John Venn's example: John Smith is a consumptive Englishman of fifty. What is the probability he lives to sixty-one? A frequentist counts: among people like him, what fraction lived eleven more years? But "people like him" isn't given. He is a man, a fifty-year-old, a consumptive, an Englishman, and each class has its own frequency. Venn makes it sharp [1]: suppose nine in ten Englishmen are harmed by living in Madeira, and nine in ten consumptives are helped by it. Both tables can be true at once, both cover Smith, and they point in opposite directions. Should he go?
Reichenbach's rule is to use the narrowest class with reliable statistics. Here neither class sits inside the other, so neither is narrower. Narrow further, to consumptive Englishmen in Madeira, and the data may not exist; narrow all the way and the class holds only Smith, whose frequency is 0 or 1. Venn's practical answer, if forced to decide, was that lungs matter more to survival than birthplace, an empirical judgment that, he admits, the given counts can't supply.
This is usually counted against frequentism. Alan Hájek's point [2] is that the Bayesian alternative inherits it. Degrees of belief guide action only if they answer to something outside the believer, and one standard link, the Principle of Direct Probability, says: if you know a fraction x of cases like this turn out A, believe A to degree x. That link is where frequencies enter a Bayesian's beliefs, and so where the choice of class enters with them. Smith's case supplies two such frequencies that disagree, and weighting them needs a reason, which brings the same choice back. A subjectivist who ties belief to nothing escapes, but then probability becomes, in Hájek's phrase, "autobiography rather than epistemology."
His diagnosis is that probability is two-place. There is P(lives to 61 | consumptive) and P(lives to 61 | Englishman), and no privileged unconditional P(lives to 61) behind them, just as there is no distance without a reference frame. That dissolves what he calls the metaphysical problem: which number is the true one. It leaves the epistemological one: which conditional probability should Smith use when he buys life insurance? Singling one out is choosing a reference class under another name.
An outside-view forecast, an insurance premium or a clinical risk score is always conditional on a chosen class, and the class has to fit the case. The same argument is running over AI risk: Scott Alexander defends probabilities with no frequencies behind them [3], and Narayanan and Kapoor reply that without a reference class such a number comes down to the analyst's intuition [4].
References
The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability [link]
Venn, J., 1888. Macmillan and Co.The reference class problem is your problem too [link]
Hájek, A., 2007. Synthese, Vol 156(3), pp. 563–585. Springer Science and Business Media LLC. DOI: 10.1007/s11229-006-9138-5In Continued Defense Of Non-Frequentist Probabilities [link]
Alexander, S., 2024. Astral Codex Ten.AI existential risk probabilities are too unreliable to inform policy [link]
Narayanan, A. and Kapoor, S., 2024. AI Snake Oil.