Chance and Credence

The single most useful distinction in the philosophy of probability is between objective chance — a probabilistic feature of the world, if there is such a thing — and credence — the degree of confidence an agent invests in a proposition. Both are measured on the interval $[0,1]$, both obey the same calculus, and ordinary language uses "probability" for each without warning. But they are claims about entirely different subject matters. That a coin has a chance of $1/2$ of landing heads is, if true, a fact about the coin and the tossing setup, and would remain true if nobody ever thought about it. That I am $1/2$ confident it will land heads is a fact about me, and would be true even if the coin were double-headed and I were badly misinformed.

This page sets out the distinction, the notation the section uses to keep it visible, the third notion that sits awkwardly between the two, and the central problem the distinction creates: if chance and credence really are different, why should learning a chance oblige anyone to adopt a matching credence? That is the coordination problem, and Lewis's Principal Principle is the canonical answer to it.

Two notions, one calculus

The contrast can be drawn along several axes at once.

The last three rows do most of the work. A chance is not relative to evidence: the atom's decay probability does not change when a physicist looks it up. A credence is essentially evidence-relative, and the same agent will have different credences at different times as evidence accumulates. Conversely, once a coin has landed, its chance of having landed heads is $1$ or $0$; an agent who did not see the result may quite reasonably retain a credence of $1/2$. Anyone who says "the probability is one half" about a coin already lying under a cup is making a claim that is false read as a chance and possibly true read as a credence.

Because both satisfy the Kolmogorov axioms, no formal test distinguishes them. The mathematics is deliberately indifferent to the difference, which is precisely why the difference has to be maintained by the interpreter.

A third notion: evidential probability

Between the two sits a notion that is neither, and that different traditions assign to one side or the other: the probability of a hypothesis on a body of evidence, written $P(h\mid e)$ and understood as measuring how strongly $e$ supports $h$, independently of whether anyone believes either.

This is objective in that it does not depend on any agent's psychology — two people with the same evidence face the same evidential probability — but it is not a feature of the physical world in the way a chance is. It is a relation between propositions. This is the notion the logical interpretation attempts to formalise, and it is what an objective Bayesian means by saying priors are rationally constrained. If a defensible notion of evidential probability exists, the epistemic side of the field looks much more objective than subjectivists allow; if it does not, credence is the only epistemic notion available and its variability across agents has to be tolerated.

The three-way distinction is worth carrying explicitly, because arguments frequently equivocate between the second and third:

• $Ch(A)=x$ — the world is so structured that $A$ has chance $x$.
• $Cr(A)=x$ — this agent, now, is confident of $A$ to degree $x$.
• $P(A\mid E)=x$ — the evidence $E$ supports $A$ to degree $x$, whoever holds it.

The coordination problem

Grant that chance and credence are distinct. Then a question arises that would not arise if they were the same: why should knowing a chance constrain belief at all?

The connection is not a logical truth, and it is not optional either. If I am told, on impeccable authority, that this coin is biased with a chance of $0.7$ for heads, and I nonetheless set my credence at $0.2$, I have made a mistake — but not a mistake of arithmetic. Nothing in the axioms links $Ch$ and $Cr$; they are simply two functions. So the link must be a normative principle about rationality, and it must be stated carefully enough to be both true and non-trivial.

Lewis's Principal Principle is the standard formulation, treated in full on its own page. In a simplified form:

$$Cr(A\mid Ch(A)=x\wedge E)=x$$

for any rational initial credence function $Cr$, provided $E$ is admissible. The proviso is essential and is where the difficulty concentrates. Without it the principle is plainly false: if $E$ includes the information that the coin in fact landed tails, no rational agent keeps credence $0.7$ in heads. Admissible evidence is, roughly, evidence that bears on $A$ only by way of bearing on $A$'s chance — historical information, information about the setup — and not evidence that bypasses the chance, such as a report from the future, a crystal ball, or inside knowledge of the outcome.

Three features of this principle deserve emphasis, and each generates its own literature.

• It is a constraint on credence, not a definition of chance. Read one way it tells us how to believe; read another it functions as an implicit definition, on the grounds that chance simply is whatever plays this role in guiding rational belief. Lewis leaned toward the second reading — "chance is the thing that satisfies the Principal Principle" — which makes the principle constitutive rather than merely regulative, and correspondingly harder to revise.
• "Admissible" resists sharp definition. Every attempt to give necessary and sufficient conditions either lets in too much or presupposes the notion of chance it was meant to help explicate. This is a genuine soft spot rather than a technicality.
• It creates trouble for Humean accounts of chance. If chances are determined by the whole history of the world — as on the best-system view — then present chances depend on what will happen, and an agent who conditioned on a full specification of the chances would sometimes have to assign non-zero credence to futures that would undermine those very chances. This is Lewis's "Big Bad Bug", and it motivated the New Principle of Hall and Thau, which conditionalises on the theory of chance rather than on the chance values directly.

The reference-class problem

The coordination problem has a mirror image on the objective side. Suppose we want to find out a chance rather than to respond to a known one. The natural method is statistical: look at what happens in similar cases. But "similar" has to be made precise, and every individual belongs to indefinitely many classes with different frequencies.

A fifty-year-old smoker with a family history of longevity, who exercises daily and lives at altitude, belongs to the class of fifty-year-olds, of smokers, of smokers who exercise, of people at altitude, and to the intersection of all of these. Each class has its own mortality rate, and the rates differ. Which one is his probability of surviving the decade?

The problem is genuinely general, and its bite differs by interpretation:

• For frequentism it is close to fatal, because a frequency is defined only relative to a class, and nothing in the theory selects one. The individual has no probability simpliciter.
• For propensity views it is less pressing in principle — the chance belongs to the concrete setup, not to a class — but returns as an epistemological problem, since we still have to estimate the propensity from data about other cases.
• For subjectivism it is not a problem about probability at all but about evidence: the agent has a credence, and the question is only which statistics are relevant to forming it. This is arguably the position's most attractive feature.

Note that the reference-class problem is not confined to philosophy. It is the same problem faced by actuaries choosing rating factors, by courts assessing statistical evidence about a defendant, and by any predictive model asked for the risk attaching to a particular individual rather than to a population.

Can credence do without chance?

A radical subjectivist — de Finetti is the exemplar — holds that objective chance is a metaphysical extravagance and that everything worth saying can be said with credence alone. The position is stronger than it first appears, because two formal results appear to supply what chance was wanted for.

De Finetti's representation theorem shows that an agent whose credences over an infinite sequence of trials are exchangeable — invariant under permutation of the trials — must represent them as a mixture of i.i.d. distributions, weighted by a measure over the possible "chances". Objective chance is thereby recovered as a mathematical artifact of a symmetry in the agent's own credences, without any commitment to its existence. Merging-of-opinions results add that agents with different but mutually absolutely continuous priors converge as evidence accumulates, so the apparent objectivity of scientific probability can be explained as intersubjective agreement in the limit rather than as correspondence to a worldly chance.

The reply, developed at length in the interpretations folder, is that these results deliver less than they appear to.

• The representation theorem is a theorem about credences, and it applies only to agents whose credences are already exchangeable. Exchangeability is a substantive assumption that itself needs justification, and in many scientific applications — where trials are known not to be symmetric — it is simply false.
• Merging requires agreement on which events have probability zero, and can be arbitrarily slow. It is an asymptotic result and is silent about the finite case, which is the only case anyone is ever in.
• Most importantly, neither result explains why we should defer to physical chances. The de Finetti reconstruction shows that chance-talk can be paraphrased away; it does not show that a physicist who says the decay chance is fixed by the nucleus is saying something about their own doxastic symmetries. The strong law is likewise a theorem about a measure and cannot supply an independent standard the measure must meet.

Where this leaves us

The distinction between chance and credence is not a piece of terminological hygiene but the structural fact that organises the field. The metaphysical question of the introduction is a question about chance; the epistemic question is a question about credence; and the Principal Principle is the bridge, which is why so much weight rests on a principle whose central term nobody can define precisely.

Three positions on the relation are available, and most disputes in the section are downstream of the choice:

The next page maps the interpretations that fill in these columns, and shows how each handles the single case, the choice of priors, and the relation to laws.