In the Accuracy Dominance argument for Probabilism, we show that, if your credence function does not obey the axioms of the probability calculus, there is an alternative credence function, defined on exactly the same propositions, that has greater accuracy than yours for sure. In Briggs and Pettigrew’s Accuracy Dominance argument for Conditionalization, we show that, if you plan to update your prior in some way other than by Conditionalization, then there is an alternative prior and an alternative updating plan that have greater total accuracy than your prior and updating plan for sure. In the Accuracy Dominance argument for Weak GRP, we show that, if your prior is not a convex combination of your possible posteriors, there is an alternative prior and alternative possible posteriors, each paired with one of your possible posteriors, such that if you were to replace your prior with the alternative prior and each of the posteriors with its paired alternative, you’d have greater total accuracy for sure.
First, a couple of quick words about measuring accuracy. Briggs and Pettigrew’s argument, like the increasingly standard Accuracy Dominance argument for Probabilism, assumes that our measures of inaccuracy have three properties (Predd et al. 2009):
Additivity The inaccuracy of a whole credence function is the sum of the inaccuracy of the credences it assigns.
More precisely: If \({\mathfrak {I}}\) is a legitimate measure of the inaccuracy of a credence function at a world, then there is, for each X in \({\mathcal {F}}\), a scoring rule \({\mathfrak {s}}_X : \{0, 1\} \times [0, 1] \rightarrow [0, \infty ]\) such that, for any credence function c defined on \({\mathcal {F}}\) and any world w,
$$\begin{aligned} {\mathfrak {I}}(c, w) = \sum _{X \in {\mathcal {F}}} {\mathfrak {s}}_X(v_w(X), c(X)) \end{aligned}$$
where \(v_w(X) = 0\) if X is false at w and \(v_w(X) = 1\) if X is true at w. In this case, we say that \({\mathfrak {s}}\) generates \({\mathfrak {I}}\).
Continuity The inaccuracy of a credence is a continuous function of that credence.
More precisely: If \({\mathfrak {I}}\) is a legitimate measure of inaccuracy that is generated by the scoring rule \({\mathfrak {s}}\), then, for all X in \({\mathcal {F}}\), \({\mathfrak {s}}_X(1, x)\) and \({\mathfrak {s}}_X(0, x)\) are continuous functions of x.
Strict Propriety Each credence expects itself to be most accurate.
More precisely: If \({\mathfrak {I}}\) is a legitimate measure of inaccuracy that is generated by the scoring rule \({\mathfrak {s}}\), then, for all X in \({\mathcal {F}}\) and \(0 \le p \le 1\),
$$\begin{aligned} p{\mathfrak {s}}_X(1, x) + (1-p){\mathfrak {s}}_X(0, x) \end{aligned}$$
is uniquely minimized, as a function of x, at \(x = p\).
When \({\mathfrak {I}}\) satisfies these three properties, we say that it is an additive and continuous strictly proper inaccuracy measure.
Next, let’s specify a flaw that a prior and set of possible posteriors might jointly have.
Accuracy Domination A pair \((c, R = \{c_1, \ldots , c_n\})\) consisting of a prior c and a set R of possible posteriors is accuracy dominated iff, for all legitimate inaccuracy measures \({\mathfrak {I}}\), there is an alternative pair \((c^\star , R^\star = \{c^\star _1, \ldots , c^\star _n\})\) such that, for all possible worlds w and all \(i = 1, \ldots , n\),
$$\begin{aligned} {\mathfrak {I}}(c^\star , w) + {\mathfrak {I}}(c^\star _i, w) < {\mathfrak {I}}(c, w) + {\mathfrak {I}}(c_i, w) \end{aligned}$$
And now our theorem:
Theorem 2
(c, R) violates Weak GRP iff it is accuracy dominated.
Proof of Theorem 2
Given a set of possible posteriors, \(R = \{c_1, \ldots , c_n\}\), define the following set of \((n+1)\)-dimensional vectors of credence functions:
$$\begin{aligned} {\overline{R}} = \{(v_w, c_1, \ldots , c_{i-1}, v_w, c_{i+1}, \ldots , c_n) : w \in {\mathcal {W}}, i = 1, \ldots , n\} \end{aligned}$$
where, again, \(v_w\) is the credence function with \(v_w(X) = 0\) if X is false at w and \(v_w(X) = 1\) if X is true at w. Then:
Lemma 3
If c violates Weak GRP, then
$$\begin{aligned} (c, c_1, \ldots , c_n) \not \in {\overline{R}}^+ \end{aligned}$$
where \({\overline{R}}^+\) is the convex hull of \({\overline{R}}\).Footnote 8
Proof of Theorem Lemma 3.
To prove this, we prove the contrapositive. Suppose \((c, c_1, \ldots , c_n) \in {\overline{R}}^+\). Then there are \(0 \le \lambda _{w, i} \le 1\) such that \(\sum _w \sum _i \lambda _{w, i} = 1\) and
$$\begin{aligned} (c, c_1, \ldots , c_n) = \sum _w \sum _i \lambda _{w, i} (v_w, c_1, \ldots , c_{i-1}, v_w, c_{i+1}, \ldots , c_n) \end{aligned}$$
So,
$$\begin{aligned} c = \sum _w \sum _i \lambda _{w, i} v_w \end{aligned}$$
and
$$\begin{aligned} c_j = \sum _w \lambda _{w, j} v_w + \sum _w \sum _{i \ne j} \lambda _{w, i} c_j \end{aligned}$$
So
$$\begin{aligned} \left( \sum _w \lambda _{w, j}\right) c_j = \sum _w \lambda _{w, j} v_w \end{aligned}$$
So let \(\lambda _j = \sum _w \lambda _{w, j}\). Then, for \(j = 1, \ldots , n\),
$$\begin{aligned} \lambda _j c_j = \sum _w \lambda _{w, j} v_w \end{aligned}$$
And thus
$$\begin{aligned} \sum _i \lambda _i c_i = \sum _i \sum _w \lambda _{w, i} v_w = c \end{aligned}$$
Thus, c is in \({\overline{R}}^+\), and c satisfies Weak GRP.
Return to Proof of Theorem 2. Now, we appeal to two central facts about additive and continuous strictly proper inaccuracy measures:
Lemma 4
(Proposition 2, Predd et al. 2009) Suppose \({\mathfrak {I}}\) is an additive and continuous strictly proper inaccuracy measure. Then there is a Bregman divergence \({\mathfrak {D}}\) such that, for any credence function c and any world w,Footnote 9
$$\begin{aligned} {\mathfrak {I}}(c, w) = {\mathfrak {D}}(v_w, c) \end{aligned}$$
Thus, the inaccuracy of c at w is the divergence from \(v_w\) to c.
Lemma 5
(Proposition 3, Predd et al. 2009) Suppose \({\mathfrak {D}}\) is a Bregman divergence and \({\mathcal {P}}\) is a set of credence functions. Then, if c is not in \({\mathcal {P}}^+\), then there is \(c^\star \) in \({\mathcal {P}}^+\) such that, for all p in \({\mathcal {P}}\),Footnote 10
$$\begin{aligned} {\mathfrak {D}}(p, c^\star ) < {\mathfrak {D}}(p, c) \end{aligned}$$
Now, suppose \({\mathfrak {I}}\) is an additive and continuous strictly proper inaccuracy measure and \({\mathfrak {D}}\) is its accompanying Bregman divergence. Now, suppose (c, R) violates Weak GRP. Then
$$\begin{aligned} (c, c_1, \ldots , c_n) \not \in {\overline{R}}^+ \end{aligned}$$
So, by Lemma 5, there is
$$\begin{aligned} (c^\star , c^\star _1, \ldots , c^\star _n) \in {\overline{R}}^+ \end{aligned}$$
such that, for all w and i,
$$\begin{aligned}&{\mathfrak {D}}(v_w, c^\star ) + {\mathfrak {D}}(c_1, c^\star _1) + \cdots + {\mathfrak {D}}(c_{i-1}, c^\star _{i-1}) +\\&\quad {\mathfrak {D}}(v_w, c^\star _i) + {\mathfrak {D}}(c_{i+1}, c^\star _{i+1}) + \cdots + {\mathfrak {D}}(c_n, c^\star _n) <\\&\quad {\mathfrak {D}}(v_w, c) + {\mathfrak {D}}(c_1, c_1) + \cdots + {\mathfrak {D}}(c_{i-1}, c_{i-1}) + \\&\quad {\mathfrak {D}}(v_w, c_i) + {\mathfrak {D}}(c_{i+1}, c_{i+1}) + \cdots + {\mathfrak {D}}(c_n, c_n) \end{aligned}$$
Now, \({\mathfrak {D}}(c_i, c^\star _i) \ge 0\) for all \(i = 1, \ldots , n\). Furthermore, \({\mathfrak {D}}(c_i, c_i) = 0\). So we can infer:
$$\begin{aligned} {\mathfrak {D}}(v_w, c^\star ) + {\mathfrak {D}}(v_w, c^\star _i) < {\mathfrak {D}}(v_w, c) + {\mathfrak {D}}(v_w, c_i) \end{aligned}$$
And so, by Lemma 4,
$$\begin{aligned} {\mathfrak {I}}(c^\star , w) + {\mathfrak {I}}(c^\star _i, w) < {\mathfrak {I}}(c, w) + {\mathfrak {I}}(c_i, w) \end{aligned}$$
as required.
Next, suppose (c, R) does satisfy Weak GRP. Then
$$\begin{aligned} c = \sum _i \lambda _i c_i \end{aligned}$$
Now consider \(c^\star , c^\star _1, \ldots , c^\star _n\). Then, since \({\mathfrak {I}}\) is strictly proper:
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\(\sum _w c(w) {\mathfrak {I}}(c, w) \le \sum _w c(w) {\mathfrak {I}}(c^\star , w)\)
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\(\sum _w c_i(w) {\mathfrak {I}}(c_i, w) \le \sum _w c(w) {\mathfrak {I}}(c^\star _i, w)\), for \(i = 1, \ldots , n\).
Then
$$\begin{aligned}&\sum _{w, i} \lambda _i c_i(w) ({\mathfrak {I}}(c, w) + {\mathfrak {I}}(c_i, w)) \\&\quad = \sum _{w, i} \lambda _i c_i(w) {\mathfrak {I}}(c, w) + \sum _{w, i} \lambda _i c_i(w){\mathfrak {I}}(c_i, w) \\&\quad = \sum _w c(w) {\mathfrak {I}}(c, w) + \sum _{w, i} \lambda _i c_i(w) {\mathfrak {I}}(c_i, w) \text{ since } c = \sum _i \lambda _i c_i \\&\quad = \sum _w c(w) {\mathfrak {I}}(c, w) + \sum _i \lambda _i \sum _w c_i(w) {\mathfrak {I}}(c_i, w) \\&\quad \le \sum _w c(w) {\mathfrak {I}}(c^\star , w) + \sum _i \lambda _i \sum _w c_i(w) {\mathfrak {I}}(c^\star _i, w) \\&\quad = \sum _w c(w) {\mathfrak {I}}(c^\star , w) + \sum _{w, i} \lambda _i c_i(w){\mathfrak {I}}(c^\star _i, w) \\&\quad = \sum _{w, i} \lambda _i c_i(w) {\mathfrak {I}}(c^\star , w) + \sum _{w, i} \lambda _i c_i(w){\mathfrak {I}}(c^\star _i, w) \text{ since } c = \sum _i \lambda _i c_i \\&\quad = \sum _{w, i} \lambda _i c_i(w) ({\mathfrak {I}}(c^\star , w) + {\mathfrak {I}}(c^\star _i, w)) \end{aligned}$$
So it cannot be that \({\mathfrak {I}}(c^\star , w) + {\mathfrak {I}}(c^\star _i, w) < {\mathfrak {I}}(c, w) + {\mathfrak {I}}(c_i, w)\) for all w in \({\mathcal {W}}\). \(\Box \)