Bayesian learning with multiple priors and nonvanishing ambiguity

Abstract

The existing models of Bayesian learning with multiple priors by Marinacci (Stat Pap 43:145–151, 2002) and by Epstein and Schneider (Rev Econ Stud 74:1275–1303, 2007) formalize the intuitive notion that ambiguity should vanish through statistical learning in an one-urn environment. Moreover, the multiple priors decision maker of these models will eventually learn the “truth.” To accommodate nonvanishing violations of Savage’s (The foundations of statistics, Wiley, New York, 1954) sure-thing principle, as reported in Nicholls et al. (J Risk Uncertain 50:97–115, 2015), we construct and analyze a model of Bayesian learning with multiple priors for which ambiguity does not necessarily vanish in an one-urn environment. Our decision maker only forms posteriors from priors that survive a prior selection rule which discriminates, with probability one, against priors whose expected Kullback–Leibler divergence from the “truth” is too far off from the minimal expected Kullback–Leibler divergence over all priors. The “stubbornness” parameter of our prior selection rule thereby governs how much ambiguity will remain in the limit of our learning model.

Appendix: formal proofs

Existence of an emerging posterior To see that the limit (43 ) exists, rewrite, for any $\Theta ^{\prime }$, $\pi _{\mu _{0}}\left( \Theta ^{\prime }\mid X_{1},\ldots ,X_{t}\right)$ as conditional expectation of the indicator function of $\Theta ^{\prime }$ with respect to the induced probability measure P on the joint index and parameter space $\left( \Theta \times \Omega ^{\infty },\mathcal {F\otimes }\Sigma ^{\infty }\right)$. To be precise, for the notation of our setup, it holds that, for all $\theta \in \Theta$ and $B\in \Sigma ^{\infty }$,

$$\begin{aligned} P_{\theta }\left( B\right) \equiv P\left( B\mid \theta \right) \equiv P\left( \Theta \times B\mid \left\{ \theta \right\} \times \Omega ^{\infty }\right) \end{aligned}$$

as well as, for all $\Theta ^{\prime }\in \mathcal {F}$,

$$\begin{aligned} \mu _{0}\left( \Theta ^{\prime }\right) \equiv P\left( \Theta ^{\prime }\right) \equiv P\left( \Theta ^{\prime }\times \Omega ^{\infty }\right) \text {.} \end{aligned}$$

By Theorem 35.6 in Billingsley (1995) (which is an implication of the martingale convergence theorem), we obtain

$$\begin{aligned} \pi _{\mu _{0}}\left( \Theta ^{\prime }\mid X_{1},\ldots ,X_{t}\right)\equiv & {} E \left[ I_{\Theta ^{\prime }}\left( \theta \right) ,P\left( \theta \mid X_{1},\ldots ,X_{t}\right) \right] \\\rightarrow & {} E\left[ I_{\Theta ^{\prime }}\left( \theta \right) ,P\left( \theta \mid X_{1},X_{2},\ldots \right) \right] \equiv \pi _{\mu _{0}}^{\infty }\left( \Theta ^{\prime }\right) \end{aligned}$$

whereby convergence happens with P probability one.$\square$

Proof of Proposition 2

For notational convenience, we write

$$\begin{aligned} \pi _{\mu _{0}}^{t}\equiv \pi _{\mu _{0}}\left( \cdot \mid X_{1},\ldots ,X_{t}\right) \text {.} \end{aligned}$$

• Step 1 Suppose that there exists some $\pi _{\mu _{0}^{*}}^{\infty }\in \Pi _{R}^{\infty }$ but

  $$\begin{aligned} \pi _{\mu _{0}^{*}}^{\infty }\notin \overline{\lim }\bigcup \limits _{\mu _{0}\in \mathcal {M}_{0,R}^{t}}\left\{ \pi _{\mu _{0}}^{t}\right\} \text {, a.s. }P_{\theta ^{*}}\text {.} \end{aligned}$$

  (113)

  Since $\mu _{0}^{*}\in \mathcal {M}_{0,R}^{\infty }$ and $\mathcal {M}_{0}$ is compact, there must be (a.s. $P_{\theta ^{*}}$) some subsequence $\left\{ \mu _{0}^{t_{k}}\right\} _{k\in \mathbb {N}}$ such that $\mu _{0}^{t_{k}}\in \mathcal {M}_{0,R}^{t_{k}}$ for all $k=1,2,\ldots$ which converges to $\mu _{0}^{*}$. But then $\pi _{\mu _{0}^{t_{k}}}^{t}\in \Pi _{R}^{t}$ for all $k=1,2,\ldots$, implying

  $$\begin{aligned} \pi _{\mu _{0}^{*}}^{\infty }\in \overline{\lim }\bigcup \limits _{\mu _{0}\in \mathcal {M}_{0,R}^{t}}\left\{ \pi _{\mu _{0}}^{t}\right\} \text {, a.s. }P_{\theta ^{*}}\text {,} \end{aligned}$$

  (114)

  a contradiction to (113).

• Step 2 Next suppose that there exists some

  $$\begin{aligned} \pi ^{\infty }\in \overline{\lim }\bigcup \limits _{\mu _{0}\in \mathcal {M} _{0,R}^{t}}\left\{ \pi _{\mu _{0}}^{t}\right\} \text {, a.s. }P_{\theta ^{*}} \end{aligned}$$

  (115)

  but

  $$\begin{aligned} \pi ^{\infty }\notin \Pi _{R}^{\infty }. \end{aligned}$$

  (116)

  By (115), there must be (a.s. $P_{\theta ^{*}}$) some subsequence $\left\{ \pi _{\mu _{0}^{t_{k}}}^{t_{k}}\right\} _{k\in \mathbb {N}}$ such that $\pi _{\mu _{0}^{t_{k}}}^{t_{k}}\in \Pi _{R}^{t_{k}}$ for all $k=1,2,\ldots$ which converges to $\pi ^{\infty }$. By compactness of $\mathcal { M}_{0}$, we can extract a subsequence $\left\{ \mu _{0}^{t_{k^{\prime }}}\right\} _{k^{\prime }\in \mathbb {N}}$ from $\left\{ \mu _{0}^{t_{k}}\right\} _{k\in \mathbb {N}}$ which converges to some $\mu _{0}^{*}\in \mathcal {M}_{0}$. As this $\mu _{0}^{*}$ is a cluster point of $\left\{ \mathcal {M}_{0,R}^{t}\right\} _{t\in \mathbb {N}}$, we have that $\mu _{0}^{*}\in \mathcal {M}_{0,R}^{\infty }$ so that $\pi _{\mu _{0}^{*}}^{\infty }\in \Pi _{R}^{\infty }$. Since $\left\{ \pi _{\mu _{0}^{t_{k^{\prime }}}}^{t_{k^{\prime }}}\right\} _{k^{\prime }\in \mathbb {N} }$ converges to $\pi _{\mu _{0}^{*}}^{\infty }$ as well as to $\pi ^{\infty }$, we must have that $\pi _{\mu _{0}^{*}}^{\infty }=\pi ^{\infty }$, a contradiction to (116).

Collecting Steps 1 and 2 proves the proposition. $\square$

Proof of Theorem 2

• Step 1 By Definition 7, the expected loglikelihood maximizing prior(s) will be in $\mathcal {M}_{0,\gamma }^{t}$ for all t, i.e.,

  $$\begin{aligned} \arg \max _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }\ln \prod _{i=1}^{t}\frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}\left( \theta \right) \subseteq \mathcal {M}_{0,\gamma }^{t}\text {.} \end{aligned}$$

  (117)

  By a similar formal argument as under Step 3 below, it can be shown that $\mathcal {M}_{0,\gamma }^{\infty }$ (a.s. $P_{\theta ^{*}}$) is never empty since

  $$\begin{aligned} \arg \min _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }D_\mathrm{{KL}}(\varphi _{\theta ^{*}}||\varphi _{\theta })\cdot \mu _{0}\left( \theta \right) \subseteq \mathcal {M}_{0,\gamma }^{\infty }\text { a.s. } P_{\theta ^{*}}\text {,} \end{aligned}$$

  (118)

  i.e., the expected Kullback–Leibler divergence minimizers belong asymptotically to the expected $\gamma$-loglikelihood maximizers for any value of $\gamma$.

• Step 2 Observe that any

  $$\begin{aligned} \mu _{0}^{\prime }\in \arg \min _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }D_\mathrm{{KL}}(\varphi _{\theta ^{*}}||\varphi _{\theta })\cdot \mu _{0}\left( \theta \right) \end{aligned}$$

  (119)

  belongs to (88) if (88) is non-empty.

• Step 3 Suppose now that

  $$\begin{aligned} \mu _{0}^{\prime }\in \mathcal {M}_{0,\gamma }^{\infty } \end{aligned}$$

  (120)

  but

  $$\begin{aligned} \mu _{0}^{\prime }\notin \arg \min _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }D_\mathrm{{KL}}(\varphi _{\theta ^{*}}||\varphi _{\theta })\cdot \mu _{0}\left( \theta \right) \text {.} \end{aligned}$$

  (121)

  This is only possible if there exists some subsequence $\left\{ t_{k}\right\} _{k\in \mathbb {N}}\subseteq \left\{ t\right\} _{t\in \mathbb {N} }$ such that

  $$\begin{aligned} \sum _{\theta \in \Theta }\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}^{\prime }\left( \theta \right)\ge & {} \gamma \cdot \max _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}\left( \theta \right) \Leftrightarrow \end{aligned}$$

  (122)
  $$\begin{aligned} \sum _{\theta \in \Theta }\frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}^{\prime }\left( \theta \right)\ge & {} \gamma \cdot \max _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta } \frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m} \cdot \mu _{0}\left( \theta \right) \Rightarrow \nonumber \\ \end{aligned}$$

  (123)
  $$\begin{aligned} \lim _{t_{k}\rightarrow \infty }\sum _{\theta \in \Theta }\frac{1}{t_{k}} \sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}^{\prime }\left( \theta \right)\ge & {} \lim _{t_{k}\rightarrow \infty }\gamma \cdot \max _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta } \frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m} \cdot \mu _{0}\left( \theta \right) \text {.}\nonumber \\ \end{aligned}$$

  (124)

Focus on the l.h.s. term of (124). Because $\Theta$ is finite, we can switch the sum and the limit to obtain

$$\begin{aligned} \lim _{t_{k}\rightarrow \infty }\sum _{\theta \in \Theta }\frac{1}{t_{k}} \sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}^{\prime }\left( \theta \right) =\sum _{\theta \in \Theta }\lim _{t_{k}\rightarrow \infty }\frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}^{\prime }\left( \theta \right) \text {.} \end{aligned}$$

(125)

Turn now to the r.h.s. term of (124). We are going to argue, via Berge’s (1997) maximum theorem, that we can switch the max and the limit. To this purpose, define the following inner product

$$\begin{aligned} f\left( y_{t_{k}},\mu _{0}\right)\equiv & {} \sum _{\theta \in \Theta }\frac{1}{ t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}\left( \theta \right) \end{aligned}$$

(126)

$$\begin{aligned}= & {} y_{t_{k}}\cdot \mu _{0} \end{aligned}$$

(127)

where

$$\begin{aligned} y_{t_{k}}=\left( \frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta _{1}}}{\mathrm{{d}}m},\ldots ,\frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{ \mathrm{{d}}\varphi _{\theta _{n}}}{\mathrm{{d}}m}\right) \end{aligned}$$

(128)

and

$$\begin{aligned} \mu _{0}=\left( \mu _{0}\left( \theta _{1}\right) ,\ldots ,\mu _{0}\left( \theta _{n}\right) \right) \text {.} \end{aligned}$$

(129)

Next define the value function of (126) as

$$\begin{aligned} M\left( y_{t_{k}}\right) =\max _{\mu _{0}\in \mathcal {M}_{0}}f\left( y_{t_{k}},\mu _{0}\right) \text {.} \end{aligned}$$

(130)

Since $\mathcal {M}_{0}$ is, as a closed subset of $\triangle ^{n}$, compact and f is continuous, we know from Berge’s (1997, p. 116)²¹ maximum theorem that the value function $M\left( y_{t_{k}}\right)$ is continuous. Consequently, if $\lim _{t_{k}\rightarrow \infty }y_{t_{k}}$ exists, then

$$\begin{aligned} \lim _{t_{k}\rightarrow \infty }\gamma \cdot M\left( y_{t_{k}}\right) =\gamma \cdot M\left( \lim _{t_{k}\rightarrow \infty }y_{t_{k}}\right) \text {.} \end{aligned}$$

(131)

In other words, if

$$\begin{aligned} \lim _{t_{k}\rightarrow \infty }\frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m} \end{aligned}$$

(132)

exists (a.s. $P_{\theta ^{*}}$) for all $\theta$, which we will show in a moment, then

$$\begin{aligned}&\lim _{t_{k}\rightarrow \infty }\gamma \cdot \max _{\mu _{0}\in \mathcal {M} _{0}}\sum _{\theta \in \Theta }\frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}\left( \theta \right) \end{aligned}$$

(133)

$$\begin{aligned}&\quad =\gamma \cdot \max _{\mu _{0}\in \mathcal {M}_{0}}\lim _{t_{k}\rightarrow \infty }\sum _{\theta \in \Theta }\frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}} \ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}\left( \theta \right) \end{aligned}$$

(134)

$$\begin{aligned}&\quad =\gamma \cdot \max _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }\lim _{t_{k}\rightarrow \infty }\frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\cdot \mu _{0}\left( \theta \right) \text {.} \end{aligned}$$

(135)

Recall that the law of large numbers implies for the i.i.d.

$$\begin{aligned} \ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\left( X_{1}\right) ,\ldots ,\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\left( X_{n}\right) \end{aligned}$$

(136)

that

$$\begin{aligned} \lim _{t_{k}\rightarrow \infty }\frac{1}{t_{k}}\sum \limits _{i=1}^{t_{k}}\ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}=E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\right] \text { a.s. }P_{\theta ^{*}} \end{aligned}$$

(137)

for any $\theta$. By (137) and using (125) and (135), we obtain that (124) is (a.s. $P_{\theta ^{*}}$) equivalent to

$$\begin{aligned}&\sum _{\theta \in \Theta }E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{ \mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\right] \cdot \mu _{0}^{\prime }\left( \theta \right) \ge \gamma \cdot \max _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\right] \cdot \mu _{0}\left( \theta \right) \Leftrightarrow \end{aligned}$$

(138)

$$\begin{aligned}&\quad \sum _{\theta \in \Theta }E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{ \mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\right] \cdot \mu _{0}^{\prime }\left( \theta \right) \ge \gamma \cdot \left( -\min _{\mu _{0}\in \mathcal {M} _{0}}\sum _{\theta \in \Theta }-E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\right] \cdot \mu _{0}\left( \theta \right) \right) \qquad \qquad \end{aligned}$$

(139)

$$\begin{aligned}&\quad \Leftrightarrow \nonumber \\&\quad -\sum _{\theta \in \Theta }E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{ \mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\right] \cdot \mu _{0}^{\prime }\left( \theta \right) +\gamma \cdot E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{\mathrm{{d}}\varphi _{\theta ^{*}}}{\mathrm{{d}}m}\right] \end{aligned}$$

(140)

$$\begin{aligned}&\quad \le \gamma \cdot \min _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }-E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{\mathrm{{d}}\varphi _{\theta }}{ \mathrm{{d}}m}\right] \cdot \mu _{0}\left( \theta \right) +\gamma \cdot E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{\mathrm{{d}}\varphi _{\theta ^{*}}}{\mathrm{{d}}m}\right] \nonumber \\&\quad \Leftrightarrow \nonumber \\&\quad \gamma \cdot \sum _{\theta \in \Theta }D_\mathrm{{KL}}(\varphi _{\theta ^{*}}||\varphi _{\theta })d\mu _{0}^{\prime }\left( \theta \right) -\left( 1-\gamma \right) \sum _{\theta \in \Theta }E_{\varphi _{\theta ^{*}}} \left[ \ln \frac{\mathrm{{d}}\varphi _{\theta }}{\mathrm{{d}}m}\right] \cdot \mu _{0}^{\prime }\left( \theta \right) \end{aligned}$$

(141)

$$\begin{aligned}&\quad \le \gamma \cdot \min _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }D_\mathrm{{KL}}(\varphi _{\theta ^{*}}||\varphi _{\theta })d\mu _{0}\left( \theta \right) \nonumber \\&\quad \Leftrightarrow \nonumber \\&\quad \sum _{\theta \in \Theta }D_\mathrm{{KL}}(\varphi _{\theta ^{*}}||\varphi _{\theta })\cdot \mu _{0}^{\prime }\left( \theta \right) \\&\quad \le \min _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }D_\mathrm{{KL}}(\varphi _{\theta ^{*}}||\varphi _{\theta })d\mu _{0}\left( \theta \right) +\frac{\left( 1-\gamma \right) }{\gamma }\sum _{\theta \in \Theta }E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{\mathrm{{d}}\varphi _{\theta }}{ \mathrm{{d}}m}\right] \cdot \mu _{0}^{\prime }\left( \theta \right) \text {.} \nonumber \end{aligned}$$

(142)

This proves that

$$\begin{aligned} \mu _{0}^{\prime }\notin \arg \min _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }D_\mathrm{{KL}}(\varphi _{\theta ^{*}}||\varphi _{\theta })\cdot \mu _{0}\left( \theta \right) \end{aligned}$$

(143)

is in $\mathcal {M}_{0,\gamma }^{\infty }$ if, and only if, $\mu _{0}^{\prime }$ is in (88).

Step 4 Combining the last argument with Step 1 shows that

$$\begin{aligned} \arg \min _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta \in \Theta }D_\mathrm{{KL}}(\varphi _{\theta ^{*}}||\varphi _{\theta })\cdot \mu _{0}\left( \theta \right) =\mathcal {M}_{0,\gamma }^{\infty } \end{aligned}$$

(144)

whenever (88) is empty.

Collecting results proves the theorem. $\square$

Proof of Proposition 4

• Step 1 Consider the a priori decision situation. Analogous to the argumentation for Proposition 1, we have that

  $$\begin{aligned} \mathrm{{MEU}}\left( f_{E}h,\mathcal {M}_{0}\circ \Phi \right) =\frac{1}{3}\text { and } \mathrm{{MEU}}\left( g_{E}h^{\prime },\mathcal {M}_{0}\circ \Phi \right) =\frac{2}{3} \text {.} \end{aligned}$$

  (145)

  Further, note that

  $$\begin{aligned} \mathrm{{MEU}}\left( g_{E}h,\mathcal {M}_{0}\circ \Phi \right)\le & {} \mathrm{{MEU}}\left( g_{E}h,\sum _{\theta \in \Theta }\varphi _{\theta }\left( \omega \right) \mu _{0}^{\prime }\left( \theta \right) \right) \end{aligned}$$

  (146)
  $$\begin{aligned}\le & {} EU\left( g_{E}h,\sum _{\theta \in \Theta }\varphi _{\theta }\left( \omega \right) \delta _{29}\right) \end{aligned}$$

  (147)
  $$\begin{aligned}= & {} EU\left( g_{E}h,\varphi _{29}\right) \end{aligned}$$

  (148)
  $$\begin{aligned}= & {} \frac{29}{90}<\frac{1}{3} \end{aligned}$$

  (149)

  as well as

  $$\begin{aligned} \mathrm{{MEU}}\left( f_{E}h^{\prime },\mathcal {M}_{0}\circ \Phi \right)\le & {} \mathrm{{MEU}}\left( f_{E}h^{\prime },\sum _{\theta \in \Theta }\varphi _{\theta }\left( \omega \right) \mu _{0}^{\prime \prime }\left( \theta \right) \right) \end{aligned}$$

  (150)
  $$\begin{aligned}\le & {} \mathrm{{EU}}\left( f_{E}h^{\prime },\sum _{\theta \in \Theta }\varphi _{\theta }\left( \omega \right) \delta _{31}\right) \end{aligned}$$

  (151)
  $$\begin{aligned}= & {} \mathrm{{EU}}\left( f_{E}h^{\prime },\varphi _{31}\right) \end{aligned}$$

  (152)
  $$\begin{aligned}= & {} \frac{1}{3}+\frac{29}{90}<\frac{2}{3}\text {.} \end{aligned}$$

  (153)

  Consequently, the inequalities (22–23) hold.

• Step 2 Consider the a posteriori decision situation. Note that

  $$\begin{aligned} \mathrm{{MEU}}\left( f_{E}h,\Pi _{\gamma }^{\infty }\circ \Phi \right) =\frac{1}{3} \text { and }\mathrm{{MEU}}\left( g_{E}h^{\prime },\Pi _{\gamma }^{\infty }\circ \Phi \right) =\frac{2}{3}\text {.} \end{aligned}$$

  (154)

  The specifications of $\mu _{0}^{\prime }$ and $\mu _{0}^{\prime \prime }$ imply, by Corollary 1, for some sufficiently large $\gamma$ the existence of some

  $$\begin{aligned} \delta _{\theta ^{\prime }},\delta _{\theta ^{\prime \prime }}\in \Pi _{\gamma }^{\infty } \end{aligned}$$

  (155)

  such that $\theta ^{\prime }<30<\theta ^{\prime \prime }$. Consequently,

  $$\begin{aligned} \mathrm{{MEU}}\left( g_{E}h,\Pi _{Z}^{\infty }\circ \Phi \right)\le & {} EU\left( g_{E}h,\sum _{\theta \in \Theta }\varphi _{\theta }\left( \omega \right) \delta _{\theta ^{\prime }}\right) \end{aligned}$$

  (156)
  $$\begin{aligned}\le & {} EU\left( g_{E}h,\sum _{\theta \in \Theta }\varphi _{\theta }\left( \omega \right) \delta _{29}\right) \end{aligned}$$

  (157)
  $$\begin{aligned}< & {} \frac{1}{3} \end{aligned}$$

  (158)

  as well as

  $$\begin{aligned} \mathrm{{MEU}}\left( f_{E}h^{\prime },\Pi _{Z}^{\infty }\circ \Phi \right)\le & {} \mathrm{{EU}}\left( f_{E}h^{\prime },\sum _{\theta \in \Theta }\varphi _{\theta }\left( \omega \right) \delta _{\theta ^{\prime \prime }}\right) \end{aligned}$$

  (159)
  $$\begin{aligned}\le & {} \mathrm{{EU}}\left( f_{E}h^{\prime },\sum _{\theta \in \Theta }\varphi _{\theta }\left( \omega \right) \delta _{31}\right) \end{aligned}$$

  (160)
  $$\begin{aligned}< & {} \frac{2}{3}\text {,} \end{aligned}$$

  (161)

  which proves the inequalities (24–25).

$\square$

Proof of Proposition 5

By the proof of Theorem 2 (cf., inequality (138) as well as Step 2.), $\mu _{0}^{\prime }\in \mathcal {M} _{0,\gamma }^{\infty }$ if, and only if,

$$\begin{aligned} \sum _{\theta ^{\prime }\in \Theta }E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{\mathrm{{d}}\varphi _{\theta ^{\prime }}}{\mathrm{{d}}m}\right] \cdot \mu _{0}^{\prime }\left( \theta ^{\prime }\right) \ge \gamma \cdot \max _{\mu _{0}\in \mathcal {M}_{0}}\sum _{\theta ^{\prime }\in \Theta }E_{\varphi _{\theta ^{*}}}\left[ \ln \frac{\mathrm{{d}}\varphi _{\theta ^{\prime }}}{\mathrm{{d}}m}\right] \cdot \mu _{0}\left( \theta ^{\prime }\right) \text {.} \end{aligned}$$

(162)

By Assumption 2, we have, for any $\theta \in \Theta$,

$$\begin{aligned} \delta _{\theta }\in \Pi _{\gamma }^{\infty } \end{aligned}$$

(163)

if, and only if,

(164)

(165)

(166)

$$\begin{aligned} \frac{\mathrm{{d}}\varphi _{\theta ^{*}}\left( \omega _{2}\right) \cdot \ln \mathrm{{d}}\varphi _{\theta }\left( \omega _{2}\right) +\left( \frac{2}{3}-\mathrm{{d}}\varphi _{\theta ^{*}}\left( \omega _{2}\right) \right) \cdot \ln \left( \frac{2 }{3}-\mathrm{{d}}\varphi _{\theta }\left( \omega _{2}\right) \right) }{\mathrm{{d}}\varphi _{\theta ^{*}}\left( \omega _{2}\right) \cdot \ln \mathrm{{d}}\varphi _{\theta ^{*}}\left( \omega _{2}\right) +\left( \frac{2}{3}-\mathrm{{d}}\varphi _{\theta ^{*}}\left( \omega _{2}\right) \right) \cdot \ln \left( \frac{2}{3} -\mathrm{{d}}\varphi _{\theta ^{*}}\left( \omega _{2}\right) \right) }\\\le & {} \gamma \nonumber \\\Leftrightarrow & {} \nonumber \end{aligned}$$

(167)

$$\begin{aligned} \frac{\frac{1}{3}\cdot \ln \frac{\theta }{90}+\frac{1}{3}\cdot \ln \left( \frac{60-\theta }{90}\right) }{\frac{2}{3}\cdot \ln \frac{1}{3}}\le & {} \gamma \text {,}\nonumber \\ \end{aligned}$$

(168)

which proves the proposition. $\square$