Coherence (Statistics)

In Bayesian epistemology and the foundations of statistics, coherence is the normative requirement that an agent's credences (degrees of belief) satisfy the Kolmogorov axioms of probability. A coherent agent assigns non-negative credences, gives credence 1 to tautologies, and respects finite additivity for mutually exclusive propositions. The primary justification for coherence is the Dutch book theorem: an agent whose credences violate the probability axioms is vulnerable to a "Dutch book" — a combination of bets, each individually fair or favorable by the agent's own lights, that together guarantee a net loss regardless of how the world turns out.

Coherence is the minimal rationality constraint in Bayesian epistemology. It says nothing about which specific credences to hold — only that they must fit together consistently. Two agents with wildly different beliefs about the world can both be coherent, provided each individually satisfies the probability axioms. Coherence is a synchronic constraint (applying at a single time); the corresponding diachronic constraint is conditionalization.

The Dutch Book Argument

The Dutch book argument is the classic justification for coherence. It proceeds in two steps. First, the Dutch book theorem (Ramsey 1926, de Finetti 1937) shows that if an agent's credences violate the probability axioms, a bookie can construct a set of bets — each acceptable to the agent — whose combined payoff is negative in every possible state of the world. The agent is guaranteed to lose money.

Second, the converse Dutch book theorem (Kemeny 1955, Lehman 1955) shows that if an agent's credences do satisfy the probability axioms, no such Dutch book can be constructed. Coherence is both necessary and sufficient for Dutch book immunity.

Extensions: Diachronic Coherence

The Dutch book argument can be extended from synchronic to diachronic coherence. The diachronic Dutch book theorem (Lewis 1999, Teller 1973) shows that an agent who plans to update her credences by any rule other than conditionalization is vulnerable to a diachronic Dutch book: a sequence of bets placed before and after the evidence arrives that guarantee a net loss. This provides a pragmatic justification for conditionalization as the uniquely rational updating rule.

Specifically, suppose an agent plans that upon learning E, she will adopt new credences P_new that differ from P_old(· | E). Then there exist bets, placed before and after learning E, that guarantee a combined loss. Only the conditionalization plan — setting P_new(H) = P_old(H | E) for all H — is immune to diachronic Dutch books.

Accuracy-Based Arguments

James Joyce (1998) pioneered an alternative justification for coherence that avoids the pragmatic assumptions of the Dutch book argument. The key idea is that credences can be evaluated by their accuracy — their closeness to the truth values (0 or 1) of the propositions they concern. Joyce proved that for any incoherent credence function, there exists a coherent credence function that is strictly more accurate in every possible world. Incoherent credences are accuracy-dominated.

This result is striking because it justifies probabilism on purely epistemic grounds, without appealing to betting, money, or decision theory. An agent who cares only about having accurate beliefs — about getting her credences as close to the truth as possible — is compelled by that concern alone to be coherent.

Coherence and Bayesian Statistics

In applied Bayesian statistics, coherence manifests as the requirement that priors, likelihoods, and posteriors form a valid probability model. A statistician who specifies a prior that is not a valid probability distribution, or who updates by a rule other than Bayes' theorem, is incoherent in the technical sense. The entire machinery of Bayesian inference — conjugate priors, MCMC sampling, variational approximations — is designed to maintain coherence while making computation tractable.

Coherence also connects to de Finetti's theorem: if an agent's credences about an exchangeable sequence are coherent, they can be represented as a mixture of i.i.d. distributions — effectively implying a prior over parameters. The prior is not an arbitrary choice but a consequence of coherence combined with a symmetry judgment.

"Your probability for an event is your willingness to take either side of a bet at the corresponding odds. If your probabilities don't satisfy the standard axioms, I can make a book against you — a set of bets which you are willing to take but which result in a sure loss for you." — Frank Ramsey, "Truth and Probability" (1926)

Beyond Classical Coherence

Recent work has extended the coherence concept in several directions. Probabilistic coherence under imprecise probabilities requires that the agent's credal set (set of probability functions) be convex and closed. Quantum coherence in QBism extends Dutch book arguments to quantum settings, where the Born rule plays the role of a quantum coherence constraint. And scoring rule coherence requires that an agent's reported credences minimize expected loss under a proper scoring rule — a constraint that turns out to be equivalent to standard probabilistic coherence.

These extensions demonstrate that coherence, far from being a narrow technical requirement, is a deep structural principle that recurs wherever rational agents must manage uncertainty — whether classical, quantum, or somewhere in between.

Example: A Bookmaker Who Guarantees Losing Money

A bookmaker offers odds on three horses in a race. He assigns the following probabilities to each horse winning:

The Dutch Book

Because these probabilities are incoherent — they don't obey the axioms of probability — a clever bettor can construct a "Dutch book": a set of bets that guarantees the bookmaker loses money no matter which horse wins.

The Dutch book theorem proves that any agent whose credences violate the probability axioms is vulnerable to a guaranteed loss. Conversely, if your credences are coherent — if they satisfy the axioms — no Dutch book can be constructed against you.