On Accuracy and Coherence with Infinite Opinion Sets

Abstract

There is a well-known equivalence between avoiding accuracy dominance and having probabilistically coherent credences (see, e.g., de Finetti 1974; Joyce 2009; Predd et al. 2009; Pettigrew 2016). However, this equivalence has been established only when the set of propositions on which credence functions are defined is finite. In this article, I establish connections between accuracy dominance and coherence when credence functions are defined on an infinite set of propositions. In particular, I establish the necessary results to extend the classic accuracy argument for probabilism to certain classes of infinite sets of propositions, including countably infinite partitions.

1. Introduction

A central norm in the epistemology of partial belief is probabilism: a person’s degrees of belief—or credences—should satisfy the laws of probability.¹ There is a long tradition in the spirit of Savage (1971) and de Finetti (1974) of appealing to the epistemic virtue of accuracy to justify probabilism (also see Rosenkrantz 1981). One particular form of argument is the accuracy dominance argument for probabilism introduced by Joyce (1998). Let a set ${\mathcal{F}}$ of propositions be an opinion set and a function $c:{\mathcal{F}}\to [0,1]$ be a credence function on ${\mathcal{F}}$ . Let a credence function be coherent if it satisfies the axioms of probability. A credence function $c'$ on ${\mathcal{F}}$ accuracy dominates a credence function $c$ on ${\mathcal{F}}$ if $c$ is more inaccurate than $c'$ , no matter how the world turns out to be (where inaccuracy is precisified as in Section 2). Then the existing accuracy dominance arguments purport to vindicate probabilism by showing that a credence function is not accuracy dominated if and only if it is coherent.

However, there is a limitation to almost all of the literature on accuracy dominance arguments for probabilism: the opinion set is assumed to be finite.² Indeed, de Finetti (1974), Lindley (1987), Joyce (1998, 2009), Predd et al. (2009), Leitgeb and Pettigrew (2010a, b), and Pettigrew (2016) all establish their dominance results only for finite opinion sets.³ In this paper, I remove this assumption and prove dominance results that I hope to be useful in evaluating the extent to which accuracy dominance arguments for probabilism succeed when the opinion set is infinite.

I begin in Section 2 by reviewing the mathematical framework and the standard dominance result for finite opinion sets. Sections 3–5 are concerned with accuracy and coherence in the infinite setting. In Sections 3 and 4, I make headway on characterizing the opinion sets and accuracy measures for which there is an equivalence between coherence and avoiding dominance as in the finite case. In Section 5, I extend the accuracy framework to the uncountable setting and prove that coherence is necessary to avoid dominance on uncountable opinion sets. I conclude in Section 6 with a discussion of the results established in Sections 3–5.

2. The finite case

I first set up the framework that will be used throughout the article. Fix a set $W$ (not necessarily finite) that represents the set of possible worlds and, for now, a finite set ${\mathcal{F}}\subseteq {\mathcal{P}}(W)$ of propositions that represents an opinion set—the set of propositions that an agent has beliefs about.

Definition 2.1 An algebra over $W$ is a subset ${\mathcal{F}}^*\!\subseteq {\mathcal{P}}(W)$ such that:

1. 1. $W\in {\mathcal{F}}^*$ ;

2. 2. if $p,p'\in {\mathcal{F}}^*$ , then $p\cup p'\!\in {\mathcal{F}}^*$ ; and

3. 3. if $p\in {\mathcal{F}}^*$ , then $W\setminus p\in {\mathcal{F}}^*$ .

Definition 2.2

1. i. A credence function on an opinion set ${\mathcal{F}}$ is a function from ${\mathcal{F}}$ to $[0,1]$ .

2. ii. A credence function $c$ is coherent if it can be extended to a finitely additive probability function on an algebra ${\mathcal{F}}^*$ over $W$ containing ${\mathcal{F}}$ . That is, there is an algebra ${\mathcal{F}}^*\supseteq {\mathcal{F}}$ over $W$ and a function $c^*:{\mathcal{F}}^*\to [0,1]$ such that:

   1. (a) $c^*(p)=c(p)$ for all $p\in {\mathcal{F}}$ ;

   2. (b) $c^*(p\cup p')=c^*(p)+c^*(p')$ for $p,p'\in {\mathcal{F}}^*$ with $p\cap p'=\varnothing$ ; and

   3. (c) $c^*(W)=1$ .

3. iii. A credence function that is not coherent is incoherent.

Remark 2.3 If ${\mathcal{F}}=\{p_1,\ldots,p_n\}$ , a credence function $c$ over ${\mathcal{F}}$ can be identified with the vector $(c(p_1),\ldots,c(p_n))\in [0,1]^n$ . Thus, the space of all credence functions over ${\mathcal{F}}$ can be identified with $[0,1]^n\subseteq {\mathbb{R}}^n$ . We often simplify notation by setting $c_i:\ =c(p_i)$ .

We introduce an important subclass of the class of all credence functions, namely, the (coherent) credence functions that match the truth values of ${\mathcal{F}}$ at a world $w$ exactly.

Definition 2.4 Fix an opinion set ${\mathcal{F}}$ . For each $w\in W$ , let $v_{w}:{\mathcal{F}}\to \{0,1\}$ be defined by $v_w(p)=1$ if and only if $w\in p$ . We call $v_w$ the omniscient credence function at world w. We let ${\mathcal{V}}_{{\mathcal{F}}}$ denote the set of all omniscient credence functions on ${\mathcal{F}}$ . Note that $|{\mathcal{V}}_{{\mathcal{F}}}|\leq 2^{|{\mathcal{F}}|}$ .

Next, we specify the inaccuracy measures that we will be concerned with in this section. Fix a finite opinion set ${\mathcal{F}}$ , and let ${\mathcal{C}}$ denote the set of credence functions on ${\mathcal{F}}$ . We define an inaccuracy measure to be a function of the form

$${\mathscr {I}}:{\mathcal {C}}\times W\to [0,\infty ].$$

The class of inaccuracy measures we consider is a generalization of the class defended by Pettigrew (2016): the inaccuracy measures defined in terms of what we call a quasi-additive Bregman divergence. It is a subclass of the inaccuracy measures assumed by Predd et al. (2009). ⁴

Definition 2.5 Suppose ${\mathfrak{D}}:[0,1]^n\times [0,1]^n\to [0,\infty ]$ .

1. 1. ${\mathfrak{D}}$ is a divergence if ${\mathfrak{D}}({\mathbf{x}},{\mathbf{y}})\geq 0$ for all ${\mathbf{x}},{\mathbf{y}}\in [0,1]^n$ with equality if and only if ${\mathbf{x}}={\mathbf{y}}$ .

2. 2. ${\mathfrak{D}}$ is quasi-additive if there exists a function ${\mathfrak{d}}:[0,1]^2\to [0,\infty ]$ and a sequence of elements $\{a_i\}_{i=1}^n$ from $(0,\infty )$ such that

   $${\mathfrak {D}}({\mathbf {x}},{\mathbf {y}})=\sum _{i\,=\,1}^na_i{\mathfrak {d}}(x_i,y_i),$$
   in which case we say ${\mathfrak{D}}$ is generated by ${\mathfrak{d}}$ and $\{a_i\}_{i\,=\,1}^n$ .

3. 3. ${\mathfrak{D}}$ is a quasi-additive Bregman divergence if ${\mathfrak{D}}$ is a quasi-additive divergence generated by ${\mathfrak{d}}$ and $\{a_i\}_{i=1}^n$ , and in addition, there is a function $\varphi :[0,1]\to {\mathbb{R}}$ such that:

   1. (a) $\varphi$ is continuous and strictly convex on $[0,1]$ ; ⁵

   2. (b) $\varphi$ is continuously differentiable on $(0,1)$ with the formal definition

      $$\varphi '(i):=\lim _{x\to i}\varphi '(x)$$
      for $i\in \{0,1\}$ ; ⁶ and

   3. (c) for all $x,y\in [0,1]$ , we have

      $${\mathfrak {d}}(x,y)=\varphi (x)-\varphi (y)-\varphi '(y)(x-y).$$
      We call such a ${\mathfrak{d}}$ a one-dimensional Bregman divergence.

We take the inaccuracy of a credence function $c$ at a world $w$ to be the distance between $c$ and the omniscient credence function $v_w$ , where distance is measured with a quasi-additive Bregman divergence.

Definition 2.6 Let a legitimate inaccuracy measure be an inaccuracy measure given by

$${\mathscr {I}}(c,w)={\mathfrak {D}}(v_w,c),$$

where ${\mathfrak{D}}$ is a quasi-additive Bregman divergence.

By allowing different weights depending on the proposition, we can accommodate the intuition that some propositions are more important to know than others. ⁷ Even if one thinks that inaccuracy measures should be additive, as Pettigrew (2016) does, relaxing this restriction makes our results more widely relevant. A popular example of an additive legitimate inaccuracy measure is the Brier score (see Section 12, “Homage to the Brier Score,” in Joyce 2009):

$${\mathscr {I}}(c,w)=\sum _{i\,=1}^n(v_w(p_i)-c(p_i))^2.$$

Remark 2.7 The class of additive Bregman divergences is the class of additive and continuous strictly proper scoring rules (See Pettigrew [2016, 66]; also see, for example, Banerjee et al. [2005] and Gneiting and Raftery [2007] for more details on Bregman divergences, as well as their connection to strictly proper scoring rules).

We now recall the dominance result connecting coherence to accuracy dominance when the opinion set is finite. It was first proved for the Brier score by de Finetti (1974, 87–90) and extended to any legitimate inaccuracy measure by Predd et al. (2009). ⁸ See Schervish et al. (2009) and Pettigrew (2022) (see also Nielsen 2022) for further generalizations of the finite result.

Definition 2.8 For each pair of credence functions $c, c^*$ over ${\mathcal{F}}$ :

1. 1. $c^*$ weakly dominates $c$ relative to an inaccuracy measure ${\mathscr{I}}$ if ${\mathscr{I}}(c,w)\geq {\mathscr{I}}(c^*,w)$ for all $w\in W$ and ${\mathscr{I}}(c,w)\gt {\mathscr{I}}(c^*,w)$ for some $w\in W$ ;

2. 2. $c^*$ strongly dominates $c$ relative to ${\mathscr{I}}$ if ${\mathscr{I}}(c,w)\gt {\mathscr{I}}(c^*,w)$ for all $w\in W$ .

Theorem 2.9 (de Finetti 1974; Predd et al. 2009) Let ${\mathcal{F}}$ be a finite opinion set, ${\mathscr{I}}$ be a legitimate inaccuracy measure, and $c$ be a credence function on ${\mathcal{F}}$ . Then the following are equivalent:

1. 1. $c$ is not strongly dominated;

2. 2. $c$ is not weakly dominated;

3. 3. $c$ is coherent.

Further, if $c$ is incoherent, then $c$ is strongly dominated by a coherent credence function.

On the basis of Theorem 2.9, authors in the accuracy literature conclude that an incoherent credence function is objectionable because there is an undominated coherent credence function that does strictly better in terms of accuracy, no matter how the world turns out to be, whereas coherent credence functions are not accuracy dominated in this way. Because it is the basis of the accuracy dominance argument for probabilism in the finite case, Theorem 2.9 is the result we would like to extend to infinite opinion sets. We now make progress toward this goal when ${\mathcal{F}}$ is countably infinite.

Note that any missing proofs in Sections 3 – 5 can be found in the appendix.

3. The countably infinite case: Coherence is necessary

3.1 Generalized legitimate inaccuracy measures

We begin with a discussion of how to measure inaccuracy in the countably infinite setting. Fix a countably infinite opinion set ${\mathcal{F}}$ over a set $W$ of worlds (of arbitrary cardinality). Let ${\mathcal{C}}$ be the set of credence functions over ${\mathcal{F}}$ , which can be identified with $[0,1]^\infty$ (see Remark 2.3). An inaccuracy measure remains a map from ${\mathcal{C}}\times W$ to $[0,\infty ]$ .

The class of inaccuracy measures that we use is defined in terms of generalizations of quasi-additive Bregman divergences.

Definition 3.1 Suppose ${\mathfrak{D}}:[0,1]^\infty \times [0,1]^\infty \to [0,\infty ]$ . Then we call ${\mathfrak{D}}$ a generalized quasi-additive Bregman divergence if

$${\mathfrak {D}}({\mathbf {x}},{\mathbf {y}})=\sum _{i=1}^\infty a_i {\mathfrak {d}}(x_i,y_i),$$

where ${\mathfrak{d}}$ is a bounded ⁹ one-dimensional Bregman divergence as in Definition 2.5.3 and $\{a_i\}_{i=1}^\infty$ is a sequence of elements from $ (0,\infty )$ with $\sup _{i}a_i\lt \infty$ . ¹⁰

Remark 3.2 Note that ${\mathfrak{d}}$ —defined in terms of $\varphi$ —being bounded is equivalent to $\varphi '$ being bounded on $[0,1]$ . Further, we may assume that $\varphi (0)=\varphi '(0)=0$ because ${\mathfrak{d}}_\varphi ={\mathfrak{d}}_{\bar{\varphi }}$ if $\varphi$ and $\bar{\varphi }$ differ by a linear function. ¹¹

In the appendix, I show that generalized quasi-additive Bregman divergences are examples of what Csiszár (1995) calls Bregman distances, which are generalizations of quasi-additive Bregman divergences defined on spaces of nonnegative functions.

We consider the suggestively named inaccuracy measures:

Definition 3.3 Given an enumeration of ${\mathcal{F}}$ , ¹² let a generalized legitimate inaccuracy measure be an inaccuracy measure ${\mathscr{I}}:{\mathcal{C}}\times W\to [0,\infty ]$ given by

(1) $$ {\mathscr{I}}(c,w)={\mathfrak{D}}(v_w,c) $$

for ${\mathfrak{D}}$ a generalized quasi-additive Bregman divergence. ¹³

Notice that the Brier score extends to a generalized legitimate inaccuracy measure, namely, the squared $\ell ^2({\mathcal{F}})$ norm

(2) $$ {\mathscr{I}}(c,w)=||v_w-c||_{\ell ^2({\mathcal{F}})}^2=\sum _{i\,=1}^\infty (v_w(p_i)-c(p_i))^2. $$

We call (2) the generalized Brier score.

The name “generalized legitimate inaccuracy measure” is motivated by the observation that a generalized legitimate inaccuracy measure naturally restricted to the finite opinion sets is a legitimate inaccuracy measure. This is because 1) for both the generalized and finite legitimate inaccuracy measures, the score of an individual proposition is defined by a one-dimensional Bregman divergence, and 2) for both the generalized and finite legitimate inaccuracy measures, the scores of individual propositions are combined in a weighted, additive way to give a score for the entire credence function. To use the terminology of Leitgeb and Pettigrew (2010a), in the finite and countably infinite setting, the local scores are the same, and the global scores relate to the local scores in the same way. These observations support the view that insofar as quasi-additive Bregman divergences are the appropriate functions to use for measuring inaccuracy in the finite setting, generalized quasi-additive Bregman divergences are the appropriate functions to use for measuring inaccuracy in the countably infinite setting (see Section 6 for further discussion of generalized legitimate inaccuracy measures). ¹⁴

3.2 Coherence is necessary

I now state one of the main results: coherence is necessary to avoid accuracy dominance in the countably infinite case.

Theorem 3.4 Let ${\mathcal{F}}$ be a countably infinite opinion set, ${\mathscr{I}}$ a generalized legitimate inaccuracy measure, and $c$ an incoherent credence function. Then:

1. 1. $c$ is weakly dominated relative to ${\mathscr{I}}$ by a coherent credence function; and

2. 2. if ${\mathscr{I}}(c,w)\lt \infty$ for each $w\in W$ , then $c$ is strongly dominated relative to ${\mathscr{I}}$ by a coherent credence function.

Remark 3.5 By analyzing the proof of Theorem 3.4, one can see that the most general way to state the theorem is as follows: assume $c$ is incoherent; if ${\mathscr{I}}(c,w)\lt \infty$ for some $w$ , then there is a coherent credence function $d$ such that ${\mathscr{I}}(d,w)\lt {\mathscr{I}}(c,w)$ for all $w$ such that ${\mathscr{I}}(c,w)\lt \infty$ ; if ${\mathscr{I}}(c,w)=\infty$ for all $w\in W$ , then any omniscient credence function weakly dominates $c$ .

Remark 3.6 The following is easy to prove from the results of Schervish et al. (2009): any incoherent credence function $c$ over a countably infinite opinion set is weakly dominated but not necessarily by a coherent credence function; and if ${\mathscr{I}}(c,w)\lt \infty$ for each $w\in W$ , then $c$ is strongly dominated but not necessarily by a coherent credence function. ¹⁵ Thus, the value in the proof strategy to come is that the dominating credence function is proven to be coherent, which is analogous to the finite case. ^{16 , 17}

Note that one direction of Walsh’s (ms) accuracy dominance result follows immediately from Theorem 3.4. We first recall his result.

Theorem 3.7 (Walsh ms) Let ${\mathcal{F}}$ be a countably infinite opinion set. Let

(3) $$ {\mathscr{I}}(c,w)=\sum _{i=1}^\infty 2^{-i}(v_w(p_i)-c(p_i))^2. $$

Then:

1. 1. if $c$ is incoherent, then $c$ is strongly dominated relative to ${\mathscr{I}}$ by a coherent credence function; and

2. 2. if $c$ is coherent, then $c$ is not weakly dominated relative to ${\mathscr{I}}$ by any credence function $d\ne c$ .

Part 1 of this result follows from Theorem 3.4 by defining ${\mathscr{I}}$ in terms of the generalized quasi-additive Bregman divergence generated by $\{2^{-i}\}_{i=1}^\infty$ and

$${\mathfrak {d}}(x,y)=x^2-y^2-2y(x-y)=\varphi (x)-\varphi (y)-\varphi '(y)(x-y),$$

where $\varphi (x)=x^2$ . Note that ${\mathscr{I}}(c,w)\lt \infty$ for all $c\in {\mathcal{C}}$ and $w\in W$ as $\sum _{i=1}^\infty 2^{-i}\lt \infty$ .

4. The countably infinite case: The sufficiency of coherence

Unlike coherent credence functions on finite opinion sets, coherent credence functions on countably infinite opinion sets can be strongly dominated.

Example 4.1 Let ${\mathcal{F}}=\{\{n\geq N:n\in {\mathbb{N}}\}:N\in {\mathbb{N}}\}$ be an opinion set over ${\mathbb{N}}$ (including zero). Let

$$c(\{n\geq N\})=\frac {{1}\over{\sqrt {N+1}}.$$

Then $c$ is coherent—in fact, countably coherent (see Definition 4.6)—but ${\mathscr{I}}(c,w)=\infty$ for all $w\in W$ when ${\mathscr{I}}$ is the generalized Brier score. Thus, any omniscient credence function strongly dominates $c$ .

In fact, the classic example of a merely finitely additive probability function—the 0-1 function defined on the finite-cofinite algebra over ${\mathbb{N}}$ taking value zero on finite sets—restricts to a coherent dominated credence function.

Example 4.2 Let ${\mathcal{F}}=\{\{n\leq N:n\in {\mathbb{N}}\}:N\in {\mathbb{N}}\}$ be an opinion set over ${\mathbb{N}}$ (including zero). Let

$$c(\{n\leq N\})=0.$$

Then $c$ is coherent—as well as finitely supported and not countably coherent—but ${\mathscr{I}}(c,w)=\infty$ for all $w\in W$ when ${\mathscr{I}}$ is the generalized Brier score. Thus, any omniscient credence function strongly dominates $c$ .

The goal of this section is to characterize the opinion sets and inaccuracy measures for which some variant of Theorem 2.9 holds. We can extend Theorem 2.9 by proving dominance results for countably coherent credence functions and using an opinion set compactification construction to transfer these results to merely coherent credence functions. At points, the results will only apply to the generalized Brier score. I conjecture that any such result extends to any generalized legitimate inaccuracy measure. In any case, this is a well-motivated restriction because the Brier score has been defended by many—including Horwich (1982), Maher (2002), Joyce (2009), and Leitgeb and Pettigrew (2010a)—as being a particularly appropriate way to measure inaccuracy.

A summary of the main results from Sections 3-4 can be found in Figure 4.1.

Figure 4.1. Summary of the results presented in Sections 3 and 4. Each column summarizes a main result by specifying conditions under which a particular conclusion about a credence function $c$ holds. The column titled “Opinion Space” specifies conditions on the opinion space on which $c$ is defined, the column titled “Inaccuracy Measure” specifies conditions on the generalized legitimate inaccuracy measure relative to which dominance is defined, and the column titled “Conclusion” specifies what can be deduced regarding the relationship between coherence and dominance for $c$ under the given conditions. “–” in a box means no additional condition is imposed.

Throughout the rest of Section 4 , we assume that the opinion set ${\mathcal{F}}$ is countably infinite, unless otherwise stated.

4.1 Countable coherence

I begin by introducing the notion of a countably coherent credence function and establishing a characterization theorem regarding countable coherence on countably discriminating opinion sets that extends a result of de Finetti (1974).

Definition 4.3 For an opinion set ${\mathcal{F}}\subseteq {\mathcal{P}}(W)$ , we define an equivalence relation $\sim$ on $W$ such that $w\sim w'$ if and only if $\{p\in {\mathcal{F}}:w\in p\}=\{p\in {\mathcal{F}}:w'\in p\}$ . We call the set of equivalence classes of $W$ the quotient of W relative to ${\mathcal{F}}$ . If the quotient of $W$ relative to ${\mathcal{F}}$ is countable, then we call ${\mathcal{F}}$ countably discriminating.

Clearly, any countable opinion set over a countable set of worlds is countably discriminating.

The following characterization of the coherent credence functions on finite opinion sets is due to de Finetti (1974). Recall that ${\mathcal{V}}_{{\mathcal{F}}}$ is the set of omniscient credence functions on ${\mathcal{F}}$ , which is finite when ${\mathcal{F}}$ is finite.

Theorem 4.4 (de Finetti 1974) $c$ is a coherent credence function on a finite opinion set ${\mathcal{F}}$ if and only if there are $\lambda _w\in [0,1]$ with $\sum _{v_w\in {\mathcal{V}}_{{\mathcal{F}}}}\lambda _w=1$ such that

$$c(p)=\sum _{\upsilon_w\in {\mathcal {V}}_{{\mathcal {F}}}}\lambda _w \upsilon_{w}(p)$$

for all $p\in {\mathcal{F}}$ .

Theorem 4.4 is integral to Predd et al.’s (2009) proof that coherence is sufficient to avoid dominance in Theorem 2.9. I now show de Finetti’s (1974) characterization of the coherent credence functions on finite opinion sets extends to countably coherent credence functions on countably infinite opinion sets that are countably discriminating.

Definition 4.5 A $\sigma$ -algebra over $W$ is a subset ${\mathcal{F}}^*\subseteq {\mathcal{P}}(W)$ such that:

1. 1. $W\in {\mathcal{F}}^*$ ;

2. 2. if $\{p_i\}_{i\,=1}^\infty \subseteq {\mathcal{F}}^*$ , then $\bigcup _{i\,=1}^\infty p_i\in {\mathcal{F}}^*$ ; and

3. 3. if $p\in {\mathcal{F}}^*$ , then $W\setminus p\in {\mathcal{F}}^*$ .

Definition 4.6 Let a credence function $c$ be countably coherent if $c$ extends to a countably additive probability function on a $\sigma$ -algebra ${\mathcal{F}}^*$ containing ${\mathcal{F}}$ . ¹⁸ That is, there is a $c^*:{\mathcal{F}}^*\to [0,1]$ such that:

1. 1. $c^*(p)=c(p)$ for all $p\in {\mathcal{F}}$ ;

2. 2. $c^*(\bigcup _{i\,=1}^\infty p_i)=\sum _{i\,=1}^\infty c^*(p_i)$ for $\{p_i\}_{i\,=1}^\infty \subseteq {\mathcal{F}}^*$ with $p_i\cap p_j=\varnothing$ for $i\ne j$ ; and

3. 3. $c^*(W)=1$ .

Otherwise, a credence function is countably incoherent.

Proposition 4.7 Let ${\mathcal{F}}$ be a countably infinite opinion set that is countably discriminating (thus, $V_{{\mathcal{F}}}$ is countable). Then a credence function $c$ is countably coherent if and only if there are $\lambda _{v_w}\in [0,1]$ with $\sum _{v_w\in {\mathcal{V}}_{{\mathcal{F}}}}\lambda _{v_w}=1$ such that

$$c(p)=\sum _{\upsilon_w\in {\mathcal {V}}_{{\mathcal {F}}}} \lambda _{\upsilon_w} v_w(p)$$

for all $p\in {\mathcal{F}}$ .

4.2 Compactification of an opinion space

In this section, I introduce the compactification construction of what is called an opinion space. The construction will be relevant to transferring dominance results for countably coherent credence functions to merely coherent credence functions, the reason being that merely coherent credence functions become countably coherent if the underlying set of worlds is “compactified.”

Definition 4.8 An opinion space is a pair $(W,{\mathcal{F}})$ , where $W$ is a nonempty set and ${\mathcal{F}}\subseteq {\mathcal{P}}(W)$ .

From here on out, I will speak in terms of opinion spaces as opposed to opinion sets in order to keep track of the underlying set of worlds. We continue to assume that ${\mathcal{F}}$ is countably infinite.

Borkar et al. (2003) proved that the opinion spaces that satisfy a certain compactness property are precisely those where the set of coherent credence functions and the set of countably coherent credence functions coincide. ¹⁹

Definition 4.9 Let $(W,{\mathcal{F}})$ be an opinion space. Let $f(n)\in \{0,1\}$ and set $p_n^{f(n)}=p_n$ if $f(n)=0$ and $p_n^{f(n)}=p_n^c$ if $f(n)=1$ . Then $(W,{\mathcal{F}})$ is compact if for any choice of $\{p_n\}_{n=1}^\infty \subseteq {\mathcal{F}}$ and $f:{\mathbb{N}}\to \{0,1\}$ , if $\bigcap _{n=1}^Np_n^{f(n)}$ is nonempty for every $N$ , then $\bigcap _{n=1}^\infty p_n^{f(n)}$ is nonempty.

As an example, note that the opinion spaces from Examples 4.1 and 4.2 are not compact. Indeed, for the first example, $\bigcap _{n\,=1}^\infty p_n=\varnothing$ , and yet every finite subset of ${\mathcal{F}}$ has a nonempty intersection; for the second example, $\bigcap _{n\,=1}^\infty p_n^c=\varnothing$ , whereas $\bigcap _{n\,=1}^Np_n^c\ne \varnothing$ for every $N$ .

Theorem 4.10 (Borkar et al. 2003) The following are equivalent:

1. 1. $(W,{\mathcal{F}})$ is compact;

2. 2. for every credence function $c$ on $(W,{\mathcal{F}})$ , $c$ is coherent if and only if $c$ is countably coherent.

I now show how to turn any space into a compact space and, in light of Theorem 4.10, any coherent credence function into a countably coherent credence function. Let $(W,{\mathcal{F}})$ be an opinion space. Let $S$ denote the set of sequences of the form $\{p_n^{f(n)}\}$ (as in Definition 4.9) such that $\bigcap _{n=1}^Np_n^{f(n)}\ne \varnothing$ for every $N$ but $\bigcap _{n=1}^\infty p_n^{f(n)}=\varnothing$ . Define $W^*=W\cup \{x_s:s\in S\}$ , where each $x_s$ is a formal point corresponding to the element $s\in S$ . Define ${\mathcal{F}}^*\subseteq {\mathcal{P}}(W^*)$ as follows: for each $p\in {\mathcal{F}}$ , let $S_p$ denote the set of sequences $s$ of the form $\{p_n^{f(n)}\}$ (as in Definition 4.9) such that $s\in S$ , $p_n=p$ for some $n$ , and $f(n)=0$ . Then define

$$p^*=p\cup \{x_s:s\in S_p\}.$$

Finally, let ${\mathcal{F}}^*=\{p^*:p\in {\mathcal{F}}\}$ . We call $(W^*, {\mathcal{F}}^*)$ the compactification of $(W,{\mathcal{F}})$ . We always denote the compactification of $(W,{\mathcal{F}})$ by $(W^*,{\mathcal{F}}^*)$ . Further, we let $\Psi$ denote the natural bijection from ${\mathcal{F}}$ to ${\mathcal{F}}^*$ given by $\Psi (p)=p^*$ .

We first note that $(W^*,{\mathcal{F}}^*)$ is in fact compact.

Lemma 4.11 For $(W,{\mathcal{F}})$ an opinion space, $(W^*,{\mathcal{F}}^*)$ is compact.

Next, we note that, as suggested, we can naturally turn a coherent credence function into a countably coherent credence function by compactifying the underlying opinion space.

Lemma 4.12 Let $(W,{\mathcal{F}})$ be an opinion space and $c$ be a coherent credence function on $(W,{\mathcal{F}})$ . Let $(W^*,{\mathcal{F}}^*)$ be the compactification of $(W,{\mathcal{F}})$ , and define $c^*(\Psi (p)):=c(p)$ for each $p\in {\mathcal{F}}$ . Then $c^*$ is a countably coherent credence function on $(W^*,{\mathcal{F}}^*)$ , and ${\mathscr{I}}(c,w)={\mathscr{I}}(c^*,w)$ for $w\in W$ .

For a coherent credence function $c$ defined on an opinion space $(W,{\mathcal{F}})$ , we let $c^*$ denote the countably coherent credence function on $(W^*,{\mathcal{F}}^*)$ given as in Lemma 4.12.

Example 4.13 As an example, let us compute the compactification of the opinion space from Example 4.2 and show how to identify a coherent credence function on the space with a countably coherent credence function on its compactification. We note that only for $f(n)=1$ for all $n\in {\mathbb{N}}$ is $\bigcap _{n=1}^Np_n^{f(n)}$ nonempty for every $N$ while $\bigcap _{n=1}^\infty p_n^{f(n)}=\varnothing$ . Indeed, assume $f(m)=0$ for some $m$ . If $f(i)=1$ for some $i\geq m+1$ , then because $p_i^c\cap p_m=\varnothing$ , we have $\bigcap _{n=1}^ip_n^{f(n)}=\varnothing$ which contradicts our assumption. So $f(i)=0$ for all $i\geq m+1$ . But then because $\bigcap _{n=1}^mp_n^{f(n)}\ne \varnothing$ and $\bigcap _{n=1}^mp_n^{f(n)}\subseteq p_i$ for all $i\geq m+1$ , it also follows that $\bigcap _{n=1}^\infty p_n^{f(n)}\ne \varnothing$ , which contradicts our assumption. So $S$ is a single point $x$ , $W^*=W\cup \{w^*\}$ , and ${\mathcal{F}}^*=\{\{n\leq N\}:N\in {\mathbb{N}}\}$ . ${\mathcal{F}}^*$ is identical to ${\mathcal{F}}$ , except that there is a point in the complement of every proposition in ${\mathcal{F}}^*$ . For a coherent credence function $c$ on $(W,{\mathcal{F}})$ , $c^*$ on $(W^*,{\mathcal{F}}^*)$ is identical to $c$ and is a countably coherent credence function on the compact opinion space $(W^*,{\mathcal{F}}^*)$ . For example, for credence function $c$ in Example 4.2, $c^*$ extends to the countably additive omniscient credence function $v_{w^*}$ on the $\sigma$ -algebra generated by ${\mathcal{F}}^*$ .

Using Theorem 4.10, Lemma 4.11, and Lemma 4.12, the proof strategy for extending Theorem 2.9 is more precisely as follows. First, we establish dominance results for countably coherent credence functions. Second, we transform each coherent credence function on $(W,{\mathcal{F}})$ into a countably coherent credence function on $(W^*,{\mathcal{F}}^*)$ , as in Lemma 4.12. Lastly, we use the dominance results for countably coherent credence functions to establish dominance results for coherent credence functions in certain cases where there is “accuracy dominance stability” in compactifying.

4.3 W-stable opinion spaces

In this section, we establish the equivalence between coherence and avoiding weak dominance for certain countably infinite opinion spaces (Theorem 4.19), as well as additional results extending Theorem 2.9 to the countably infinite setting (corollary 4.22 and Theorem 4.23). We first note that under certain circumstances, countably coherent credence functions are not weakly dominated (Proposition 4.15 and Proposition 4.16); then I use the compactification construction from the previous section and a property of an opinion space—W-stability (Definition 4.17)—to establish that for certain opinion spaces, mere coherence is also sufficient to avoid weak dominance.

I first prove that if a countably coherent credence function $c$ has finite expected inaccuracy, then $c$ is not weakly dominated.

Definition 4.14 For $c$ , a countably coherent credence function, and ${\mathscr{I}}$ , a generalized legitimate inaccuracy measure, we say that $c$ has finite expected inaccuracy relative to ${\mathscr{I}}$ if $c$ has a countably additive extension $\bar{c}$ defined on the opinion space $(W,\sigma ({\mathcal{F}}))$ such that ${\mathbb{E}}_{\bar{c}}{\mathscr{I}}(c,\cdot )\lt \infty$ . ²⁰ For $c$ , a coherent but not countably coherent credence function, and ${\mathscr{I}}$ , a generalized legitimate inaccuracy measure, we say that $c$ has finite expected inaccuracy relative to ${\mathscr{I}}$ if $c^*$ has finite expected inaccuracy relative to ${\mathscr{I}}$ .

Proposition 4.15 Let $(W,{\mathcal{F}})$ be an opinion space and ${\mathscr{I}}$ be a generalized legitimate inaccuracy measure. If $c$ is a countably coherent credence function on $(W, {\mathcal{F}})$ with finite expected inaccuracy relative to ${\mathscr{I}}$ , then $c$ is not weakly dominated relative to ${\mathscr{I}}$ .

Here is another dominance result for countably coherent credence functions where we assume ${\mathcal{F}}$ is point-finite $(|\{p\in {\mathcal{F}}:w\in p\}|\lt \infty$ for all $w\in W$ ) but weaken the assumption that $c$ has finite expected inaccuracy considerably, namely, to somewhere finitely inaccurate (there is a $w\in W$ such that ${\mathscr{I}}(c,w)\lt \infty$ ). We also restrict to the generalized Brier score ${\mathscr{B}}$ .

Proposition 4.16 Let $(W,{\mathcal{F}})$ be a point-finite opinion space. If a credence function $c$ is countably coherent and somewhere finitely inaccurate relative to ${\mathscr{B}}$ , then $c$ is not weakly dominated relative to ${\mathscr{B}}$ .

I now introduce the notion of W-stability, which will allow the use of Propositions 4.15 and 4.16 to prove extensions of Theorem 2.9. Intuitively, an opinion space is W-stable if coherence is preserved under compacitification.

Definition 4.17 Let $(W,{\mathcal{F}})$ be W-stable relative to ${\mathscr{I}}$ if for any coherent credence function $c$ on $(W,{\mathcal{F}})$ , if $c$ is weakly dominated relative to ${\mathscr{I}}$ , then $c^*$ on $(W^*,{\mathcal{F}}^*)$ is weakly dominated relative to ${\mathscr{I}}$ .

Remark 4.18 Not all opinion spaces are W-stable relative to every generalized legitimate inaccuracy measure. Indeed, consider the opinion space from Example 4.2. The credence function $c$ in that example, which assigns $0$ to each proposition, is strongly dominated. However, where $w^*$ is the single world added to the complement of each proposition when compactifying (see Example 4.13) and $v_{w^*}:{\mathcal{F}}^*\to [0,1]$ is the omniscient credence function at world $w^*$ , we have that $c^*=v_{w^*}$ , and so ${\mathscr{B}}(c^*,w^*)=0$ . Further, for any credence function $d\ne c^*$ , we have ${\mathscr{B}}(d,w^*)\gt 0$ because ${\mathscr{B}}(d,w^*)=0$ if and only if $d=v_{w^*}=c^*$ . Thus, once the underlying space is compactified, $c$ is no longer weakly dominated relative to ${\mathscr{B}}$ .

Using Proposition 4.15, we establish one of the main results: sufficient and partly necessary conditions on an opinion space for coherence to be equivalent to not being weakly dominated.

Theorem 4.19 Let ${\mathscr{I}}$ be a generalized legitimate inaccuracy measure and $(W,{\mathcal{F}})$ a W-stable opinion space relative to ${\mathscr{I}}$ where all coherent credence functions have finite expected inaccuracy relative to ${\mathscr{I}}$ . Then the following are equivalent:

1. 1. $c$ is coherent;

2. 2. $c$ is not weakly dominated.

Proof We prove that if $c$ is coherent, then $c$ is not weakly dominated. Let $(W^*,{\mathcal{F}}^*)$ be the compactification of $(W,{\mathcal{F}})$ . If $c$ is coherent on $(W,{\mathcal{F}})$ , then $c^*$ is countably coherent by Lemma 4.12. Further, $c^*$ has finite expected inaccuracy by definition and the assumption that $c$ has finite expected inaccuracy. Thus, by Proposition 4.15, $c^*$ is not weakly dominated. But since $(W,{\mathcal{F}})$ is W-stable, this implies that $c$ is not weakly dominated. The other direction follows from Theorem 3.4. □

Remark 4.20 It is trivial to see that W-stability is necessary for the equivalence of coherence and not being weakly dominated. How far finite expected inaccuracy can be weakened is an open question.

Remark 4.21 If ${\mathscr{I}}$ is defined with summable weights, that is, $\{a_i\}_{i=1}^\infty$ such that $\sum _{i=1}^\infty a_i\lt \infty$ , then there is a $C\lt \infty$ such that ${\mathscr{I}}(c,w)\lt C$ for all credence functions $c$ and $w\in W$ . So, in particular, all coherent credence functions have finite expected inaccuracy relative to ${\mathscr{I}}$ .

If we add in an additional finiteness assumption, then we get the full equivalence of Theorem 2.9.

Corollary 4.22 In Theorem 4.19, if in addition, all coherent credence functions $c$ have ${\mathscr{I}}(c,w)\lt \infty$ for all $w\in W$ , then the following are equivalent:

1. 1. $c$ is coherent;

2. 2. $c$ is not weakly dominated;

3. 3. $c$ is not strongly dominated.

We combine W-stability and Proposition 4.16 to get another set of sufficient conditions on $(W,{\mathcal{F}})$ for Theorem 2.9 to go through for the generalized Brier score.

Theorem 4.23 Let $(W,{\mathcal{F}})$ be a W-stable opinion space with $(W^*,{\mathcal{F}}^*)$ point-finite such that all coherent credence functions on $(W,{\mathcal{F}})$ are somewhere finitely inaccurate relative to ${\mathscr{B}}$ . Then the following are equivalent:

1. 1. $c$ is coherent;

2. 2. $c$ is not weakly dominated relative to ${\mathscr{B}}$ ;

3. 3. $c$ is not strongly dominated relative to ${\mathscr{B}}$ .

Remark 4.24 We can drop the assumption that all coherent credence functions are somewhere finitely inaccurate in Theorem 4.23 if we strengthen W-stable to compact so that $(W,{\mathcal{F}})=(W^*,{\mathcal{F}}^*)$ . Indeed, compactness alongside point-finiteness implies that coherent credence functions on $(W,{\mathcal{F}})=(W^*,{\mathcal{F}}^*)$ are somewhere finitely inaccurate: if there were a coherent (and thus countably coherent) credence function infinitely inaccurate at all worlds, then it would be strongly dominated by an omniscient credence function, contradicting Proposition 4.27 later in this section.

4.3.1 Partitions

As an application of Theorem 4.19, I establish Theorem 2.9 for countably infinite partitions. ²¹ In parts of the existing literature (e.g., Joyce 2009), credence functions are assumed to be defined on a (finite) partition of $W$ to begin with, and so such a result might be especially relevant to extending the accuracy dominance argument for probabilism to countably infinite opinion sets.

Lemma 4.25 A countably infinite partition is W-stable relative to any generalized legitimate inaccuracy measure.

Theorem 4.26 Let $(W,{\mathcal{F}})$ be a countably infinite partition and ${\mathscr{I}}$ be a generalized legitimate inaccuracy measure. Then the following are equivalent:

1. 1. $c$ is coherent;

2. 2. $c$ is not weakly dominated;

3. 3. $c$ is not strongly dominated.

4.4 S-stable opinion spaces

In this section, I establish the equivalence between coherence and avoiding strong dominance for certain countably infinite opinion spaces (Theorem 4.29). The conditions are in terms of the analogous stability condition—S-stability (Definition 4.28)—but a different finiteness assumption, and the proof strategy is the same as for Theorem 4.19.

I begin by establishing that on compact countably infinite opinion spaces, coherent and thus countably coherent credence functions (recall Theorem 4.10) are not strongly dominated.

Proposition 4.27 Let $(W,{\mathcal{F}})$ be a compact opinion space and ${\mathscr{I}}$ be a generalized legitimate inaccuracy measure. If $c$ is coherent (and thus countably coherent), then $c$ is not strongly dominated relative to ${\mathscr{I}}$ .

I now introduce S-stability and the main theorem of this section.

Definition 4.28 Let $(W,{\mathcal{F}})$ be S-stable relative to ${\mathscr{I}}$ if for any coherent credence function $c$ on $(W,{\mathcal{F}})$ , if $c$ is strongly dominated relative to ${\mathscr{I}}$ , then $c^*$ on $(W^*,{\mathcal{F}}^*)$ is strongly dominated relative to ${\mathscr{I}}$ .

Theorem 4.29 Let ${\mathscr{I}}$ be a generalized legitimate inaccuracy measure and $(W,{\mathcal{F}})$ be an S-stable opinion space relative to ${\mathscr{I}}$ . Assume that ${\mathscr{I}}(c,w)\lt \infty$ for each coherent credence function $c$ and $w\in W$ . Then the following are equivalent:

1. 1. $c$ is coherent;

2. 2. $c$ is not strongly dominated.

Proof Assume $c$ is coherent. $c^*$ defined on the compact opinion space $(W^*,{\mathcal{F}}^*)$ is countably coherent by Lemma 4.12. Thus, by Proposition 4.27, $c^*$ is not strongly dominated. But since ${\mathcal{F}}$ is S-stable, this implies that $c$ is not strongly dominated relative to ${\mathscr{I}}$ . The other direction follows from Theorem 3.4. □

Remark 4.30 It is trivial to see that S-stability is necessary for the equivalence of coherence and avoiding strong dominance. How much the assumption that coherent credence functions satisfy ${\mathscr{I}}(c,w)\lt \infty$ for all $w$ can be weakened is an open question.

Remark 4.31 Schervish et al. (2009) take a different approach to dropping the assumption that the opinion set is finite: they apply weak and strong dominance notions to finite subsets of opinion sets of arbitrary cardinality. They also explore connections between the two notions of dominance considered here—weak and strong dominance—and what they call coherence $_1$ , which amounts to avoiding being susceptible to a finite Dutch book. ²²

Remark 4.32 Theorem 4.29 is related to Theorem 1 of Schervish et al. (2014). However, 1) their assumptions are in some ways weaker and in some ways stronger than those in Theorem 4.29, ²³ and 2) although Schervish et al. (2014) establish that coherence is sufficient for avoiding strong dominance in certain cases, unlike Theorems 4.19 and 4.29, their results do not show that coherence is sufficient for avoiding even weak dominance in certain cases or that incoherence always entails being weakly dominated (and sometimes strongly dominated) by a coherent credence function (see Remark 3.6).

For a summary of the results established thus far, see Figure 4.1.

4.5 Further directions

Although Theorems 4.19 and 4.29 come close to characterizing the countably infinite opinion spaces on which not being weakly and strongly dominated, respectively, are equivalent to coherence, how far the finiteness assumptions in the theorems can be weakened is an open question. This is a natural next line of inquiry. In addition, it would be useful to determine characterizations of W- and S-stability in terms of the inaccuracy measure that would make it relatively easy to check whether an opinion set is W- or S-stable. For example, might it be that an opinion space is W- and S-stable relative to a generalized legitimate inaccuracy measure if the generalized legitimate inaccuracy measure only outputs finite scores for credence functions on that opinion space? Also, there are natural ways to generalize the previous results to more closely match the finite results: allow different one-dimensional Bregman divergences for different propositions and allow unbounded one-dimensional Bregman divergences.

Another direction one could go in exploring the sufficiency of coherence for avoiding dominance is as follows: instead of characterizing the countably infinite opinion sets on which Theorem 2.9 goes through, one could characterize the kinds of coherent credence functions for which Theorem 2.9 goes through on any countably infinite opinion set. ²⁴ Doing so might show that although coherence is not enough to avoid dominance in all cases, coherence along with additional plausible constraints is sufficient. In particular, although restricting to finitely supported credence functions is not enough to establish the sufficiency of coherence for avoiding strong dominance (as shown in Example 4.2), whether countable coherence is equivalent to avoiding weak or strong dominance on the restricted class is an open question.

5. The uncountable case

So far, we have been concerned with credences defined on countable opinion sets. We now consider what can be said in favor of probabilism when credences are defined on uncountable opinion sets, though much of what is said will be preliminary. When extending from the finite to the countably infinite setting, we used inaccuracy measures that naturally restrict to legitimate inaccuracy measures in the finite case. When extending from the countable to the uncountable setting, we will use inaccuracy measures defined as integration against a measure on the uncountable set of propositions. This is a natural generalization of (generalized) legitimate inaccuracy measures, for (generalized) legitimate inaccuracy measures are defined as integration against a particular kind of measure on a countable set of propositions. Indeed, upon inspection, one can see that in the finite and countably infinite setting, a (generalized) legitimate inaccuracy measure defined in terms of weights $\{a_i\}$ and one-dimensional Bregman divergence ${\mathfrak{d}}$ is given by integrating ${\mathfrak{d}}(v_w(\cdot ),c(\cdot ))$ as a function of ${\mathcal{F}}=\{p_1, p_2,\ldots \}$ against the measure $\mu$ on ${\mathcal{F}}$ defined by $\mu (A)=\sum _{p_i\in A}a_i$ for $A\in {\mathcal{P}}({\mathcal{F}})$ . We generalize this construction for uncountable ${\mathcal{F}}$ by defining the legitimate inaccuracy measures to be given by integration of a one-dimensional Bregman divergence ${\mathfrak{d}}(v_w(\cdot ),c(\cdot ))$ as a function of ${\mathcal{F}}$ against a measure $\mu$ defined on ${\mathcal{F}}$ .

Due to the measure-theoretic construction of the inaccuracy measures we consider, we restrict our attention to measurable credence functions and equate credence functions that are equal almost everywhere. On some measure spaces, such as the weighted counting measure spaces (with all nonzero weights) underlying (generalized) legitimate inaccuracy measures, we lose nothing since every credence function is measurable and only the empty set is measure zero. However, in other cases, these assumptions are substantive. I discuss this issue further after stating the main theorem of this section (Theorem 5.3).

We now formally extend the accuracy framework to the measure theoretic setting.

Definition 5.1 Let ${\mathcal{F}}\subseteq {\mathcal{P}}(W)$ be an opinion set, $({\mathcal{F}},{\mathcal{A}},\mu )$ a measure space, ²⁵ and $c:{\mathcal{F}}\to {\mathbb{R}}^+$ . If $c$ is ${\mathcal{A}}$ -measurable and $\mu (\{p:c(p)\notin [0,1]\})=0$ , we call $c$ a $\mu$ -credence function. We say that a $\mu$ -credence function $c$ is $\mu$ -coherent if there is a coherent (in the usual sense) credence function $c'$ on ${\mathcal{F}}$ with $c=c'$ $\mu$ -a.e. We say a $\mu$ -credence function $c$ is $\mu$ -incoherent if there is no coherent credence function $c'$ such that $c=c'$ $\mu$ -a.e.

Definition 5.2 Let ${\mathcal{F}}$ be an opinion set (of arbitrary cardinality) over a set $W$ of worlds. Let $({\mathcal{F}},{\mathcal{A}},\mu )$ be a $\sigma$ -finite measure space over the opinion set ${\mathcal{F}}$ . Let ${\mathcal{C}}$ be the space of all $\mu$ -credence functions. Assume ${\mathscr{I}}:{\mathcal{C}}\times W\to [0,\infty ]$ is such that, for all $(c,w)\in {\mathcal{C}}\times W$ , we have

$${\mathscr {I}}(c,w)=B_{\varphi,\mu }(\upsilon_w,c),$$

where $B_{\varphi,\mu }$ is a Bregman distance relative to $\varphi$ ²⁶ and $({\mathcal{F}},{\mathcal{A}},\mu )$ (see Definition A.1). In particular, each $v_w$ is a $\mu$ -credence function. Then we call ${\mathscr{I}}$ an integral inaccuracy measure on $({\mathcal{F}},{\mathcal{A}},\mu )$ .

Here is a dominance result for integral inaccuracy measures. The proof is essentially a measure theoretic version of the proof of Theorem 3.4.

Theorem 5.3 Let ${\mathscr{I}}$ be an integral inaccuracy measure on a finite measure space $({\mathcal{F}},{\mathcal{A}},\mu )$ . ²⁷ Then for every $\mu$ -credence function $c$ , if $c$ is $\mu$ -incoherent, then there is a $\mu$ -coherent $\mu$ -credence function $c'$ that strongly dominates $c$ relative to ${\mathscr{I}}$ .

It is worth noting that this result does not show that every incoherent credence function is strongly dominated because not every incoherent credence function is a $\mu$ -incoherent credence function: there can be incoherent credence functions that are $\mu$ -a.e. equivalent to coherent credence functions, even ones that are undominated insofar as any coherent credence function is undominated. ²⁸ This is to be expected because, as discussed earlier, in moving to a measure-theoretic framework, measure-zero differences between credence functions will not be detected as far as accuracy is concerned, and thus incoherent credence functions that deviate from coherence by a measure-zero set will have the same inaccuracy scores as coherent credence functions. Thus, an accuracy dominance argument in this measure-theoretic setting will at most establish that one ought to have a credence function that is coherent off a measure-zero set (i.e., is $\mu$ -coherent). Theorem 5.3 is a first step toward such an argument. I leave for future work a consideration of different ways to extend inaccuracy measures to the uncountable setting that might establish not just $\mu$ -coherence but (strict) coherence.

Here is an example of how Theorem 5.3 can be used to give an accuracy argument in a concrete uncountable setting. Assume we have a coin with unknown bias $\theta \in [0,1]$ and a set of propositions of the form “ $a \leq \theta \leq b$ ” for each $a,b\in [0,1]$ with $a\leq b$ . Then a credence function on this uncountable opinion set can be represented by a function

$$c:X\to [0,1],$$

where $X=\{(a,b):0\leq a\leq b\leq 1\}\subseteq [0,1]^2$ . We put the Lebesgue measure on $X$ to generalize the additive constraint often assumed in the finite case. We let

$${\mathscr {I}}(c,w)=\int _X {\mathfrak {d}}(v_w({\bf {X}}),c({\bf {X}}))\lambda (d{\bf {X}})$$

for a bounded one-dimensional Bregman divergence ${\mathfrak{d}}$ . Then the assumptions of Theorem 5.3 hold, so we get the following dominance result: for any $\lambda$ -credence function $c$ , if $c$ is $\lambda$ -incoherent, then there is a $\lambda$ -coherent $\lambda$ -credence function that strongly dominates $c$ .

6. Discussion

I now briefly consider the difficult question of what normative conclusions to draw from the results of Sections 3–5.

6.1 Extending the finite accuracy dominance argument for probabilism

To begin, it has been shown by Schervish et al. (2009, Theorem 1) that a credence function on an opinion set of arbitrary cardinality is coherent if and only if its restriction to any finite subset of the opinion set is not strongly dominated. One might wonder whether this result suffices for an accuracy-based argument for probabilism on an infinite opinion set. ²⁹ Why care, in addition, about dominance relations when all infinitely many of the agent’s credences are scored at once?

We need to separate the questions of whether there is an accuracy-based argument for probabilism in the infinite setting and whether there is an extension to the infinite setting of the accuracy dominance arguments for probabilism already established in the finite setting. Although there are differences between the various accuracy dominance arguments for probabilism in the finite setting, they share the key feature of appealing to the following mathematical fact: relative to some legitimate way of scoring the inaccuracy of credence functions over a given opinion set, every incoherent credence function is accuracy dominated by a coherent credence function, and no coherent credence function is accuracy dominated by any other credence function. ³⁰ Motivating the appeal to this mathematical fact is a commitment to the epistemic value of having an overall accurate epistemic state, where the epistemic state in question is represented by a credence function.

By suggesting that the epistemic value underlying the finite accuracy dominance arguments is that of overall accuracy, I do not mean to suggest that inaccuracy measures ought to be sensitive to what Pettigrew (2016) calls the “irreducibly global features” of a credence function (see 49–50). To suggest so would likely conflict with allowing an inaccuracy measure to be additive, which I do here. Indeed, Pettigrew motivates additivity by suggesting that inaccuracy need not be sensitive to global features because credence functions are not unified doxastic states but simply agglomerations of individual credences. ³¹ The thought here, however, is that in the finite accuracy dominance argument, all credences in the agglomeration contribute to the accuracy scores of the representing credence function (perhaps differentially weighted), which are then analyzed for dominance relations. Doing so is motivated by the epistemic ideal of an accuracy undominated epistemic state, where every credal state in the epistemic state is accounted for in scoring. By similarly requiring every credal state in an infinite agglomeration to contribute to the accuracy scores of the representing credence function that are then analyzed for dominance relations, I prove results that allow one to extend the accuracy dominance argument while retaining this underlying motivating ideal.

Now, Schervish et al.’s Theorem 1 alone does not establish the cited crucial mathematical fact when credence functions are defined over an infinite opinion set, for it suggests no way to score credence functions over infinite opinion sets in the first place. Therefore, an appeal to their result would not be motivated by a commitment to the epistemic value of having an overall accurate epistemic state, a commitment that underwrites the existing accuracy dominance arguments for probabilism. Moreover, while their Theorem 1 establishes that for an incoherent credence function $c$ , the restriction of $c$ to any finite subset on which it is incoherent is strictly dominated by a coherent credence function, the dominating credence function may depend on the finite subset. ³² Thus, their result does not provide even an alternative sense in which every incoherent credence function is dominated by a single coherent credence function, which is the kind of result appealed to in the argument for finite accuracy dominance.

Of course, even if a local dominance result like that of Schervish et al. cannot be used to extend to the infinite setting what is often referred to as the accuracy dominance argument for probabilism, it could be used to establish a different accuracy-based argument for probabilism in the infinite setting. Thus, we should not assume that from an accuracy perspective, probabilism in the infinite setting will stand or fall based on the results in this article—that depends, for instance, on whether it is irrational in itself to be in an epistemic state where restrictions of one’s representing credence function are accuracy dominated. My point is simply that whatever accuracy-based arguments one can give for probabilism in the infinite setting, it is important to determine whether the influential set of arguments that are often referred to under the single heading of “the accuracy dominance argument for probabilism” must be restricted to finite opinion sets. In light of these considerations, to make clear that I will be concerned with comparing the “total” inaccuracy scores of credence functions, let us call the inaccuracy (resp., accuracy) of an agent’s entire epistemic state—that is, the inaccuracy of the agent’s full credence function—the total inaccuracy (resp., total accuracy) of the agent’s epistemic state.

6.2 The status of the accuracy dominance argument for probabilism on infinite opinion sets

Given the results of the last three sections, what should we conclude about the accuracy dominance argument for probabilism and more generally about the accuracy framework applied to credence functions defined on infinite opinion sets? When an agent’s epistemic state contains only finitely many credences, the concern for total accuracy leads to a dominance justification for probabilism (using Theorem 2.9). However, we saw earlier that when the agent’s epistemic state includes even countably infinitely many credences, the concern for total accuracy does not so clearly lead to a dominance justification for probabilism because coherence is not sufficient to avoid accuracy dominance in all cases (as in Examples 4.1 and 4.2). I suggest that the main normative challenges at this point are i) to clarify this asymmetry in the relationship between coherence and total accuracy dominance for finite and infinite opinion sets and ii) to determine what this asymmetry means for the accuracy framework more generally.

One response is to deny that real-world agents ever have infinitely many credences at a time, and thus this entire discussion is mere ideal theory. The empirical claim that real-world agents are restricted to finite opinion sets is not obvious, however. For example, if there were a coin in front of me of unknown bias, could I not have credences in each of the propositions “the bias of the coin is $x$ ” for $x\in [0,1]$ ? ³³ Or could I not have credence $2^{-n}$ in the proposition “the coin would land heads $n$ times in a row if I flipped it $n$ times in a row” for each $n\in {\mathbb{N}}$ ? More generally, it seems we can have credences parameterized by some infinite set such as the natural numbers. Thus, this kind of objection to considering infinite opinion sets at all seems unlikely to work or, at the very least, requires further defense.

Alternatively, one could argue that any asymmetry is unimportant with regard to probabilism: while the results show that some coherent credence functions are ruled out as irrational on the basis of total accuracy dominance, all incoherent credence functions are ruled out on the same basis (in light of Theorems 3.4 and 5.3). Thus, probabilism is justified similarly in both the finite and infinite case: one ought to have at least coherent credences so as to avoid total accuracy dominance. The only difference is that in the infinite case, coherence is not enough to avoid total accuracy dominance. I find this response promising. However, one complication is that although Theorems 3.4 and 5.3 together guarantee that incoherent credence functions are dominated by coherent credence functions, no matter the infinite opinion set, they do not guarantee dominance by an undominated coherent credence function in general. ³⁴ But as discussed in Footnote 17, it has been argued that being accuracy dominated is not irrational in itself, but rather, being dominated by a credence function that is itself not dominated is irrational. ³⁵ Thus, this line of response would require proving additional results or appealing to a controversial decision-theoretic principle.

A more radical response would be to claim that upon examination, there is simply no reasonable formal measure of total inaccuracy for an agent with infinitely many credences. In particular, no generalized legitimate inaccuracy measure, as I have been calling them, is in fact a legitimate measure of total inaccuracy in the infinite setting. Thus, to justify probabilism on any opinion set using accuracy considerations, there is no other option but to appeal to a local dominance result like that of Schervish et al. (2009). I see three challenges with this response. First, if there is no way to evaluate epistemic states with infinitely many credences for total inaccuracy, i) is the ability to justify norms on epistemic states in terms of total inaccuracy restricted in scope, and ii) if so, might this restriction in scope have negative consequences for the more general project of justifying epistemic norms by appealing to total inaccuracy? For example, if concern for total accuracy can only motivate epistemic norms in the finite case so that one must appeal to other epistemic virtues in the infinite case anyway, why care so much about total accuracy in the finite case?

Second, there are infinite opinion sets for which the natural extensions of the inaccuracy measures used in the finite case seem to behave as they do in the finite case with respect to accuracy dominance (e.g., for countably infinite partitions). Do we reject the legitimacy of generalized legitimate inaccuracy measures even in these cases where the measures behave as we would like? If so, why? And if not, then work must be done to spell out precisely for which infinite opinion sets there is no reasonable way to measure total inaccuracy. Presumably, Theorems 4.19 and 4.29 would be helpful toward this end.

Third, if one is to deny the legitimacy of what I have called generalized legitimate inaccuracy measures, then one needs to explain what exactly is wrong with them, given their structural and axiomatic similarity to the inaccuracy measures often used to measure the total accuracy of credence functions on finite opinion sets. ³⁶ Although one might think that a simple response is that no credence function should be infinitely inaccurate according to a reasonable inaccuracy measure, this response is not so obviously compelling. First, as Pettigrew (2016) points out, it is hard to assess the plausibility of this claim because whether inaccuracy can be infinite or not “is not something which our concept of accuracy contains much information about” (37). Another issue with this response, also raised by Pettigrew, is that the logarithmic inaccuracy measure—often seen as a reasonable way to measure inaccuracy in the finite case (see, e.g., McCutcheon 2019)—can output an infinite inaccuracy score even for finite opinion sets. So if one rejects generalized legitimate inaccuracy measures by virtue of their outputting infinite values, this will have consequences for what the legitimate inaccuracy measures are in the finite case. Lastly, insisting on only inaccuracy measures that output a finite inaccuracy score would justify focusing on a subclass of the generalized legitimate inaccuracy measures rather than justify rejecting all of them. ³⁷ In fact, it is an interesting open question whether this restriction to the finite generalized legitimate inaccuracy measures picks out a set of generalized legitimate inaccuracy measures for which the finite dominance result goes through to all countably infinite opinion sets. ³⁸

To conclude, I do not claim to have provided a full analysis of the options available for responding to the results in this paper, but I hope to have shown that further philosophical work is in order if we are to understand their implications for probabilism and accuracy-based justifications of epistemic norms more generally.

7. Conclusion

As discussed in the previous section, there is plenty of normative work to be done using the results established in this article. In light of the failure of coherence being sufficient to avoid strong dominance on certain countably infinite opinion sets, the most pressing question seems to be this: is there an accuracy dominance argument for probabilism on at least all countable opinion sets? If not, what does this mean for the accuracy project as a whole? Can we give some sort of privileged status to certain kinds of opinion sets or inaccuracy measures for which coherence is equivalent to not being dominated (e.g., partitions)? What is the normative status of the stronger condition of countable coherence? Further, although the measure theoretic framework introduced in Section 5 to score the inaccuracy of credence functions over opinion sets of arbitrary cardinality seems like a natural extension of the finite and countably infinite frameworks, is it well motivated that inaccuracy does not track the behavior of a credence function on measure zero sets? The hope with this paper is to start a conversation about these questions by first establishing relevant mathematical results.