Alan Hajek's Destruction of Finite Frequentism

A common line you hear from the hoi polloi on trains, buses, and street corners: “probability is just about frequencies of recurrent events.” Often this line is uttered as if it’s a tautology. As if anyone in the know would know it’s so and abandon the silly Bayesian practice of assigning credences to non-repeatable events.

Now, even if you think this is the right way to think about probability in some class of repeatable events, it’s not the right way to think about credences—degrees of belief. Continental drift is a one-off non-repeatable event. Nonetheless, our credence in it should be very high.

But Alan Hajek—one of the leading philosophers of probability in the world—thinks that it’s not even the right way to think about probability. He has two papers that each present fifteen arguments against one of the two standard frequentist theories. Hajek writes:

To philosophers or philosophically inclined scientists, the demise of frequentism is familiar, I admit, even though it hasn't quite been universally accepted. Familiar too are many of the arguments that I will present here — indeed, some of them were inspired by Richard Jeffrey's "Mises Redux" (1977) — though I hope it will be useful to have them gathered in one place. Other arguments in this paper are new, as far as I am aware. So even if the fact that there is bad news for frequentism is old news, I hope it is newsworthy just how much bad news there really is.

There are two kinds of frequentism: finite and hypothetical. Finite frequentism says that the probability of some event A in the reference class B is the relative frequency of actual occurrences of A in B. For example, the probability of a coin coming up heads is the proportion of coins that come up heads out of all those flipped.

Why doesn’t Hajek like finite frequentism?

First, frequentism depends on a reference class. When ascertaining the probability of some event, the frequentist looks at how often events of some type turn out that way. When analyzing a particular coinflip, they ask: what portion of overall coins come up heads? But what determines the reference class? Why coins in general, rather than coins flipped in the present month or on the present day? These could give very different answers. Any choice of reference class will seem horribly arbitrary.

Second, frequentism depends on grouping together unique events. In order to group events together, their differences must not make any difference to their probability. You cannot group tosses of weighted coins with unweighted ones. But insofar as the frequentist excludes all other probabilistically-relevant factors, the reference class will end up very small, so that probabilities may turn out strange. If there are only a few dozen coins in the reference class, it’s likely to be small enough to be distorted by randomness.

Third, frequentism leaves probability bizarrely dependent on how other distant objects happen to behave. Suppose I hold a coin in my hand and flip it. The probability of it coming up heads seems to be 1/2, and this seems independent of coins being flipped in other galaxies. However, for the finite frequentist, frequency is just about statistical tendencies: the coin only has a 1/2 chance of coming up heads if coins come up heads half the time. Thus, if, by chance, most distant coins happened to come up heads, then the odds of this fair coin coming up heads would be other than 1/2. This seems wrong.

In fact, for the finite frequentist, each time one tosses a coin, it has some small impact on the probability of other coins turning out other ways. By affecting the frequency, it affects the probability of them turning out some way. But surely some alien flipping a coin a trillion light-years away has no effect on the probability of my coin coming up heads.

Fourth, finite frequentism seems to imply the vacuousness of all sorts of ordinary probability judgments. Suppose I wish to know the probability that I will die next year. The finite frequentist answers: there is no such probability, because there is no class of sufficiently similar people to me. But surely it is a defect of a theory if it holds that there is no fact—not even an imprecise one—about my probability of dying. Surely statements like “the probability I will die next year will increase if I’m diagnosed with cancer” are not meaningless.

Fifth, frequentism implies that statements about probability allow you to infer things about other minds. If you know that the probability of a coin coming up heads is .5, then you can know that there are many coin flips. But this seems wrong. Knowing how likely a coin is to come up heads doesn’t seem to tell you anything deep about the external world. As Hajek puts it:

What's troubling, though, is that statements of probability about a mind, an object, or an event, seem to be simply irrelevant to the existence of other minds, other objects, other events of the same sort, right here in the actual world.

Sixth, finite frequentism must deny all sorts of ordinary sentences like “the probability of the coin landing heads really was 1/2, but there was an unusually high portion of heads in the tosses.” To the finite frequentist, probability just is the portion of heads in the tosses.

Seventh, finite frequentism holds that chances reduce to frequencies. But then frequencies cannot explain chances. Thus, one can no longer say “half the coins came up heads because the probability of a fair coin coming up heads is 1/2.”

Eighth, the view implies that if an event never occurs, it has no probability. But this is false. Suppose various dice were manufactured but none had ever been tossed. It would still be perfectly correct to say, “the probability of a die coming up five is 1/6.”

Ninth, the view implies that if an event only occurs once, its probability is either 0 or 1. Thus, if only one coin was tossed, the probability of coins coming up heads would be 1 if it came up heads and 0 if it didn’t. Problem: this is false.

Ten, the view cannot assign probabilities to universal generalizations like “no actual president will be named Gregory.” Either this is always true or always false. But surely the right probability to assign to this given our evidence is some amount other than 0 or 1.

Eleven, frequentism assigns intermediate probabilities even in cases that are deterministic. E.g. even if it’s deterministic that a coin comes up heads, so long as half of coins come up tails, the probability of a coin coming up tails will be .5 rather than 0 or 1. I don’t really buy this argument. Hajek finds this objectionable because it assigns non-extreme probabilities to deterministic events. But this just seems right to me. Even in a deterministic world, the odds of a coin coming up heads are .5.

Twelve, finite frequentism implies that if coins are flipped an odd number of times, then they cannot be fair, for they cannot have come up heads precisely half the time. In fact, to the frequentist, the notion of a fair coin tossed an odd number of times is rather like a square circle—an in-principle impossibility.

Thirteen, finite frequentism generates myriad spurious correlations. A is spuriously correlated with B if the likelihood of A given B is higher than the general likelihood of A, but B doesn’t causally affect A. The portion of people with green shirts who live to sixty is no doubt different from the portion of the population in general who live to sixty. Thus, finite frequentism implies that wearing a green shirt affects the odds of living to sixty.

Fourteen, finite frequentism implies that probabilities cannot be irrational, because irrational numbers cannot describe frequencies.

Thus, any theory which gives such values to probabilities is necessarily false, according to finite frequentism, irrespective of its subject matter. That's certainly a quick refutation of quantum mechanics! For example, according to finite frequentism, the radioactive law for radium is false for all time periods that have irrational probabilities for decay - which is to say that it is false almost everywhere.

Fifteen, finite frequentists ought to hold that the only core scientific virtue is consistency with frequency judgments. But this doesn’t describe how actual scientific practice works, and it doesn’t describe how it should work.

Hajek calls finite frequentism “about as close to being refuted as a serious philosophical position ever gets.” I’m inclined to agree.