Accuracy and Scoring Rules

The accuracy argument for probabilism replaces the pragmatic premises of the Dutch-book argument with an epistemic one. Beliefs aim at truth. A credence, being graded, cannot be simply true or false, but it can be more or less accurate — closer to or further from the truth value of the proposition it concerns. Given a measure of accuracy, one can ask which credence functions do well by it, and the answer turns out to be: only the coherent ones.

The result, due to Joyce (1998), is that every incoherent credence function is accuracy-dominated — there is a coherent function that is strictly more accurate in every possible world. Since it is irrational to adopt a position guaranteed to be worse than an available alternative however things turn out, credences should be coherent.

This is the strongest available argument for probabilism, and it is not vulnerable to the objections that trouble Dutch books. Its own difficulty lies elsewhere: in justifying the measure of accuracy.

Measuring inaccuracy

Let $w$ be a possible world and let $v_w(A)=1$ if $A$ is true at $w$ and $0$ otherwise. The omniscient credence function at $w$ is $v_w$: it assigns $1$ to every truth and $0$ to every falsehood. Accuracy is closeness to this ideal, and inaccuracy is distance from it.

The standard measure is the Brier score, the squared Euclidean distance:

$$\mathcal{B}(Cr,w)=\sum _{A\in \mathcal{F}}(Cr(A)-v_w(A){)}^2$$

If it rains and you had credence $0.9$ in rain, your contribution is $(0.9-1{)}^2=0.01$; had you been at $0.2$ it would be $0.64$. Lower is better.

The Brier score is not the only option, and the choice matters. The logarithmic score $-\log Cr(A_w)$, where $A_w$ is the true cell of a partition, is the other standard measure and has a natural information-theoretic reading: it is the surprisal of the truth. The absolute-value score $|Cr(A)-v_w(A)|$ is equally intuitive and, crucially, gives different verdicts.

Propriety

The key technical notion is strict propriety. A scoring rule is strictly proper if, by the lights of any probabilistic credence function $p$, the expected score of reporting $p$ itself is strictly better than that of reporting any other function $q$:

$${\mathbb{E}}_p[\mathcal{S}(p,\cdot )]<{\mathbb{E}}_p[\mathcal{S}(q,\cdot )]\text{for all }q\ne p$$

The condition says the rule gives no incentive to misreport. An agent with credence $0.7$ minimises expected inaccuracy by announcing $0.7$, not by hedging toward $0.5$ or exaggerating toward $1$.

The Brier and logarithmic scores are strictly proper. The absolute-value score is not: under it, an agent with any credence above $0.5$ minimises expected inaccuracy by announcing $1$. That is plainly the wrong verdict for an epistemic norm — it recommends certainty on the basis of a bare majority of evidence — which is why improper rules are excluded. But note what this does to the argument's structure: propriety is itself justified partly by the fact that improper rules give intuitively wrong answers, and the intuitions in question are already probabilistic ones.

The dominance argument

With a strictly proper scoring rule fixed, the argument runs:

1. Accuracy is the fundamental epistemic virtue. Credences aim at truth, and their epistemic value is measured by closeness to the truth values.
2. Inaccuracy is measured by a strictly proper scoring rule, satisfying reasonable structural conditions (continuity, and treating propositions symmetrically).
3. Dominance. For any incoherent $Cr$, there is a coherent $C{r}^{\ast }$ such that $\mathcal{S}(C{r}^{\ast },w)<\mathcal{S}(Cr,w)$ for every world $w$.
4. It is irrational to adopt an option dominated in every state when an undominated alternative is available.
5. Therefore credences should be coherent.

The geometry behind step 3 is illuminating and worth stating, because it explains why the result holds rather than merely that it does. Represent credence functions over $n$ propositions as points in $[0,1{]}^n$. The omniscient functions $v_w$ are among the vertices of this cube. The set of coherent credence functions is exactly the convex hull of the $v_w$ — a closed convex set. An incoherent function is a point outside that set. By elementary convex geometry, for any point outside a closed convex set there is a point inside it closer to every point of the set, in particular closer to every vertex. That closer point is the dominating coherent credence function.

So the theorem is, at bottom, the observation that coherence is convexity, and that points outside a convex hull are dominated relative to the vertices. This is why the result is robust across scoring rules satisfying the structural conditions: it is a fact about the geometry, not about the Brier score specifically.

Objections

The choice of scoring rule

The argument requires a measure of inaccuracy, and different measures are available. Joyce's original paper imposed axioms intended to characterise legitimate measures; critics noted these axioms are strong, and that some are motivated by the very probabilistic intuitions the argument is supposed to establish.

The sharpest form of the worry concerns propriety. Why should an epistemic norm be strictly proper? The natural justification is that an improper rule gives agents reason to misrepresent their credences — but "reason" here is expected value, computed with a probability function. If the justification for the measure presupposes that expectations are computed probabilistically, the argument is closer to circular than it appears.

Joyce's later work responds by deriving propriety from more basic considerations about what it is for one credence to be more truth-directed than another. Whether this fully escapes the objection is disputed.

Is accuracy the only epistemic value?

Step 1 is a substantive commitment — a form of epistemic value monism. Other candidate epistemic goods include explanatory power, coherence with background theory, and informativeness. Note that accuracy and informativeness pull apart: a credence of $0.5$ in every proposition is uninformative but never badly inaccurate, while bold credences risk more. The scoring-rule framework can accommodate a trade-off between accuracy and boldness, but doing so introduces a further parameter with no principled setting.

Dominance for credences

Step 4 borrows a principle from decision theory. Dominance reasoning is uncontroversial when the states are independent of the acts; when they are not, it can fail. In the epistemic case the "states" are worlds and the "acts" are credence functions, and worlds do not depend on what one believes — so the application looks safe. But there is a subtlety: the argument compares an agent's actual credence with an alternative that the agent may have no way of identifying, and "you should adopt the dominating function" is idle advice unless the agent can locate it.

It proves too little

The argument establishes coherence and nothing else. It does not tell an agent which coherent function to adopt, and every coherent function is undominated. So accuracy considerations leave the problem of the priors entirely untouched — the same limitation the Dutch-book argument has.

The wider program

Accuracy-first epistemology has become a research program rather than a single argument, and its interest lies in how much it recovers from the same premises.

• Conditionalization can be given an accuracy-based defence: Greaves and Wallace show that among all update rules, planning to conditionalize maximises expected accuracy from the current perspective.
• The Principal Principle has been argued for on accuracy grounds by Pettigrew — deferring to known chances is the policy that maximises expected accuracy when chance is understood as the objective expert. See the Principal Principle.
• Indifference and equivocation can be defended by minimising worst-case inaccuracy, which favours the maximum-entropy distribution and connects the program to objective Bayesianism.

The ambition is to derive the whole Bayesian package from the single premise that belief aims at truth. That the derivations go through is a genuine achievement; that each requires its own auxiliary assumptions about the measure of accuracy is the standing reservation.

Where this sits

The accuracy argument supplies what the Dutch-book argument could not: an epistemic justification for an epistemic norm, with no detour through betting behaviour or practical loss. It is the better argument, and it is why probabilism is now defended on accuracy grounds in most contemporary work.

Both arguments share a limitation that shapes the rest of this folder. They constrain the structure of credence without constraining its content: coherence is necessary, and it is nearly all that these arguments give. What fixes the particular numbers is the subject of conditionalization for the dynamics and priors for the starting point, and it is there that the real disagreement lies.