Demystifying Dilation

Abstract

Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that E is stochastically independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (2) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence between dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models.

Appendix

We now return to our discussion of the technical machinery for the general case from Sect. 5. In the general setting, where infinitely many events inhabit the algebra ${\fancyscript{A}}$, closure with respect to the total variation norm topology (i.e., strong topology) or with respect to the weak topology (distinct from the weak*-topology) introduces more closed sets than the weak*-topology. The former topologies are stronger and indeed too strong for the purposes of imprecise probabilities, which demand a very weak topology, the weak*-topology. Hence, the discussion in Seidenfeld and Wasserman (1993, pp. 1141–1443), which employs the total variation norm, suits sets of probabilities over an algebra consisting of finitely many events.

A set of probabilities in this general setting is always norm-bounded (with respect to the norm-dual), so a weak*-closed set of probabilities is weak*-compact. Accordingly, every weak*-continuous functional on the dual space achieves its minimum at the extreme points of a closed convex (i.e., a weak*-closed and convex) set of probabilities, and since all and only evaluation functionals are weak*-continuous, the lower probability of an event is the minimum number assigned to the event by all probability functions in the set, where as in the finite setting a probability function from the collection of extreme points of the set witnesses the lower probability of the event.

Much as in the finite case, any compact convex set of probabilities is the closed convex hull of its extreme points (i.e, any convex weak*-compact set of probabilities is the weak*-closed convex hull of its extreme points). However, while in the finite setting a compact convex set of probabilities is identified with the convex hull of its extreme points, in the general setting a compact convex set of probabilities is identified with the closed convex hull of its extreme points. In both the finite and general setting, the convex hull of the set of extreme points is dense in the compact convex set in question, but in the general setting, a compact convex set may properly contain the convex hull of its extreme points and so the hull must in addition be closed.

Proof of Proposition 4.3

Let ${\mathbb{P}({\fancyscript{A}})}$ be the set of all probability functions on ${\fancyscript{A}}$, and let ${\mathbb{D}=_{df}\{p\in\mathbb{P}({\fancyscript{A}}):p(E)\geq\underline{\hbox{P}}(E)}$ for all ${E\in\fancyscript{A}\}}$. Observe that ${\mathbb{P}\subseteq\mathbb{D}}$ and that ${\mathbb{D}}$ is convex and weak*-closed and so weak*-compact. In addition, since ${\hbox{co}(\mathbb{P})\subseteq \mathbb{D}}$, it follows that the ${\overline{\hbox{co}}({\mathbb{P}})\subseteq\mathbb{D}}$, a weak*-compact set, whence ${\underline{\hbox{P}}(E) =\inf\{p(E): p \in \mathbb{P}\}=\min\{p(E): p \in\overline{\hbox{co}}(\mathbb{P})\}}$, the minimum being achieved at an extreme point of ${\overline{\hbox{co}}(\mathbb{P})}$.

Now given $\underline{\hbox{P}}(H)>0$, let ${\mathbb{D}[H]}$ be defined by:

$$ {\mathbb{D}}[H] \quad=_{df}\quad \{p\in{\mathbb{P}}({{\fancyscript{A}}}): p(E\cap H) \geq \underline{\hbox{P}}(E|H)p(H)\;\hbox{for all}\; E\in{\fancyscript{A}}\}. $$

As before, observe that ${\mathbb{P}\subseteq\mathbb{D}[H]}$ and that ${\mathbb{D}[H]}$ is convex and weak*-closed and so weak*-compact. Hence, ${\overline{\hbox{co}}(\mathbb{P})\subseteq\mathbb{D}[H]}$. Since $\underline{\hbox{P}}(H)>0$, by the first part it follows that ${\overline{\hbox{co}}(\mathbb{P})\subseteq\{p\in\mathbb{P}({\fancyscript{A}}):p(H)>0\}}$. Since with respect to the weak* topology on the dual space every evaluation functional f ^* is a real-valued continuous linear functional on the dual space with f ^*(p) = p(f) for each p from the dual space, it follows that $\frac{(E\cap H)^*}{H^*}$ is a continuous explicitly quasiconcave function of p on ${\overline{\hbox{co}}(\mathbb{P})}$, so it attains a minimum at an extreme point of ${\overline{\hbox{co}}(\mathbb{P})}$ (thus, ${\min\{\frac{(E\cap H)^*(p)}{H^*(p)}:p\in\overline{\hbox{co}}(\mathbb{P})\}=\min\{\frac{p(E\cap H)}{p(H)}:p\in\overline{\hbox{co}}(\mathbb{P})\}}$ exists and is an extreme point of ${\overline{\hbox{co}}(\mathbb{P})}$), whence ${\underline{\hbox{P}}(E|H)= \inf\{p(E|H): p \in \mathbb{P}\}=\min\{p(E|H):p\in\overline{\hbox{co}}(\mathbb{P})\}}$, as desired. □

Proof of Proposition 5.1

We first show that (i) $\Longleftrightarrow$ (iii), and we then show that (i) $\Longleftrightarrow$ (ii).

(i)⇒(iii):

Suppose that ${\fancyscript{B}}$ dilates E. Then for each $i\in I,\;\underline{\hbox{P}}(E|H_{i})<\underline{\hbox{P}}(E)\leq\overline{\hbox{P}}(E)<\overline{\hbox{P}}(E|H_{i})$. For each $i\in I$, consider the real-valued function $\underline{\varepsilon}_{i}(p)=_{df}|p(E|H_{i})-\underline{\hbox{P}}(E|H_{i})|$ and the real-valued function S _p,i (E, H _i ). We recall that the weak* topology on the dual space is a locally convex topological vector space with respect to which every evaluation functional f ^* is a real-valued continuous linear functional on the dual space. It follows that $\underline{\varepsilon}_{i}(p)$ and S _p,i (E, H _i ) are continuous functions of p on ${\mathbb{P}}$ for each $i\in I$.

Now let $i\in I$. By hypothesis, there is ${p_{1}\in \mathbb{P}}$ such that S _{p_1}, i(E, H _i ) > 1, so importantly, C ⁺ _i is nonempty. Then since ${C^{+}_{i}=_{df}\{p\in\mathbb{P}:\hbox{S}_p(E,H_{i})\geq 1\}}$ is a weak*-closed and so weak*-compact set, it follows that $\underline{\varepsilon}_{i}$ achieves a minimum value on C ⁺ _i (and the set of minimizers of $\underline{\varepsilon}_{i}$ is also compact). Choosing a minimizer ${p_{i}\in \mathbb{P}}$ of $\underline{\varepsilon}_{i}$, we see that for every ${p\in\mathbb{P}}$, if $|p(E|H_{i})- \underline{\hbox{P}}(E|H_{i}) |<\underline{\varepsilon}_{i}(p_{i})= |p_{i}(E| H_{i}) - \underline{\hbox{P}}(E|H_{i})|$, then S _p (E, H _i ) < 1. We have accordingly shown that ${\underline{\mathbb{P}}(E|H_{i},\underline{\varepsilon}_{i}(p_{i}))\subseteq \hbox{S}_{\mathbb{P}}^{-}(E,H_{i})}$. Of course, we may suppress reference to the minimizer p _i in $\underline{\varepsilon}_{i}(p_{i})$. The other inclusion ${\overline{\mathbb{P}}(E|H_{i},\overline{\varepsilon}_{i})\subseteq\hbox{S}_{\mathbb{P}}^{+}(E,H_{i})}$ is established by a similar argument.

(iii) $\Leftarrow$ (i):

Suppose that ${\fancyscript{B}}$ does not dilate E. Then there is $i\in I$ such that $\underline{\hbox{P}}(E|H_{i})\geq \underline{\hbox{P}}(E)$ or $\overline{\hbox{P}}(E)\geq \overline{\hbox{P}}(E|H_{i})$. We may assume without loss of generality that $\underline{\hbox{P}}(E|H_{i})\geq \underline{\hbox{P}}(E)$ for some $i\in I$. First, if $\underline{\hbox{P}}(E)\leq\overline{\hbox{P}}(E)\leq \underline{\hbox{P}}(E|H_{i})$, then choosing a minimizer ${p\in\mathbb{P}}$ of $\underline{\hbox{P}}(E|H_{i})$, we see that S _p (E, H _i ) ≥ 1. Second, if $\underline{\hbox{P}}(E)< \underline{\hbox{P}}(E|H_{i})<\overline{\hbox{P}}(E)$, then for every $\varepsilon>0$ we can find a convex combination ${p\in \mathbb{P}}$ of ${p_{0},p_{1}\in\mathbb{P}}$ assigning a probability to E within $\varepsilon$-distance below $\underline{\hbox{P}}(E|H_{i})$, where $\underline{\hbox{P}}(E)\leq p_{0}(E)< \underline{\hbox{P}}(E|H_{i})< p_{1}(E)\leq\overline{\hbox{P}}(E)$, so S _p (E, H _i ) > 1. Third, if $\underline{\hbox{P}}(E)= \underline{\hbox{P}}(E|H_{i})<\overline{\hbox{P}}(E)$, then choosing a minimizer ${p\in \mathbb{P}}$ of $\underline{\hbox{P}}(E)$, we see that S _p (E, H _i ) ≥ 1. Evidently, the conditions of the main claim cannot be jointly satisfied.

(i) ⇔ (ii):

On the one hand, suppose that (i) obtains. Then since (iii) accordingly obtains, define $(\varepsilon_{i})_{i\in I}$ by setting $\varepsilon_{i}=_{df}\min(\underline{\varepsilon}_{i},\overline{\varepsilon}_{i})$ for each $i\in I$. Clearly the inclusions still obtain for the $\varepsilon_{i}$. On the other hand, if (ii) obtains, obviously by setting $\underline{\varepsilon}_{i}=_{df}\varepsilon_{i}$ and $\overline{\varepsilon}_{i}=_{df}\varepsilon_{i}$ for each $i\in I$, condition (iii) obtains and so (i) obtains.

□

Proof of Corollary 5.2

Only (i)$\Longleftrightarrow$(iii) requires proof. On the one hand, suppose that ${\fancyscript{B}}$ dilates E. Then for each $i\in I,\;\underline{\hbox{P}}(E|H_{i})<\underline{\hbox{P}}(E)\leq\overline{\hbox{P}}(E)<\overline{\hbox{P}}(E|H_{i})$. By Proposition 4.3 we have ${\underline{\hbox{P}}(A) = \min\{p(A): p \in\mathbb{P}_{*}\},\;\underline{\hbox{P}}(A|B) = \min\{p(A|B): p \in\mathbb{P}_{*}\},\;\overline{\hbox{P}}(A) = \max\{p(A): p \in\mathbb{P}_{*}\}}$, and ${\overline{\hbox{P}}(A|B) = \max\{p(A|B): p \in\mathbb{P}_{*}\}}$ for every ${A,B\in\fancyscript{A}}$ with $\underline{\hbox{P}}(B)>0$, so ${\fancyscript{B}}$ dilates E with respect to ${\mathbb{P}_{*}}$. It follows from Proposition 5.1 that there are positive $(\underline{\varepsilon}_{i},\overline{\varepsilon}_{i})_{i\in I}$ in ${\mathbb{R}}$ such that ${\underline{\mathbb{P}}_{*}(E|H_{i},\underline{\varepsilon}_{i})\subseteq\hbox{S}_{*}^{-}(E,H_{i})}$ and ${\overline{\mathbb{P}}_{\ast}(E|H_{i},\overline{\varepsilon}_{i})\subseteq\hbox{S}_{\ast}^{+}(E,H_{i})}$ for every $i\in I$.

On the other hand, suppose that there are positive $(\underline{\varepsilon}_{i},\overline{\varepsilon}_{i})_{i\in I}$ in ${\mathbb{R}}$ such that for every ${i\in I,\;\underline{\mathbb{P}}_{*}(E|H_{i},\underline{\varepsilon}_{i})\subseteq\hbox{S}_{*}^{-}(E,F)}$ and ${\overline{\mathbb{P}}_{*}(E|H_{i},\overline{\varepsilon}_{i})\subseteq\hbox{S}_{*}^{+}(E,H_{i})}$. Then by Proposition 5.1, ${\fancyscript{B}}$ dilates E with respect to ${\mathbb{P}_{*}}$, so for every event $i\in I,\;\underline{\hbox{P}}(E|H_{i})<\underline{\hbox{P}}(E)\leq\overline{\hbox{P}}(E)<\overline{\hbox{P}}(E|H_{i})$, whence again by Proposition 4.3 it follows that ${\fancyscript{B}}$ dilates E with respect to ${\mathbb{P}}$, as desired.

Clearly, that the radii $\underline{\varepsilon}_{i}$ and $\overline{\varepsilon}_{i}$ may be chosen in the way described follows from Proposition 5.1. The other implications are trivial consequences of what we have just shown.□

Proof of Proposition 5.3

On the one hand, if ${\fancyscript{B}}$ strictly dilates E, then by Corollary 5.2 there are ${(\underline{\varepsilon}_{i})_{i\in I}\in{\mathbb{R}}_{+}^{I}}$ and ${(\overline{\varepsilon}_{i})_{i\in I}\in{\mathbb{R}}_{+}^{I}}$ such that for every $i\in I,$ $\underline{\mathbb{P}}_{*}(E|H_{i},\underline{\varepsilon}_{i})\subseteq\hbox{S}_{*}^{-}(E,H_{i})$ and $\overline{\mathbb{P}}_{*}(E|H_{i},\overline{\varepsilon}_{i})\subseteq\hbox{S}_{*}^{+}(E,H_{i})$. Let

$$ \varepsilon \quad=_{df}\quad \min\left\{\delta: \delta=\underline{\varepsilon}_{i}\hbox{ or }\delta=\overline{\varepsilon}_{i} \hbox{ for some } i\in I\right\}. $$

Then $\varepsilon>0$, and clearly the inclusions still obtain. On the other hand, if (ii) obtains, then part (ii) of Corollary 5.2 obtains, so ${\fancyscript{B}}$ strictly dilates E. □