability doesn’t explain the uncertainty, except in a weak sense. Evidence can explain. And when the evidence is fully explanatory, the probability is 1 (or 0). Because with a full explanation we have given the cause of Y, and there is no knowledge more complete than that.

H: Probability is not real. It follows that probability is not ontic. It doesn’t exist separate from the mind that entertains it. It is not a physical thing, like say electric charge is. It is purely a measure of information.

I: Nothing has a probability. It follows that you do not have a probability of being struck by lightning, or of having cancer, or of anything. Because it all depends on which evidence you use.

More! There is, as I said, lots more to be said of probability. Lots and lots more. But that’s enough for today.

If there is interest, and I can figure out how, one of these days I’ll do the whole class online.

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