The Subjective Turn: Bayesianism and the Erosion of Objective Predictive Inference — Epoche C1
Bayes' theorem is an uncontroversial piece of arithmetic about conditional probabilities. For a hypothesis $H$ and a body of evidence $E$, $$P(H \mid E) \;=\; \frac{P(E \mid H)\,P(H)}{P(E)},$$ where $P(H)$ is the prior — the probability assigned to the hypothesis before the evidence arrives — $P(E \mid H)$ is the likelihood , the probability the hypothesis assigns to the evidence turning out as it did, and $P(H \mid E)$ is the posterior , the probability of the hypothesis given that evidence. The denominator is fixed by the rest: $P(E) = P(E \mid H)P(H) + P(E \mid \neg H)P(\neg H)$, where $\neg H$ is the denial of $H$. Nothing here is in dispute; it follows in two lines from the definition of conditional probability. What is in dispute is the philosophical proposal built on it: that $P$ should be read as a person's degree of belief, that rationality consists in having such degrees obey the probability axioms and updating them by the theorem, and that this constitutes an answer to David Hume's problem of induction — the argument of the Enquiry concerning Human Understanding (1748, §IV) that any inference from the observed to the unobserved must presuppose that nature is uniform, a premise for which the only available support is an inference of exactly that kind, so that the justification is circular. The claim defended here is that Bayesianism does not supply the objective grounds for prediction often attributed to it. That claim is correct, but the reasons usually given for it are loose, and one of them — that there is no objective method for fixing priors — is simply false. Making the argument exact strengthens it. Where the subjectivity enters, and where it is constrained The subjective reading of probability was given its canonical statement by Bruno de Finetti in 1937. On competing interpretations, a probability is something impersonal: on the frequency interpretation it is the limiting relative frequency of an outcome in a long run of trials, and on the classical interpretation it is the ratio of favourable to equally possible cases, fixed by a symmetry. De Finetti's proposal is that probability is instead a feature of a person — the rate at which that person would exchange a stake for a conditional prospect — and that there is nothing else for it to be. The proposal is not licence to believe anything. Frank Ramsey in 1926 and de Finetti independently proved the Dutch book theorem : if a person's betting rates violate the probability axioms, there exists a set of bets, each of which that person regards as fair by their own rates, whose combined effect is a guaranteed loss whatever happens. Incoherence is thus not merely inelegant; it is a demonstrable disposition to be fleeced. A converse was later established: if the rates do satisfy the axioms, no such book can be made. Notice exactly what this establishes and what it leaves alone. The axioms are constraints of consistency among degrees of belief held at one time. They fix ratios and sums; they do not fix values. A person who assigns 0.9 to a severe drought and a person who assigns 0.1 are both fully coherent, in the same way that two people who assert contradictory premises may both be reasoning validly. This is the point at which the essay's original argument was right, and it is worth pressing rather than asserting. The worked case: two forecasters and one anomaly Take the case the original raised. A farmer in rural Thailand, from long experience of variable seasons, gives a severe drought in the coming monsoon a prior of $P(D) = 0.4$. An agricultural economist in Bangkok, working from a longer record in which severe droughts are less frequent, gives it $P(D) = 0.1$. Both then see the same evidence $E$: a sea-surface temperature anomaly of the kind associated with El Niño. Suppose — and this is the concession that makes the case sharp — that they agree completely on the likelihoods: $P(E \mid D) = 0.8$ and $P(E \mid \neg D) = 0.2$. Bayes' theorem is easiest to handle in odds form. The odds on $H$ are $P(H)/P(\neg H)$; dividing the theorem for $H$ by the theorem for $\neg H$ cancels the denominator $P(E)$ entirely and gives $$\frac{P(H \mid E)}{P(\neg H \mid E)} \;=\; \frac{P(E \mid H)}{P(E \mid \neg H)} \times \frac{P(H)}{P(\neg H)} .$$ Posterior odds equal the likelihood ratio times prior odds. Here the likelihood ratio is $0.8/0.2 = 4$: the anomaly is four times as probable if a severe drought is coming as if it is not. The farmer's prior odds are $0.4/0.6 = 2/3$, so his posterior odds are $8/3$, i.e. a probability of $8/11 \approx 0.73$. The economist's prior odds are $0.1/0.9 = 1/9$, so her posterior odds are $4/9$, i.e. $4/13 \approx 0.31$. Both have reasoned impeccably from the same data, and they end on opposite sides of any decision threshold near one half. The identity the usual debate misses This is where the received discussion, including the original version of this essay, becomes imprecise, and where an exact result is available. The standard reply is that priors "wash out" as evidence accumulates; the standard rejoinder is that washing out may be slow. Both are true, but the odds form yields something stronger and more specific. Write the two agents' prior odds as $O_1$ and $O_2$. Each multiplies by the same likelihood ratio $L$, since they agree on the likelihoods. Their posterior odds are $O_1 L$ and $O_2 L$, and the ratio between them is $$\frac{O_1 L}{O_2 L} \;=\; \frac{O_1}{O_2}.$$ The $L$ cancels. In the worked case, the ratio of the two agents' odds is $(2/3)/(1/9) = 6$ before the evidence and $(8/3)/(4/9) = 6$ after it, and it will still be 6 after a thousand further observations, provided only that the two continue to agree about likelihoods. Shared evidence never narrows a disagreement on the odds scale. It cannot: multiplying two numbers by the same factor leaves their ratio alone. What, then, is "washing out"? It is entirely an artefact of the probability scale. As evidence drives both agents' odds tow