Subjective Bayesianism

Subjective Bayesianism — personalism, in de Finetti's term — holds that probability is degree of belief, and that the only constraint reason imposes on it is coherence: degrees of belief must satisfy the probability axioms, and must change by conditioning on evidence. Beyond that, an agent's probabilities are their own. Two people with identical evidence may hold different credences without either being irrational.

De Finetti put the position at its most provocative: probability does not exist. There is no objective chance in the world waiting to be discovered — no propensity, no limiting frequency that constitutes a probability — only agents with graded expectations, some coherent and some not. The apparent objectivity of scientific probability is, on this view, something to be explained rather than assumed, and the explanation is that agents who start with different opinions but update on shared evidence converge.

This page sets out how credences are measured, the arguments that they must be probabilities, the two formal results that do most of the work for the position, and the objections — of which permissiveness is the deepest.

Credence as a measurable quantity

The position needs degrees of belief to be more than a metaphor, and the standard approach is operational: measure belief by willingness to bet.

An agent's credence in $A$ is the number $p$ such that they are indifferent between accepting and declining a bet that pays $1-p$ if $A$ and loses $p$ otherwise. Betting quotients are behavioural, elicitable, and quantitative, which is exactly what was wanted.

Ramsey saw the difficulty with the naive version and fixed it. Willingness to bet depends on the value of the stakes as well as on belief, and money has diminishing marginal utility, so betting rates in cash measure a mixture of belief and attitude to risk. Ramsey's solution, in "Truth and Probability" (1926), was to derive utility and probability together from preferences over gambles: assume the agent's preferences satisfy structural conditions (completeness, transitivity, and a continuity condition), find an "ethically neutral" proposition believed to degree $1/2$, use it to construct a utility scale, and then read off probabilities. Savage's Foundations of Statistics (1954) gives the mature version: an agent whose preferences over acts satisfy his postulates behaves as if maximising expected utility relative to a unique probability function and a utility function unique up to positive affine transformation.

This is a representation theorem, and its status should be stated carefully because it is often overstated. It does not show that rational agents have credences; it shows that agents whose preferences satisfy certain axioms can be described as having them. Whether the preference axioms are themselves requirements of rationality is a further question — Allais' and Ellsberg's paradoxes are the standard evidence that real agents systematically violate them, and it is contested whether such agents are irrational or the axioms too strong.

Why credences must be probabilities

Coherence is the position's one substantive norm, and it needs an argument. Two are standard, and both are developed further on their own pages.

Dutch books. If an agent's betting quotients violate the probability axioms, a bookmaker can offer a set of bets each of which the agent regards as fair or favourable, but which jointly guarantee a loss whatever happens. If credence in $A$ is $0.6$ and in $\neg A$ is $0.6$, bets at those rates on both lose money in every eventuality. A converse Dutch-book theorem completes the argument: if the quotients do satisfy the axioms, no such book exists. Coherence is thus necessary and sufficient for immunity to sure loss. The diachronic version, due to Lewis and Teller, extends the argument to updating: an agent who plans to update other than by conditionalization is vulnerable to a book made over time.

Accuracy. The pragmatic flavour of the Dutch book bothers many — being exploitable by a bookie looks like a practical misfortune rather than an epistemic failing — and the accuracy-based argument replaces it. Take credences to aim at truth, measure their inaccuracy by a strictly proper scoring rule such as the Brier score, and Joyce's theorem shows that any incoherent credence function is accuracy-dominated: there is a coherent function closer to the truth in every possible world. Since it is irrational to hold a position guaranteed to be worse than an available alternative however things turn out, credences should be coherent. This is a purely epistemic argument, and it is the stronger of the two, though it depends on the choice of scoring rule.

Conditionalization

The second norm is dynamic. On learning $E$ and nothing more, the agent's new credence should be the old conditional probability:

$$C{r}_{\text{new}}(A)=C{r}_{\text{old}}(A\mid E)$$

Jeffrey conditionalization generalises this to cases where experience does not deliver a proposition with certainty but redistributes credence over a partition — the important case, since observation rarely yields certainty.

Two limitations are worth flagging now. Conditionalization is undefined when $C{r}_{\text{old}}(E)=0$, so an agent who assigned zero to a hypothesis can never learn it, which is Cromwell's rule and an argument for regularity. And the rule presupposes a fixed algebra of propositions: it says nothing about what to do on encountering a genuinely new hypothesis nobody had formulated, which is the ordinary situation in science.

The two results the position leans on

De Finetti's representation theorem. Suppose an agent's credences over an infinite sequence of trials are exchangeable — invariant under permuting the order of trials. Then the theorem shows that their credence function must be representable as a mixture of i.i.d. distributions: there is a unique measure $\mu$ over the possible "chances" $\theta$ such that

$$Cr(x_1,\dots ,x_n)={\int }_0^1{\theta }^k(1-\theta {)}^{n-k}d\mu (\theta )$$

This is the position's crown jewel, and its significance is precisely stated. The agent behaves exactly as if they believed in an unknown objective chance $\theta$ and had a prior over it — but the theorem derives this appearance from a symmetry in the agent's own credences, with no chance in the ontology. Talk of unknown physical probabilities is thereby paraphrased away: it is a way of describing exchangeable belief. De Finetti took this to show that the subjectivist can say everything the objectivist can, without the metaphysics.

What it does not show — and this is the discipline the section keeps insisting on — is that objective chance does not exist, or that subjectivism is correct. It is a theorem about credence functions with a particular symmetry. Its application requires that the agent's credences be exchangeable, which is a substantive assumption and often false: an agent who suspects the coin is being warmed, or the trials are correlated, is not exchangeable, and the theorem simply does not apply. It also does not explain why we should defer to chances discovered by physics.

Merging of opinions. Blackwell and Dubins showed that two agents whose priors are mutually absolutely continuous — agreeing on which events have probability zero — converge as they conditionalize on a growing shared body of evidence. Subjectivists take this to answer the charge that their view makes science arbitrary: objectivity is not correspondence to a chance but intersubjective agreement in the limit.

The qualifications matter as much as the result. Convergence requires the absolute-continuity condition, which fails exactly when agents disagree about what is possible — the interesting case. It is asymptotic, with no guarantee about any finite stage, and the rate can be arbitrarily slow. And two agents can be made to converge to a false hypothesis if both assign the truth zero prior. Merging shows that stubbornness is not stable, not that any particular finite body of evidence compels a particular credence.

Objections

Permissiveness. The central objection. If coherence is the only constraint, an agent may assign $0.99$ to the sun failing to rise tomorrow, or hold any prior whatever about physical constants, and be perfectly rational. Nothing in the theory distinguishes the well-informed scientist from the crank with a coherent delusion, provided both do their arithmetic. The convergence results are the standard reply; the standard rejoinder is that they are asymptotic and conditional, and that scientific probability claims do not seem hostage to anyone's starting point. This is what objective Bayesianism exists to fix.

The problem of the priors. Where do initial credences come from? On the strict view, anywhere. In practice Bayesians use conventional or "uninformative" priors, which reintroduces every difficulty about indifference and parameterisation that the position had sidestepped.

Old evidence. If $E$ is already known, $Cr(E)=1$, so conditioning on $E$ changes nothing, and $E$ cannot confirm any hypothesis. Yet the perihelion of Mercury — known for decades — was powerful confirmation of general relativity. Glymour's problem is a direct consequence of modelling agents as logically omniscient with a fixed algebra, and every proposed solution modifies that idealisation.

Logical omniscience. Coherence requires assigning $1$ to every logical truth. A real agent who has not yet proved a theorem does not, and cannot without solving the halting problem in general. So the norm is not merely demanding but unsatisfiable, which raises the question of what force it has. This is the standing cost of the idealisation discussed under models and applications.

It does not capture what physicists mean. When a physicist states a half-life, they are not reporting their confidence, and would not accept that another physicist with a different coherent credence is equally correct. The subjectivist must say this appearance is misleading. Many find that the least credible commitment of the view.

Assessment

Subjectivism scores better on the formal criteria than any rival, which is why it dominates contemporary formal epistemology. Its weakness is on the one criterion not in the list: it has difficulty accounting for the objectivity that probability claims in science appear to have.

Where this sits

Subjective Bayesianism is the pure credence-first position of the chance/credence taxonomy: credence is fundamental, and chance is either eliminated or reconstructed as a feature of credal symmetry. Its formal machinery — coherence, conditionalization, representation theorems — is now common property, used by philosophers who reject its metaphysics entirely; one can accept every theorem above while holding that objective chances exist and that the Principal Principle constrains credence to track them.

The next page keeps the Bayesian apparatus and adds the constraints subjectivism refuses: objective Bayesianism, the attempt to recover as much of the logical interpretation's objectivity as survives its collapse.