Learning and Pooling, Pooling and Learning

Abstract

We explore which types of probabilistic updating commute with convex IP pooling (Stewart and Ojea Quintana 2017). Positive results are stated for Bayesian conditionalization (and a mild generalization of it), imaging, and a certain parameterization of Jeffrey conditioning. This last observation is obtained with the help of a slight generalization of a characterization of (precise) externally Bayesian pooling operators due to Wagner (Log J IGPL 18(2):336–345, 2009). These results strengthen the case that pooling should go by imprecise probabilities since no precise pooling method is as versatile.

Appendices

Appendix: Proofs

Proof of Proposition 2

Proof

We follow through Wagner’s proof for the precise case (2009, Theorem 3.3), adapting it for IP where necessary.

$(\Rightarrow )$ Assume that ${\mathcal {F}}$ is externally Bayesian, i.e., for all profiles and any likelihood function, ${\mathcal {F}}^\lambda (\varvec{p}_1,\ldots , \varvec{p}_n) = {\mathcal {F}}(\varvec{p}_1^\lambda ,\ldots , \varvec{p}_n^\lambda )$. We want to show that, for all partitions $\varvec{E} = \{E_k\}$ of $\Omega$ and all profiles in ${\mathbb {P}}^n$,

$$\begin{aligned} {\mathcal {F}}_J^{\varvec{E}}(\varvec{p}_1,\ldots , \varvec{p}_n)= & {} \left\{ \dfrac{\sum _k b_k \varvec{p}[\cdot \in E_k]}{\sum _k b_k \varvec{p}(E_k)}: \varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n)\right\} \\= & {} {\mathcal {F}}\left( \dfrac{\sum _k b_k \varvec{p}_1[\cdot \in E_k]}{\sum _k b_k \varvec{p}_1(E_k)},\ldots , \dfrac{\sum _k b_k \varvec{p}_n[\cdot \in E_k]}{\sum _k b_k \varvec{p}_n(E_k)}\right) \\= & {} {\mathcal {F}}(\varvec{p}_{1J}^{\varvec{E}},\ldots , \varvec{p}_{nJ}^{\varvec{E}}) \end{aligned}$$

where the first and last equalities are definitional. Recall the definition of $b_k$: $b_k = {\mathcal {B}}(\varvec{q},\varvec{p};E_k:E_1) = \dfrac{\varvec{q}(E_k)/\varvec{q}(E_1)}{\varvec{p}(E_k)/\varvec{p}(E_1)}$, $k = 1, 2,\ldots$ Set $\lambda (\omega ) = \sum _k b_k [\omega \in E_k]$. Wagner observes the following chain of equalities then obtains for $\varvec{p}_i, i = 1,\ldots , n$ (2009, (3.10), p. 342):

$$\begin{aligned} (\star )\sum _{\omega \in \Omega } \lambda (\omega )\varvec{p}_i(\omega ) = \sum _{\omega \in \Omega }\varvec{p}_i(\omega )\sum _k b_k [\omega \in E_k] = \sum _k b_k \sum _{\omega \in \Omega }\varvec{p}_i(\omega )[\omega \in E_k] = \sum _k b_k \varvec{p}_i(E_k) \end{aligned}$$

Since each of the terms $b_k \varvec{p}_i(E_k)$ is positive and $\sum _k b_k \varvec{p}_i(E_k) < \infty$, $\lambda$ is a likelihood function for $\varvec{p}_i,$ with $\varvec{p}^{\lambda}_{i}$ a defined, updated pmf for $i = 1,\ldots , n.$ Using $(\star )$, we can obtain

$$\begin{aligned} {\mathcal {F}}(\varvec{p}_{1J}^{\varvec{E}},\ldots , \varvec{p}_{nJ}^{\varvec{E}}) = {\mathcal {F}}\left( \frac{\varvec{p}_1\lambda (\cdot )}{\sum _{\omega ' \in \Omega }\varvec{p}_1(\omega ')\lambda (\omega ')},\ldots , \frac{\varvec{p}_n\lambda (\cdot )}{\sum _{\omega ' \in \Omega } \varvec{p}_n(\omega ')\lambda (\omega ')}\right) \end{aligned}$$

by substituting, for each $i=1,\ldots , n$, $\lambda (\cdot )$ for $\sum _k b_k [\omega \in E_k]$ in the numerator and $\sum _{\omega ' \in \Omega } \varvec{p}_i(\omega ')\lambda (\omega ')$ for $\sum _k b_k \varvec{p}_i(E_k)$ in the denominator. But by definition,

$$\begin{aligned} {\mathcal {F}}\left( \frac{\varvec{p}_1\lambda (\cdot )}{\sum _{\omega ' \in \Omega }\varvec{p}_1(\omega ')\lambda (\omega ')},\ldots , \frac{\varvec{p}_n\lambda (\cdot )}{\sum _{\omega ' \in \Omega } \varvec{p}_n(\omega ')\lambda (\omega ')}\right) = {\mathcal {F}}(\varvec{p}_1^\lambda ,\ldots , \varvec{p}_n^\lambda ) \end{aligned}$$

and by assumption ${\mathcal {F}}(\varvec{p}_1^\lambda ,\ldots , \varvec{p}_n^\lambda )={\mathcal {F}}^\lambda (\varvec{p}_1,\ldots , \varvec{p}_n)$. By definition, ${\mathcal {F}}^\lambda (\varvec{p}_1,\ldots , \varvec{p}_n) = \{\varvec{p}^\lambda : \varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n)\}$. But, for all $\varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n)$, $\varvec{p}^\lambda = \frac{\sum _k b_k \varvec{p}[\cdot \in E_k]}{\sum _k b_k \varvec{p}(E_k)}$. Hence, ${\mathcal {F}}^\lambda (\varvec{p}_1,\ldots , \varvec{p}_n) = {\mathcal {F}}_J^{\varvec{E}}(\varvec{p}_1,\ldots , \varvec{p}_n)$. So, ${\mathcal {F}}_J^{\varvec{E}}(\varvec{p}_1,\ldots , \varvec{p}_n) = {\mathcal {F}}(\varvec{p}_{1J}^{\varvec{E}},\ldots , \varvec{p}_{nJ}^{\varvec{E}})$ follows from the assumption.

$(\Leftarrow )$ Suppose that ${\mathcal {F}}$ satisfies $\textit{CJC}_W$ and that $\lambda$ is a likelihood function for $\varvec{p}_i, i = 1,\ldots , n$. Let $(\omega _1, \omega _2,\ldots )$ be a list of all of those $\omega \in \Omega$ such that $\lambda (\omega ) > 0$, and let $\varvec{E} = \{E_1, E_2,\ldots \},$ where $E_i:\,= \{\omega _i\}.$ Setting $b_k = \frac{\lambda (\omega _k)}{\lambda (\omega _1)}$ for $k = 1, 2,\ldots$, it follows that $b_k>0$ and that $b_1=1$. Since $\lambda$ is a likelihood for $\varvec{p}_i, i = 1,\ldots , n,$ we have $\sum _k b_k \varvec{p}_i(E_k)<\infty , i = 1,\ldots , n,$ and that $(\varvec{q}_1,\ldots , \varvec{q}_n) \in {\mathbb {P}}^n,$ where $\varvec{q}_i(\omega ):\,= \frac{\sum _k b_k \varvec{p}_i(\omega )[\omega \in E_k]}{\sum _k b_k \varvec{p}_i(E_k)}.$ From $\textit{CJC}_W$, it follows that $1)\ 0< \sum _k b_k \varvec{p}(E_k) < \infty$ for all $\varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n),$ and that $2)\ {\mathcal {F}}_J^{\varvec{E}}(\varvec{p}_1,\ldots , \varvec{p}_n) = {\mathcal {F}}(\varvec{p}_{1J}^{\varvec{E}},\ldots , \varvec{p}_{nJ}^{\varvec{E}})$. 1) implies that $0<\sum _{\omega \in \Omega } \lambda (\omega ) \varvec{p}(\omega ) < \infty$ for all $\varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n)$, and 2) implies that ${\mathcal {F}}^\lambda (\varvec{p}_1,\ldots , \varvec{p}_n) = {\mathcal {F}}(\varvec{p}_1^\lambda ,\ldots , \varvec{p}_n^\lambda )$ (since substituting the definition of $b_k$ in terms of $\lambda$ in $\frac{\sum _k b_k \varvec{p}_i(\omega )[\omega \in E_k]}{\sum _k b_k \varvec{p}_i(E_k)}$, the formula for obtaining the $\varvec{q}_i$, reduces that formula to the formula for updating on that $\lambda$). $\square$

Proof of Proposition 5

Proof

We provide a case in which convex IP pooling and Jeffrey conditionalization as standardly construed do not commute. Let $\varvec{q}_i$ come from $\varvec{p}_i$ by Jeffrey conditionalization, and let $\varvec{q}$ be a common posterior distribution over partition $\varvec{E}$ for $\varvec{p}_i$, $i = 1,\ldots , n$. Let ${\mathcal {F}}_{J}^{\varvec{E}}(\varvec{p}_1,\ldots , \varvec{p}_n)$ come from ${\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n)$ by Jeffrey conditionalizing each $\varvec{p}_i$ using $\varvec{q}$, the common posterior distribution over $\varvec{E}$. We offer a counterexample to commutativity in which ${\mathcal {F}}_J^{\varvec{E}}(\varvec{p}_1,\ldots , \varvec{p}_n) \ne {\mathcal {F}}(\varvec{q}_1,\ldots , \varvec{q}_n)$.

Let $\Omega = \{\omega _1, \omega _2, \omega _3, \omega _4\}$, and consider the following two pmfs listed in Table 2. Let $\varvec{E} = \{E_1, E_2\}$ with $E_1 = \{\omega _1, \omega _2\}$ and $E_2 = \{\omega _3, \omega _4\}$ be a partition of $\Omega$. Jeffrey updating both pmfs using $\varvec{q}$, where $\varvec{q}(E_1) = 2/3$ and $\varvec{q}(E_2) = 1/3$, we obtain the following posteriors listed in (Table 3).

Table 2 Priors

Table 3 Posteriors

Consider the $.50-.50$ mixture of $\varvec{p}_1$ and $\varvec{p}_2$, $\varvec{p}^\star = 0.5\varvec{p}_1 + 0.5\varvec{p}_2$. It is clear that $\varvec{p}^\star \in {\mathcal {F}}(\varvec{p}_1, \varvec{p}_2)$. Jeffrey conditionalizing $\varvec{p}^\star$ with $\varvec{q}$ gives us $\varvec{q}^\star$. In particular, $\varvec{q}^\star (\omega _1) = 2/9$ and $\varvec{q}^\star (\omega _3) = 4/21$. It is clear that $\varvec{q}^\star \in {\mathcal {F}}^J_{\varvec{E}}(\varvec{p}_1, \varvec{p}_2)$. Any $\varvec{q}_\star \in {\mathcal {F}}(\varvec{q}_1, \varvec{q}_2)$ is of the form $\varvec{q}_\star = \alpha \varvec{q}_1 + (1 - \alpha ) \varvec{q}_2$ for $\alpha \in [0, 1]$.

Suppose that ${\mathcal {F}}_J^{\varvec{E}}(\varvec{p}_1, \varvec{p}_2) = {\mathcal {F}}(\varvec{q}_1, \varvec{q}_2)$. Then, there is a $\varvec{q}_\star \in {\mathcal {F}}(\varvec{q}_1, \varvec{q}_2)$ such that $\varvec{q}^\star = \varvec{q}_\star$. In particular, $\varvec{q}_\star (\omega _1) = 2/9$ and $\varvec{q}_\star (\omega _3) = 4/21$. Letting $\varvec{q}_\star (\omega _1) = 2/9$, we can compute $\alpha$.

$$\begin{aligned} 2/9 = \varvec{q}_\star (\omega _1) = \alpha \varvec{q}_1(\omega _1) + (1 - \alpha )\varvec{q}_2(\omega _1) = \alpha 1/3 + (1 - \alpha ) 2/15 \end{aligned}$$

Solving, we get $\alpha = 4/9$. However, we are supposed to have $\varvec{q}_\star (\omega _3) = 4/21$. For $\alpha = 4/9$, that is not the case.

$$\begin{aligned} \varvec{q}_\star (\omega _3) = \alpha \varvec{q}_1(\omega _3) + (1 - \alpha ) \varvec{q}_2(\omega _3) = 4/9(1/6) + 5/9(2/9) = 16/81 > 4/21 = \varvec{q}^\star (\omega _3) \end{aligned}$$

It follows that ${\mathcal {F}}_J^{\varvec{E}}(\varvec{p}_1, \varvec{p}_2) \ne {\mathcal {F}}(\varvec{q}_1, \varvec{q}_2)$. $\square$

Proof of Proposition 6

Proof

We want to show that ${\mathcal {F}}(\varvec{q}_1,\ldots , \varvec{q}_n) = {\mathcal {F}}_I^E(\varvec{p}_1,\ldots , \varvec{p}_n)$, where $\varvec{q}_i$ comes from $\varvec{p}_i$ by general imaging on E, and ${\mathcal {F}}_I^E(\varvec{p}_1,\ldots , \varvec{p}_n)$ comes from ${\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n)$ by general imaging each $\varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n)$ on E. Again, we show both inclusions. In the proofs, we appeal to the fact any element of a convex set is some convex combination of the generating, extreme points: For any $\varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n), \varvec{p}=\sum _{i=1}^n \alpha _i\varvec{p}_i$, where $\alpha _i \ge 0$ for $i = 1,\ldots , n$, and $\sum _{i=1}^n \alpha _i= 1$ (see, e.g., Stewart & Ojea Quintana 2017, Lemma 1).

Let $\varvec{q}\in {\mathcal {F}}(\varvec{q}_1,\ldots , \varvec{q}_n)$. So, $\varvec{q}= \sum _{i=1}^n \alpha _i\varvec{q}_i$. Since $\varvec{q}$ is a linear pool of $\varvec{q}_i$ for $i = 1,\ldots , n$, by Gärdenfors’ result, Theorem 5, $\varvec{q}$ is also the result of imaging $\varvec{p}= \sum _{i=1}^n\alpha _i\varvec{p}_i$ on E, because linear pooling and general imaging commute. Since $\varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n)$, it follows that $\varvec{q}\in {\mathcal {F}}_I^E(\varvec{p}_1,\ldots , \varvec{p}_n)$.

For the other direction, assume that $\varvec{q}\in {\mathcal {F}}_I^E(\varvec{p}_1,\ldots , \varvec{p}_n)$. So, $\varvec{q}$ is the result of general imaging some $\varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n)$ on E. For any $\varvec{p}\in {\mathcal {F}}(\varvec{p}_1,\ldots , \varvec{p}_n), \varvec{p}= \sum _{i=1}^n\alpha _i\varvec{p}_i$. By Gärdenfors’ result, $\varvec{q}= \sum _{i=1}^n \alpha _i \varvec{q}_i$, where the $\varvec{q}_i$ come from the $\varvec{p}_i$ by general imaging on E, because general imaging and linear pooling commute. But then it follows that $\varvec{q}\in {\mathcal {F}}(\varvec{q}_1,\ldots , \varvec{q}_n)$. $\square$