Alan Hájek delivers a devastating blow against frequentism and other theories of probability in his influential 2007 paper “The reference class problem is your problem too.” In brief, when a hypothesis H or proposition P can be classified in various ways, and when the probability of H or P varies depending on how it is classified, the reference class problem has reared its ugly head. For his part, Hájek defines the reference class problem formally as follows:
“[A hypothesis H or proposition P] may be classified as belonging to set S1, or to set S2, and so on. Qua member of S1, its probability is p1; qua member of S2, its probability is p2, where p1 ≠ p2; and so on. And perhaps qua member of some other set, its probability does not exist at all” (Hájek, 2007, p. 565).
In his reference-class paper, Hájek carefully and lucidly explains why frequentism and other leading interpretations of probability — all theories of probability, that is, except for “radical subjectivism” or pure Bayesian probability — are contaminated by the reference class problem (ibid., pp. 566-580). Nevertheless, although subjectivism is immune from the reference class problem, Hájek attempts to discredit Bayesian methods. His critique of subjectivism, in summary, is that it is atheoretical — it is a “no-theory theory of probability” (ibid., p. 577). According to Hájek, “no-theory theories of probability” are apparently bad because they “leave quite obscure why probability should function as a guide to life, a suitable basis for rational inference and action” (ibid., p. 564). But why is radical subjectivism a “no-theory theory of probability” in Hájek view? Because, according to Hájek, radical subjectivists are, well, too subjective!
Hájek correctly describes the subjectivist or Bayesian view of probability in terms of “degrees of belief.” The problem, according to Hájek, is that a person’s degree of belief in hypothesis H or proposition P is always a subjective matter. Degrees of belief are purely personal to the person holding such belief. In Hájek’s words:
“Subjectivists regard probabilities as degrees of belief, and see [Kolmogorov’s axioms] of probability as rationality constraints on degrees of belief. Your degrees of belief can be whatever you like, as long as they remain probabilistically coherent [e.g. sum to 1]” (ibid., 576).
Since radical subjectivists regard Andrey Kolmogorov’s (1956) classic axioms of probability as the only constraints on one’s degrees of belief, Hájek concludes that such subjectivism is so “spectacularly permissive” that one is free to assign an irrational or arbitrary probability value to any possible event (ibid., emphasis in original). To illustrate his argument, Hájek proffers the following fanciful example (see also the image below for an illustration of Hájek’s point): “For example, you may with no insult to [subjectivist] rationality assign probability 0.999 to George Bush turning into a prairie dog, provided that you assign 0.001 to this not being the case (and that your other assignments also obey the probability calculus)” (ibid.).
Our friend Hájek further states: “And you are no more or less worthy of praise if you assign it 0, or 1/e or whatever you like. Your probability assignments can be completely at odds with the way the world is, and thus are ‘guides’ in name only” (ibid., footnote omitted). He thus concludes: “[Radical subjectivism] becomes autobiography rather than epistemology” (ibid.). In other words, radical subjectivists offer up a “no-theory theory of probability” because subjectivism does not impose any demanding criteria for choosing one’s priors.
We now offer the following critique of Hájek’s critique of radical subjectivism. Aside from arguing that a “no-theory theory” is still, technically speaking, a theory, we would respond to Hajek’s powerful critique in two ways.
- So what? Who cares?
Our initial response to Hájek’s critique of subjectivism is to shrug our shoulders and say “so what?” or “who cares?” To some extent, theory — or the activity of armchair theorizing — is overrated. Just as you don’t need a “theory of baseball” or a “theory of opera” in order to play baseball or sing opera, so too you don’t really need a theory of probability in order to “do probability.” Perhaps a theory of baseball or opera would be useful and illuminating for any number of reasons, but theorizing about baseball or opera is not the same as playing baseball or singing opera. Similarly, radical subjectivism is a method for doing probability, not merely an armchair theory of the activity of assigning probabilities.
In any case, subjectivism is a theory of probability, not a no-theory theory. Simply put, Hájek is wrong, and he is wrong about radical subjectivism being a “no-theory theory” because he paints an incomplete picture of subjectivism.
- Why subjectivism is a theory
In summary, although subjectivism is a “permissive” theory of probability (to quote Hájek), it is nevertheless still a testable theory. Specifically, if the main purpose of theory is to explain observations or make testable predictions (see, e.g., Popkewitz, 1980), then subjectivism is, in fact, a theory by any measure. Subjectivism explains probability in terms of “degrees of belief” and subjectivist methodology is, in principle, testable.
To begin with, the subjectivist interpretation of probability has two steps, not one. In brief, subjectivism not only requires you to assign a probability to a proposition (step 1; the prior); it also requires you to continually update your priors whenever you receive new information relevant to the proposition (step 2; Bayesian updating). Hájek, however, makes no mention at all of step 2 or Bayesian updating in his description of radical subjectivism. Although he is correct to say that step 1 of the subjectivist approach is “spectacularly permissive,” he omits the second step entirely. Hájek thus presents an incomplete picture of subjectivism.
Furthermore, this glaring omission is critical because step 2 effectively constrains (and corrects) the assignment of arbitrary, biased, or irrational priors. Consider, for example, Hájek’s proffered pairie dog example: “you may with no insult to rationality assign a 0.999 probability to George Bush turning into a prairie dog …” (Hájek, 2007, p. 577; see also the image below). Once we allow for Bayesian updating, however, Hájek’s silly prairie dog example is easily disposed of. Alan Hájek, by painting a distorted and incomplete picture of subjectivism, falls into a trap of his own making.*
If you think this is Dubya, you haven’t updated your priors.
Alan Hájek, “The reference class problem is your problem too,” Synthese, vol. 156, no. 3 (2007): pp. 563-585.
Andrey Kolmogorov, Foundations of the theory of probability, 2nd ed., New York: Chelsea (1956).
Thomas S. Popkewitz, “Paradigms in Educational Science: Different Meanings and Purpose to Theory,” Journal of Education, vol. 162, no. 1 (1980): pp. 28-46.
* In a future post, we shall explain why our defense of radical subjectivism is relevant to Holmes’s prediction theory of law.