Rational Degrees of Belief in Dynamic Epistemic Contexts — Epoche C1
What coherence is, and the argument that supports it Suppose you are willing to pay 60 pence for a ticket that pays £1 if it rains tomorrow, and also 60 pence for a ticket that pays £1 if it does not. Someone sells you both. You have paid £1.20 and you will receive exactly £1, whatever the weather: a guaranteed loss of 20 pence, arrived at by subtracting what you must receive from what you chose to pay. Nothing about the world made you lose. Your own two valuations did, because they sum to 1.20 where they should sum to 1. This is the argument that underwrites the requirement of probabilistic coherence, and it is worth stating precisely before asking whether it is enough. An agent's degree of belief in a proposition is identified with the price at which she is indifferent between buying and selling a ticket that pays one unit if the proposition is true. The Dutch book theorem, which goes back to Ramsey's essay on truth and probability, states that if such prices violate the axioms of probability — non-negativity, the assignment of $1$ to a certainty, and additivity over mutually exclusive alternatives — then a set of bets exists, each of which the agent regards as fair by her own prices, which together guarantee her a loss whatever happens. The converse also holds: if the prices obey the axioms, no such book can be made. Coherence is thus exactly the condition that protects an agent from being defeated by arithmetic alone. A second and independent vindication removes the betting apparatus, which some find an odd basis for an epistemology. James Joyce showed that for a wide family of ways of measuring how far a set of credences lies from the truth, any incoherent credence function is accuracy-dominated : there is a coherent function that is strictly closer to the truth in every possible world. Incoherence is therefore not merely imprudent but epistemically defective, without any detour through money. The dynamic rule Coherence as so far described is a constraint on a single snapshot of an agent's beliefs. Inquiry involves change, and the orthodox rule for change is conditionalisation: on learning $E$ and nothing stronger, replace each old credence $P(H)$ by the old conditional credence, $$P_{\text{new}}(H) = P(H \mid E) = \frac{P(E \mid H)\,P(H)}{P(E)},$$ the last equality being Bayes' theorem, which follows in two lines from the definition of conditional probability. This rule has its own defence of the same shape as the first: Paul Teller, developing an argument he credits to David Lewis, showed that an agent who announces in advance that she will revise by any rule other than conditionalisation exposes herself to a sequence of bets, fair by her own announced standards at the time each is offered, on which she loses in every eventuality. Coherence over time is enforced by the same machinery as coherence at a time. The original version of this essay treated coherence as a static notion opposed to the dynamics of inquiry. That opposition is not quite the right one: the framework has a dynamics, and it is well motivated. The real question is whether coherence plus conditionalisation suffices for rationality, and the answer is no — but for reasons more precise and more damaging than the essay gave. Why coherence cannot be sufficient The essay claimed that an agent might "rationally maintain a consistent but factually inaccurate belief system indefinitely". This is not a worry to be entertained; it is a one-line theorem. Return to Bayes' formula. If an agent assigns prior credence $P(H) = 0$ to some hypothesis, then for any evidence $E$ with $P(E) > 0$ the numerator $P(E \mid H)P(H)$ is zero, and so is the posterior. No observation, and no accumulation of observations however long, can move a credence of zero. Assigning zero is perfectly coherent — the axioms permit any assignment that respects them — so a coherent agent may be immunised for life against the truth by a single stroke of her initial pen. The formal result that shows what is needed to avoid this is the merging theorem of Blackwell and Dubins. It states that if two agents' prior probability functions are mutually absolutely continuous — each assigns positive probability to every event to which the other does — then as they conditionalise on a common growing stream of evidence, their predictions about the future converge almost surely. This is the strongest thing that can be said in coherence's favour: agreement is guaranteed in the limit. And the condition on which it depends identifies the missing constraint exactly. Merging requires that nothing genuinely possible be given probability zero. That constraint, usually called regularity, is not a consequence of coherence; it must be imposed separately, and it is the precise formal content of the vague demand for "responsiveness to evidence" with which the essay concluded. Even short of zero, the point survives in quantitative form. Write Bayes' theorem in odds: $$\frac{P(H \mid E)}{P(\neg H \mid E)} = \frac{P(H)}{P(\neg H)} \times \frac{P(E \mid H)}{P(E \mid \neg H)} .$$ The posterior odds are the prior odds multiplied by the likelihood ratio, so evidence acts on the odds by multiplication and the prior sets the starting point. An agent whose prior odds against a hypothesis are $1$ to $1000$, receiving observations each ten times more likely if the hypothesis is true than if it is false, needs $\log_{10} 1000 = 3$ such observations merely to reach even odds. The exponent is three because a thousand is ten cubed: each observation buys one factor of ten, and three are needed to undo three. Nothing in the coherence requirement prevents her from having chosen prior odds of $1$ to $10^{30}$ instead. One further clarification is needed, because two different things travel under the name "coherence" and the original essay moved between them. There is probabilistic coherence, the obedience to the axioms just described; and there is coherence in the sense of mutual support among the members