Coherence, striking agreement, and reliability

Abstract

Striving for a probabilistic explication of coherence, scholars proposed a distinction between agreement and striking agreement. In this paper I argue that only the former should be considered a genuine concept of coherence. In a second step the relation between coherence and reliability is assessed. I show that it is possible to concur with common intuitions regarding the impact of coherence on reliability in various types of witness scenarios by means of an agreement measure of coherence. Highlighting the need to separate the impact of coherence and specificity on reliability it is finally shown that a recently proposed vindication of the Shogenji measure qua measure of coherence vanishes.

Appendices

1. Extant coherence measures violate $(\dagger )$

Recall condition $(\dagger )$: consider a set $S=\{A_1,\ldots ,A_n\}$ of propositions and two probability distributions $\Pr$ and $\Pr '$. If the posterior probability of each non-empty subset of propositions on each non-empty subset of the remaining propositions in the set for distribution $\Pr$ exceed the corresponding posterior probabilities for $\Pr '$, then $S$ is more coherent on $\Pr$ than on $\Pr '$. In order to show that extant coherence measures violate this condition, I distinguish the following cases:

1.1 The Shogenji measure

For a set $S=\{A_1,\ldots ,A_n\}$, the Shogenji measure reads as follows:

$$\begin{aligned} {\mathcal C}_\mathrm{Sh}(S)=\frac{\Pr (A_1\wedge \cdots \wedge A_n)}{\Pr (A_1)\ldots \Pr (A_n)} \end{aligned}$$

Let $\{A,B\}$ be a set of propositions and consider the following probability distributions $\Pr$ and $\Pr '$: for $\Pr$, $\Pr (A\wedge B)=0.3$ and $\Pr (A\wedge \lnot B)=0.2=\Pr (\lnot A\wedge B)$. For $\Pr '$, $\Pr '(A\wedge B)=0.525$ and $\Pr '(A\wedge \lnot B)=\Pr '(\lnot A\wedge B)=0.225$. Then $\Pr (A)=\Pr (B)=0.5$ and $\Pr '(A)=\Pr '(B)=0.75$. Hence, $\Pr (A\wedge B)=0.3>0.25=\Pr (A)\cdot \Pr (B)$ and $\Pr '(A\wedge B)=0.525<0.5625=\Pr '(A)\cdot \Pr '(B)$. Thus, ${\mathcal C}_\mathrm{Sh}(A,B)$ is higher on probability distribution $\Pr$ than on $\Pr '$. On the other hand, $\Pr (A|B)=0.6<0.7\Pr '(A|B)$ and also $\Pr (B|A)=0.6<0.7=\Pr '(B|A)$. Thus, ${\mathcal C}_\mathrm{Sh}$ violates $(\dagger )$.

1.2 Mutual support measures

The basic idea underlying these measures is that coherent propositions mutually support each other. One way to cash out this idea is due to Douven and Meijs (2007). Their recipe for probabilistic measures of coherence averages the degree of mutual support for all pairs of non-empty, non-overlapping subsets of the given set under consideration. In a nutshell, the idea is similar to the construction of the agreement measures ${{\mathcal C}_{\mathcal A}}$ except for the fact that Douven and Meijs suggest to implement probabilistic measures of incremental support. Basically, $\mathfrak {s}$ is a measure of incremental support, if there is a threshold $\beta$ such that for each pair of a hypothesis $H$ and a piece of evidence $E$,

$$\begin{aligned} \mathfrak {s}(H,E)>\!/\!=\!/\!<\beta \text { iff } \Pr (H|E)>\!/\!=\!/\!<\Pr (H) \end{aligned}$$

(11)

Hence, evidence $E$ is said to confirm $H$ iff $\mathfrak {s}(H,E)>\beta$, $E$ is neutral towards $H$ iff $\mathfrak {s}(H,E)=\beta$, and $E$ disconfirms $H$ iff $\mathfrak {s}(H,E)<\beta$. Now, given this qualitative explication of the term confirmation, it remains an open question whether two confirmatory pieces of evidence $E_1$ and $E_2$ are equally confirmatory or not. Similarly, given that $E$ confirms both hypothesis $H_1$ and $H_2$ it will often be the case, intuitively, that one of the hypothesis is more confirmed than the other. By means of a probabilistic measure of confirmation, it is possible to answer these and related question: these measures are functions that assign to each hypothesis-evidence pair $H,E$ a number $\alpha$ that represents the degree of confirmation that $E$ confers upon $H$. Obviously, there are many ways to spell out the notion of a degree of confirmation. Accordingly, a plethora of non-equivalent confirmation measures have been proposed.²³ Since all of these measures share property (11), we need not go into the details of the various measures. Instead, I will show that all measures that satisfy property (11) will inevitably violate $(\dagger )$.

Let $\Pr$ and $\Pr '$ be two probability distributions such that $\Pr (A|B)=\Pr (B|A)=0.6$ and $\Pr '(A|B)=\Pr '(B|A)=0.7$. Furthermore, assume that $\Pr (A)=\Pr (B)=0.5$ and $\Pr '(A)=\Pr '(B)=0.75$.²⁴ Now, on the one hand, given that $\Pr (A|B)<\Pr '(A|B)$ and $\Pr (B|A)<\Pr '(B|A)$, the set $\{A,B\}$ is more coherent on probability distribution $\Pr '$ than on $\Pr$ according to $(\dagger )$. On the other hand, since $\Pr (A|B)>\Pr (A)$ and $\Pr (B|A)>\Pr (B)$, whereas $\Pr '(A|B)<\Pr '(A)$ and $\Pr '(B|A)<\Pr '(B)$, $\{A,B\}$ must turn out less coherent on $\Pr '$ than on $\Pr$. This is due to the fact that for each confirmation measure $\mathfrak {s}$, the $\mathfrak {s}$-based coherence value for $\{A,B\}$ is the straight average of $\mathfrak {s}(A,B)$ and $\mathfrak {s}(B,A)$. Since both these confirmation values are larger than $\beta$ for $\Pr$ and lower than $\beta$ for $\Pr '$, the corresponding coherence values must be so likewise. Hence, $\{A,B\}$ is less coherence on $\Pr '$ than on $\Pr$ irrespective of the chosen confirmation measure $\mathfrak {s}$.

So far there is only one (family of) coherence measure with an independent standing in the literature next to ${{\mathcal C}_{\mathcal A}}$ that might satisfy $(\dagger )$. These are the overlap measure of coherence, independently proposed by Glass (2002) and Olsson (2002), and it’s refined version endorsed by Meijs (2006). However, it is an open question whether these measures do indeed satisfy $(\dagger )$. Both these measures are not ordinally equivalent to ${{\mathcal C}_{\mathcal A}}$. Furthermore, both violate the additivity-constraint mentioned in Sect. 2.

2. Proof of Theorem 1

Theorem 1: In a scenario with two testimonies that satisfies conditions (4)-(6) and fixed equal individual reliability the following comparative assertions are true:

• (i) Given fixed equal informativity ${\text {inf}}(A_1)={\text {inf}}(A_2)$, ${\Delta }(R_1)$ covaries with ${{\mathcal C}_{\mathcal A}}(A_1, A_2)$.

• (ii) Given a fixed degree of coherence ${{\mathcal C}_{\mathcal A}}(A_1,A_2)$ and assuming equal but arbitrary informativity ${\text {inf}}(A_1)={\text {inf}}(A_2)$, ${\Delta }(R_1)$ covaries with $\sigma (A_1,A_2)$.

Proof

Given the assumptions of observation 1, it is shown by Schubert (2011) that the posterior probability of reliability for witness $1$, given two witness reports $\langle A_1^i\rangle$ and $\langle A_2^i\rangle$ can be represented as follows:²⁵

$$\begin{aligned} \Pr (R_1|\langle A_1^i\rangle ,\langle A_2^i\rangle )=\frac{\frac{\Pr (A_1,A_2)}{\Pr (A_1)\cdot \Pr (A_2)} +\frac{(1-\rho )}{\rho }}{\frac{\Pr (A_1,A_2)}{\Pr (A_1)\cdot \Pr (A_2)} +2\cdot \frac{(1-\rho )}{\rho }+\frac{(1-\rho )^2}{\rho ^2}} \end{aligned}$$

(12)

Hence, the following alternative representation for the benefit function for witness $1$ given two reports $\langle A_i^1\rangle$ and $\langle A_i^2\rangle$ immediately follows:

$$\begin{aligned} {\Delta }(R_1)=\Pr (R_1|\langle A_i^1\rangle , \langle A_i^1\rangle )-\Pr (R_1|\langle A_1^i\rangle )=\frac{\left( \frac{\Pr (A_1\wedge A_2)}{\Pr (A_1)\cdot \Pr (A_2)}-1\right) \cdot \left( \rho ^2-\rho ^3\right) }{\left( \frac{\Pr (A_1\wedge A_2)}{\Pr (A_1)\cdot \Pr (A_2)}-1\right) \cdot \rho ^2+1} \end{aligned}$$

(13)

This can be seen as follows:

$$\begin{aligned}&\Pr (R_1|\langle A_i^1\rangle , \langle A_i^1\rangle )-\Pr (R_1|\langle A_1^i\rangle )= \frac{\frac{\Pr (A_1,A_2)}{\Pr (A_1)\cdot \Pr (A_2)}+\frac{(1-\rho )}{\rho }}{\frac{\Pr (A_1,A_2)}{\Pr (A_1)\cdot \Pr (A_2)}+2\cdot \frac{(1-\rho )}{\rho }+\frac{(1-\rho )^2}{\rho ^2}}-\rho \\&\quad = \frac{\frac{\Pr (A_1,A_2)}{\Pr (A_1)\cdot \Pr (A_2)}+\frac{(1-\rho )}{\rho }-\rho \frac{\Pr (A_1\wedge A_2)}{\Pr (A_1)\Pr (A_2)}-2(1-\rho )-\frac{(1-\rho )^2}{\rho }}{\frac{\Pr (A_1,A_2)}{\Pr (A_1)\cdot \Pr (A_2)}+2\cdot \frac{(1-\rho )}{\rho }+\frac{(1-\rho )^2}{\rho ^2}}\\&\quad = \frac{\frac{\Pr (A_1\wedge A_2)}{\Pr (A_1)\Pr (A_2)}\rho ^2+(1-\rho )\rho -\frac{\Pr (A_1\wedge A_2)}{\Pr (A_1)\Pr (A_2)}\rho ^3-2\rho ^2(1-\rho )-(1-\rho )^2\rho }{\frac{\Pr (A_1\wedge A_2)}{\Pr (A_1)\Pr (A_2)}\rho ^2+2\rho (1-\rho )+(1-\rho )^2}\\&\quad = \frac{(\frac{\Pr (A_1\wedge A_2)}{\Pr (A_1)\Pr (A_2)}-1)\cdot (\rho ^2-\rho ^3)}{(\frac{\Pr (A_1\wedge A_2)}{\Pr (A_1)\Pr (A_2)}-1)\cdot \rho ^2+1} \end{aligned}$$

• (i) Now given the additional assumption in (i) that $\mathrm{inf}(A_1)=\mathrm{inf}(A_2)$, it immediately follows that there is an $r\in (0,1)$ such that $\Pr (A_1)=\Pr (A_2)=r$. Furthermore, given a fixed reliability parameter $\rho$, ${\Delta }(R_1)$ is an increasing function of $\Pr (A_1, A_2)$. Given these restrictions, ${{\mathcal C}_{\mathcal A}}(A_1,A_2)=\frac{1}{r}\cdot \Pr (A_1,A_2)$. Hence, the larger the agreement between the testimonies, the higher the impact on ${\Delta }(R_1)$, and vice versa.

• (ii) Given the additional assumption, we conclude that ${{\mathcal C}_{\mathcal A}}(A_1,A_2)=r$ for some $r\in [0,1]$. It follows that $\Pr (A_1,A_2)=r\cdot \Pr (A_1)$. Thus, given that we assume the coherence value to be fixed, $\Pr (A_1,A_2)$ is an increasing function of $\Pr (A_1)$. Thus, ${\Delta }(R_1)$ reduces to

  $$\begin{aligned} {\Delta }(R_1)=\frac{\left( \frac{r}{\Pr (A_1)}-1\right) \cdot (\rho ^2-\rho ^3)}{\left( \frac{r}{\Pr (A_1)}-1\right) \cdot \rho ^2+1} \end{aligned}$$

  ${\Delta }(R_1)$ is thus a decreasing function of $\Pr (A_1)$ as is $\sigma (A_1,A_2)=-2\log _2\Pr (A_1)$. Accordingly, the larger the informations’ specificity $\sigma (A_1,A_2)$, the higher the impact on ${\Delta }(R_1)$ and vice versa.

$\square$

3. Proof of Theorem 3

Theorem 3: In a scenario with two testimonies that satisfies conditions (5)-(8) and fixed equal individual reliability, the following comparative assertions are true:

• (i) Given fixed equal informativity $\mathrm{inf}(A_1)=\mathrm{inf}(A_2)$, ${\Delta }(R_1)$ covaries with ${{\mathcal C}_{\mathcal A}}(A_1,A_2)$.

• (ii) Given a fixed degree of coherence ${{\mathcal C}_{\mathcal A}}(A_1,A_2)$ and assuming equal but arbitrary informativity $\mathrm{inf}(A_1)=\mathrm{inf}(A_2)$, ${\Delta }(R_1)$ covaries with $\sigma (A_1,A_2)$.

Proof

I utilize the following abbreviations: $\alpha _i=\Pr (A_i)$, $\alpha _{1,2}=\Pr (A_1,A_2)$. Let $\mathbf A_i, R_i, \langle A_i\rangle$ be random variables taking $A_i$ and $\lnot A_i$, $R_i$ and $\lnot R_i$, and $\langle A_i\rangle$, $\lnot \langle A_1\rangle$ as its values for $i=1,2$. Then:

$$\begin{aligned}&\Pr (\mathbf{A_1, A_2, R_1, R_2, \langle A_1\rangle , \langle A_2\rangle })\\&\quad = \Pr (\mathbf{\langle A_1\rangle , \langle A_2\rangle , R_2, A_2}\,|\, \mathbf{R_1, A_1})\cdot \Pr (\mathbf{R_1, A_1})\\&\quad = \Pr (\mathbf{\langle A_1\rangle }| \mathbf{R_1, A_1, \langle A_2\rangle , R_2, A_2})\cdot \Pr (\mathbf{\langle A_2\rangle , R_2, A_2}\,|\, \mathbf{R_1, A_1})\cdot \Pr (\mathbf{R_1, A_1}) \\&\quad = \Pr (\mathbf{\langle A_1\rangle }| \mathbf{R_1, A_1})\cdot \Pr (\mathbf{\langle A_2\rangle , R_2, A_2, R_1, A_1}) \\&\quad = \Pr (\mathbf{\langle A_1\rangle }| \mathbf{R_1, A_1})\cdot \Pr (\mathbf{\langle A_2\rangle }\,|\,\mathbf{R_2, A_2, R_1, A_1})\cdot \Pr (\mathbf{R_2, A_2, R_1, A_1}) \\&\quad = \Pr (\mathbf{\langle A_1\rangle }| \mathbf{R_1, A_1})\cdot \Pr (\mathbf{\langle A_2\rangle }\,|\,\mathbf{R_2, A_2})\cdot \Pr (\mathbf{R_2, A_2, R_1, A_1}) \\&\quad = \Pr (\mathbf{\langle A_1\rangle }| \mathbf{R_1, A_1})\cdot \Pr (\mathbf{\langle A_2\rangle }\,|\,\mathbf{R_2, A_2})\cdot \Pr (\mathbf{R_1}\,|\, \mathbf{R_2, A_1, A_2})\cdot \\&\qquad \Pr (\mathbf{R_2}| \mathbf{A_2, A_1})\cdot \Pr (\mathbf{A_1, A_2}) \\&\quad = \Pr (\mathbf{\langle A_1\rangle }| \mathbf{R_1, A_1})\cdot \Pr (\mathbf{\langle A_2\rangle }\,|\,\mathbf{R_2, A_2})\cdot \Pr (\mathbf{R_1})\cdot \Pr (\mathbf{R_2})\cdot \Pr (\mathbf{A_1, A_2}) \end{aligned}$$

Utilizing the fact that $\Pr (\langle A_i\rangle \,|\, R_i, \lnot A_i)=0$, we get the following representation of $\Pr (R_1\,|\,\langle A_1 \rangle ,\langle A_2\rangle )$:

$$\begin{aligned} \Pr (R_1\,|\,\langle A_1 \rangle ,\langle A_2\rangle )=\frac{\Pr (R_1,\langle A_1 \rangle ,\langle A_2\rangle )}{\Pr (R_1,\langle A_1 \rangle ,\langle A_2\rangle )+\Pr (\lnot R_1,\langle A_1 \rangle ,\langle A_2\rangle )}=\frac{\lambda }{\lambda +\mu }, \end{aligned}$$

(14)

where $\lambda :=\Pr (R_1,\langle A_1 \rangle ,\langle A_2\rangle )$ and $\mu =\Pr (\lnot R_1,\langle A_1 \rangle ,\langle A_2\rangle )$ can be written as follows.

$$\begin{aligned} \lambda&= \Pr (A_1, R_1, \langle A_1\rangle , A_2, R_2, \langle A_2\rangle )\,+\,\Pr (A_1, R_1, \langle A_1\rangle , A_2, \lnot R_2, \langle A_2\rangle )\\&+\,\Pr (A_1, R_1, \langle A_1\rangle , \lnot A_2, \lnot R_2, \langle A_2\rangle )\\&= \rho ^2\Pr (A_1, A_2)\,+\,\rho (1-\rho )\alpha _2 \Pr (A_1,A_2)\,+\,\rho (1-\rho )\alpha _2 \Pr (A_1,\lnot A_2)\\&= \rho ^2\alpha _{1,2}\,+\,\rho (1-\rho )\alpha _1\alpha _2\\ \mu&= \Pr (A_1, \lnot R_1, \langle A_1\rangle , A_2, R_2, \langle A_2\rangle )\,+\,\Pr (\lnot A_1,\lnot R_1, \langle A_1\rangle , A_2, R_2, \langle A_2\rangle )\\&+\,\Pr (A_1,\lnot R_1, \langle A_1\rangle , A_2,\lnot R_2, \langle A_2\rangle )\,+\,\Pr (A_1,\lnot R_1, \langle A_1\rangle ,\lnot A_2,\lnot R_2, \langle A_2\rangle )\\&+\,\Pr (\lnot A_1,\lnot R_1, \langle A_1\rangle , A_2,\lnot R_2, \langle A_2\rangle ) +\,\Pr (\lnot A_1,\lnot R_1, \langle A_1\rangle ,\lnot A_2, \lnot R_2, \langle A_2\rangle )\\&= (1-\rho )\rho \alpha _1\Pr (A_1,A_2)\,+\,(1-\rho )\rho \alpha _1\Pr (\lnot A_1, A_2)\\&+\,(1-\rho )^2\alpha _1\alpha _2\Pr (A_1,A_2)(1-\rho )^2\alpha _1\alpha _2\Pr (A_1,\lnot A_2)\\&+\,(1-\rho )^2\alpha _1\alpha _2\Pr (\lnot A_1, A_2)\,+\,(1-\rho )^2\alpha _1\alpha _2\Pr (\lnot A_1,\lnot A_2)\\&= (1-\rho )\rho \alpha _1\alpha _2\,+\,(1-\rho )^2\alpha _1\alpha _2 \end{aligned}$$

Putting things together we get:

$$\begin{aligned}&\Pr (R_1\,|\,\langle A_1 \rangle ,\langle A_2\rangle )-\rho \\&\quad =\quad \frac{\rho ^2\alpha _{1,2}\,+\,\rho (1-\rho )\alpha _1\alpha _2}{\rho ^2\alpha _{1,2}\,+\,\rho (1-\rho )\alpha _1\alpha _2\,+\,(1-\rho )\rho \alpha _1\alpha _2\,+\,(1-\rho )^2\alpha _1\alpha _2}-\rho \\&\quad =\quad \frac{\rho ^2\alpha _{1,2}\,+\,\rho (1-\rho )\alpha _1\alpha _2}{\rho ^2\alpha _{1,2}\,+\,\alpha _1\alpha _2(1-\rho ^2)}-\rho \\&\quad =\quad \frac{\rho ^2\frac{\alpha _{1,2}}{\alpha _1\alpha _2}\,+\,\rho (1-\rho )-\rho ^3\frac{\alpha _{1,2}}{\alpha _1\alpha _2}\,-\,\rho (1-\rho )(1+\rho )}{\rho ^2\frac{\alpha _{1,2}}{\alpha _1\alpha _2}\,+\,(1-{\rho ^2})}\\&\quad =\quad \frac{\rho ^2(1-\rho )\left( \frac{\alpha _{1,2}}{\alpha _1\alpha _2}-1\right) }{\rho ^2\frac{\alpha _{1,2}}{\alpha _1\alpha _2}\,+\,(1-{\rho ^2})} \end{aligned}$$

Denote “$\frac{\alpha _{1,2}}{\alpha _1\alpha _2}$” by “$\phi$” and “$\frac{\alpha '_{1,2}}{\alpha '_1,\alpha '_2}$” by “$\phi '$”, then the following equivalence establishes the desired result.

$\Pr (R_1\,|\,\langle A_1 \rangle ,\langle A_2\rangle )-\rho >\Pr (R_1\,|\,\langle A'_1 \rangle ,\langle A'_2\rangle )-\rho$

$$\begin{aligned}&\Leftrightarrow \ \frac{\left( \phi \,-\,1\right) }{\rho ^2\phi \,+\,(1-{\rho ^2})}>\frac{\left( \phi '\,-\,1\right) }{\rho ^2\phi '\,+\,(1-{\rho ^2})}\\&\Leftrightarrow \ (\phi -1)\cdot (\rho ^2\phi '+(1-\rho ^2))>(\phi '-1)\cdot (\rho ^2\phi +(1-\rho ^2))\\&\Leftrightarrow \ (1-\rho ^2)\phi -\rho ^2\phi '>(1-\rho ^2)\phi '-\rho ^2\phi \\&\Leftrightarrow \ \phi >\phi ' \end{aligned}$$

Hence, ${\Delta }(R_1)$ is an increasing function of $\frac{\phi _{1,2}}{\phi _1\phi _2}=\frac{\Pr (A_1,A_2)}{\Pr (A_1)\cdot \Pr (A_2)}$. Accordingly, an argument analogous to B yields the desired result. $\square$

4. Proof of Theorem 4

Theorem 4: In a scenario with $n$ equivalent testimonies that satisfies conditions (5)-(7) and (9), and assuming equal individual reliability, the following assertions are true irrespective of the chosen randomization parameter $0<a<1$:

• (i) Given a fixed number $n$ of testimonies, ${\Delta }(R_1)$ is strictly increasing in inf$(A_i)$.

• (ii) Given a fixed degree of informativity inf$(A_i)$, ${\Delta }(R_1)$ is a strictly increasing function of $n$.

Proof

The following representation is shown to hold by Bovens and Hartmann (2003, p. 62) for the likelihood ratio $x=\frac{\Pr (\langle A^i\rangle \,|\,\lnot A)}{\Pr (\langle A^i\rangle \,|\,A)}=\frac{a\overline{\rho }}{\rho +a\overline{\rho }}$, where $\rho =\Pr (R_i)$ and $\alpha _i=\Pr (A_i)$:

$$\begin{aligned} {\Pr }^{*}(R_1):=\Pr (R_1\,|\,\langle A^1\rangle ,\ldots ,\langle A^n\rangle )=\frac{\alpha _i\overline{x}}{\alpha _i+\overline{\alpha }_i x^n} \end{aligned}$$

(15)

• (i) Assume that the number of testimonies is fixed. In order to show that this is a strictly decreasing function of $\Pr (A_i)$, we calculate the first derivate and show that it always exceeds zero. Let $f(\alpha _i):={\alpha _i\overline{x}}$ and $g(\alpha _i):={\alpha _i+\overline{\alpha }_i x^n}$, then

  $$\begin{aligned} \frac{\partial \Pr ^*(R_1)}{\partial \alpha _i}=\frac{f'(\alpha _i)g(\alpha _i)-f(\alpha _i)g'(\alpha _i)}{g(h)^2} \end{aligned}$$

  (16)

  where $f'(\alpha _i)=1-x$ and $g'(\alpha _i)=1-x^n$. Accordingly, inserting these into the above equation yields the following result.

  $$\begin{aligned} \frac{\partial \Pr ^*(R_1)}{\partial \alpha _i} \ = \ \frac{x^n-x^{n+1}}{({\alpha _i+\overline{\alpha }_i x^n})^2}>0 \end{aligned}$$

  (17)

  With $x\in \left( 0,1\right)$, this is positive irrespective of the chosen randomization parameter $a$. Hence, given maximal coherence, the posterior probability of reliability again is strictly decreasing in the propositions priors.

• (ii) Now assume that the prior probability $\alpha _i$ is fixed. It can easily be seen that if $n$ increases, the denominator of $\Pr ^{*}(R_1)$ decreases since $x<1$. Accordingly, $\Pr ^{*}(\alpha _i)$ must be an increasing function of $n$ for fixed informativity.

$\square$