Subjective Probability Is Objective

Because certain forms of Bayesian probability, particularly so-called subjective probability, are being taken up in quantum mechanics, it’s well to understand just what subjective probability is, if it’s anything, and how belief is different from probability.

There is also the use of forms of quantum probability calculus used increasingly in economics, which is a lot about making decisions. We’ll hold on that for now and think more about QM itself.

The paper I have in mind is Christopher Fuch’s “QBism, the Perimeter of Quantum Bayesianism“. Now Fuchs is like our Moldbug: he never writes one word when he can squeeze in three, so this paper takes some time to churn through.

I don’t want to borrow from the physics notation, because while it has great facility in doing QM math, it can be downright confusing. This short article is only a precis, something to refer back to when we get to the good stuff.

Enough preamble! Let’s get to it.

Now all probabilities are properly written like this schema: Pr(Y|X). X is always there, even when it isn’t written. In X is everything considered about Y, including, and I cannot emphasize this strongly enough, the words and grammar and notions of definition of what Y is, and also about what the stuff in X is.

Some write Pr(Y), but this is always strictly a mistake, no matter how useful it may be when doodling equations. To consider any proposition Y, whether it be mathematical, physical, or fanciful, means knowing what the Y is. That goes in X. This is why it is impossible—not just unlikely: impossible—for their to be an unconditional probability.

There is no such thing as subjective probability, but the elements of probability itself can be subjective. Confusing?

It’s easiest, though not itself without potential flaws, to be rigorous in math. If we had a Y and X which are very tightly defined, then it should be clear there is no subjectively whatsoever to Pr(Y|X).

Example: Y = 2, and X = “Y = 1 + 1”; then Pr(Y = 2 | Y = 1 + 1) = 1.

Of course, the meaning of every symbol like “+” and “=”, and even of “1” and “2” are implicit on the right hand side. Once those meanings are fixed and agreed to, then probability is rigorously objective. You have no choice but to agree with the probability if you accept the equation as written, and agree with the implicit premises (or evidence), then you must, to remain rational and coherent, agree to the probability.

Thus probability is totally objective.

Probability only seems subjective because we have free choice of (1) the Y, and (2) the X. Those are indeed subjectively chosen. For instance, you could in a fit of eccentricity insist that “+” means “subtract” (where by “subtract” you mean the usual mathematical operation). Then the probability in the example is 0. To you. But that’s only because your X is not the X of anybody else. We used to say, back when it was true, “Hey, it’s a free country”.

It becomes complicated when you’re allowed to pick the Y and X freely, which is to say, subjectively. But that’s not a bad thing, either.

I’ve used the example a million times, but let Y = “My team wins the game”, then X is a host of vague, ill-defined, half- or wholly unarticulated but felt and shifting evidence. You couldn’t write down all the X you’re using, nor hope to explain them in detail, to anybody, not even yourself. X is still there, but it’s a tangled ball of premises.

Still, you can form a Pr(Y|X), or say you have. By which I mean, you can spout a number, but even you might fool yourself into thinking it’s directly related to a set of fixed X, but which instead something else.

That’s very confusing, but the idea is simple enough. If you care about the Y, it’s very difficult to keep out of X the consequences of Y and the decisions you might make about the Y. You can say, or claim, you’ve kept these matters out, and are only judging the probability based on some X that is free of these considerations, but saying and doing are different things.

This is proved by a well known example called the (so-called the) conjunction fallacy:

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more probable?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

People who give this puzzler say X is the paragraph of information (premises), and only that paragraph, to consider in Pr(Y_i|X). The Y_i is a choice between 1 and 2. Participants are supposed to say 1 is more probable than 2, because because it’s more probable to have only one state (teller) than two (teller and purple hair).

But when ordinary people read this example, they invent for themselves different X, even while acknowledging or thinking to themselves “I only used the given X”.

For instance, many will read this is X = “Linda is a bank teller…” and tacitly rename the propositions of interest Y_1 = “Linda’s not a feminist”, and Y_2 = “Linda’s a feminist”. They’re interested in guessing whether this Linda is into fat positivity, not whether she’s some boring bank teller.

Or people will equate X with “Linda is a feminist”; after all, to many, X contains the definition of what a feminist is (“social justice”). That makes the probability of Y_2 given this X-definition 1.

But it doesn’t quite make the probability of Y_1 0. Because in 2 we know feminist Linda is a bank teller, and 1 says bank teller, too. It’s also the “right” answer, in a sort of way. And people naturally want to give credence or weight to any right answer, even though 2 is the better answer.

So, when pressed, some people might give some kind of number of the probability of Y_1. Too, some people will be on the presence of the Conjunction Fallacy, particularly if this scenario is given in a test.

There are a couple of points. The first is that unless people are forewarned to use only the explicit X given, variation arises. And even with explicit X, there are always implicit parts to that X, which can too easily vary from person to person. The further one moves from rigor to the colloquial, the larger the variation.

Belief is not probability. People may be asked to give a number for Pr(Y|X), even when one cannot be explicitly deduced. This forces a person to invent implicit X, which cannot be tracked.

Example? I said above I used an example “a million times”, a figure of speech. What is the Pr(He used the example a million times| X)? Well, what’s X? My claim for one; that Y is a figure of speech is another, and so on. But if insist you give me a number, you must invent X for yourself to produce this, even if this X is just “This is BS; I’ll just say 0.001”.

Now some have invented a sort of calculus that makes you make little bets with yourself to arrive at this number. This fails. But to show why, I’ll save for another day.

Finally, as our last point and end of the introduction, suppose Y = “We will measure the particle spin down”. X will be such things as details of the measurement apparatus and “experimental” (it doesn’t have to be an experiment) design, facts about math and assumed theories of physics, and the like.

But the measurement itself interacts with the Y. It interacts in exactly the same way as asking you to give an answer to Pr(million times|X). The rest of that we’ll also leave for another time, when we look deeper in Bayesian QM.

You call this interference subjective probability if you like, because it’s you deciding subjectively to do the measurement, though the objectively of probability itself remains. Remains, that is, once the Y and X are fixed.

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