1. Introduction

Merging of opinions results (e.g., Blackwell and Dubins Reference Blackwell and Dubins1962; Gaifman and Snir Reference Gaifman and Snir1982; Schervish and Seidenfeld Reference Schervish and Seidenfeld1990; Kalai and Lehrer Reference Kalai and Lehrer1994; Huttegger Reference Huttegger2015b) have underwritten Bayesian rejoinders to complaints about the subjective nature of personal probability (e.g., Savage Reference Savage1954/1972; Schervish and Seidenfeld Reference Schervish and Seidenfeld1990; Earman Reference Earman1992). Such results establish that sufficiently similar priors achieve consensus in the long run when fed the same increasing stream of evidence. Initial subjectivity, the line goes, is of mere transient significance, giving way to intersubjective agreement eventually. Schervish and Seidenfeld (Reference Schervish and Seidenfeld1990) show that Blackwell and Dubins’s classic result (Reference Blackwell and Dubins1962) can be extended to certain sets of probability functions, while Huttegger (Reference Huttegger2015b) provides sufficient conditions for merging of opinions for Jeffrey conditioning, a generalization of Bayesian conditionalization. We establish that Huttegger’s merging result for Jeffrey conditioning can in turn be extended to sets of probability functions in analogy to Schervish and Seidenfeld’s extension (sec. 7). We also extend Huttegger’s convergence result (to a “maximally informed opinion”) for Jeffrey conditioning to convergence of a set of probabilities to a shared maximally informed opinion (sec. 8). Convergence to a maximally informed opinion is a (weak) Jeffrey conditioning analogue of Bayesian “convergence to the truth” for conditional probabilities.

There are a number of motivations for considering merging and convergence for sets of probabilities. For example, according to proponents of imprecise probabilities (IP), rationality does not always demand numerically precise probability judgments of an agent—to say nothing of the descriptive adequacy of such precise judgments (e.g., Levi Reference Levi1974; Walley Reference Walley1991). In this context, merging and convergence results show that a single agent with a certain sort of imprecise credal state fully expects to uniformly strengthen her point of view toward a consensus among the set (merging) and for the consensus point of view to stabilize (convergence). In effect, evidence reduces an agent’s imprecise credal state toward a precise probability, at least in certain cases. In section 9.2, we discuss how our results establish that “certain evidence” is unnecessary for the reduction of an imprecise credal state toward a stable, precise probability judgment. Another motivation comes from considering these points in a social setting. In such a setting, sets represent beliefs in some community. In section 9.3, we detail an application of our study to probabilistic opinion pooling with IP (Elkin and Wheeler Reference Elkin and Wheeler2018; Stewart and Ojea Quintana Reference Stewart and Quintana2018b).

We also discuss Schervish and Seidenfeld’s (Reference Schervish and Seidenfeld1990) claim that such results for sets of probabilities allow us to relax so-called dynamic coherence to a large extent. In the case of Bayesian updating, for certain asymptotic properties like convergence to certainty and long-run consensus, it is enough that posteriors take any value in a particular set. In principle, our propositions 1 and 2 establish that a similar point can be made about relaxing dynamic coherence for Jeffrey conditioning. However, we discuss some reservations about the precise significance of this point in section 9.1.

2. Preliminaries

Let Ω be a sample space, a set of elementary events or possible worlds. We let 𝔉 denote a σ-algebra of subsets of Ω, that is, a nonempty set of subsets of Ω closed under complementation and countable unions. Elements of 𝔉 can be interpreted as events or propositions. For example, Ω could be the set of all infinite sequences of tosses of a coin, and 𝔉 would be the set of relevant coin-tossing events. Included in 𝔉 would be propositions describing finite initial segments of a sequence like the first flip landing heads, as well as propositions describing the limiting behavior of the sequence such as ${{\lim }_{n\to \infty }H}_n/n=1/4$, where H_n is the total number of heads on the first n flips. Throughout the article uppercase blackboard letters like ℙ and ℚ denote (countably additive) probability measures on (Ω, 𝔉).Footnote ¹ An event $A\in F$ is said to occur almost surely with respect to ℙ, or a.s. (ℙ), if $\mathbb{P}\left[A\right]=1$.

Let 𝔈₁, 𝔈₂, … be an infinite sequence of (finite) partitions of Ω. We suppose that, for any n, $E_{n+1}$ is a refinement of 𝔈_n; that is, every element of $E_{n+1}$ is a subset of an element of 𝔈_n. For any n, the partition 𝔈_n can be thought of as the possible information an agent might receive about the actual world $\omega \in \Omega$. A single flip of a coin determines a binary partition of the set of all infinite sequences of tosses: the coin lands heads or the coin lands tails.

Let 𝔉_n be the algebra generated by 𝔈_n. Apart from ∅ and Ω, 𝔉_n contains all the unions of elements of 𝔈_n. In the coin-tossing example, the algebra 𝔉_n can be thought of as the set of all coin-tossing events up to stage n. The refinement assumption on 𝔈₁, 𝔈₂, … implies that $F_n\subseteq F_{n+1}$, $n=1$, 2, … . We say that the sequence $\left\{F_n:n\in \mathbb{N}\right\}$ of σ-algebras is a filtration. We assume that this filtration increases to the background σ-algebra 𝔉 (i.e., $\sigma \left({{\cup }_{n\ge 0}F}_n\right)=F$), so that 𝔉 is generated by a countable sequence of subsets of Ω. The informal idea behind the filtration structure is that information is always increasing and eventually captures all propositions of interest.

For any events A, $E\in F$ with $\mathbb{P}\left[E\right]>0$, we use the standard ratio definition of conditional probability given an event:

$\begin{array}{}\text{(1)}& \mathbb{P}\left[A\right|E]=\frac{\mathbb{P}\left[A\cap E\right]}{\mathbb{P}\left[E\right]}\text{.}\end{array}$

For a sub-σ-algebra 𝔉_n of 𝔉, we will also use the standard definition of conditional probability given 𝔉_n: $\mathbb{P}\left[A\right|F_n]$ is a 𝔉_n-measurable function that satisfies

$\begin{array}{}\text{(2)}& \mathbb{P}\left[A\cap E\right]={\int }_E\mathbb{P}\left[A\right|F_n]\left(\omega \right)\mathbb{P}\left[d\omega \right]\end{array}$

for all $E\in F_n$.Footnote ² If we think of ω as the “actual world” regarding coin tosses, then, after observing n tosses, 𝔉_n is the collection of all propositions that the agent can distinguish as true or false in the actual world. So, just as the conditional probability of A given an event E can be interpreted as the probability of A on the supposition that E is true, so the conditional probability of A given 𝔉_n can be interpreted as the probability of A in light of the information provided by 𝔉_n.Footnote ³ In the current framework, conditional probabilities given sub-σ-algebras can be reduced (a.s.) to conditional probabilities given events; the former merely provide a convenient tool for working around conditioning on null events without adopting extra conventions or assumptions. Since each 𝔉_n is generated by a finite partition 𝔈_n, for almost every $\omega \in \Omega$ we have

$\begin{array}{}\text{(3)}& \mathbb{P}\left[A\right|F_n]\left(\omega \right)=\mathbb{P}(A\left|E_n\left(\omega \right)\right)\text{,}\end{array}$

where E_n(ω) is the unique cell of 𝔈_n that contains ω. If $\mathbb{P}\left(E_n\left(\omega \right)\right)=0$, then the left-hand side of (3) can be defined arbitrarily.

3. Merging of Opinions

What does it mean to say that two opinions merge, or that ℙ and ℚ agree in the limit? Following Blackwell and Dubins (Reference Blackwell and Dubins1962), we adopt the total variation distanced as a measure of the distance between ℙ and ℚ.

$d\left(\mathbb{P},\mathbb{Q}\right)=\underset{A\in F}{\sup }|\mathbb{P}\left[A\right]-\mathbb{Q}\left[A\right]|\mathrm{.}$

So if ℙ and ℚ are within ε according to the metric d, then they are within ε for every event $A\in F$. Now we can formulate a natural sense in which two sequences {p_n} and {q_n} of probability measures might become (and stay) close. We say that {p_n} and {q_n} merge if

$d\left(p_n,q_n\right)\to 0$

as $n\to \infty$. If {p_n} and {q_n} are updates (not necessarily Bayesian) of ℙ and ℚ respectively, we say that ℙ and ℚ merge if {p_n} and {q_n} do.

The main result of Blackwell and Dubins (Reference Blackwell and Dubins1962) concerns the merging of conditional probabilities. If learning goes by conditionalization, and the priors ℙ and ℚ do not disagree too drastically, then the Blackwell-Dubins result says ℙ and ℚ must assign probability 1 to merging. To make this precise, we say that ℚ is absolutely continuous with respect to ℙ, denoted $\mathbb{Q}\ll \mathbb{P}$, if for all $A\in F$

$\mathbb{P}\left[A\right]=0\Rightarrow \mathbb{Q}\left[A\right]=0.$

So ℚ agrees with ℙ about events bearing probability 0. If ℙ also agrees with ℚ about probability 0 events, so that $\mathbb{P}\ll \mathbb{Q}$, we say that ℙ and ℚ are equivalent or mutually absolutely continuous. In that case, we have for all $A\in F$

$\mathbb{P}\left[A\right]=0\iff \mathbb{Q}\left[A\right]=0.$

It turns out that absolute continuity is sufficient for the merging of conditional probabilities.Footnote ⁴ We write ${\mathbb{P}}_{F_n}=\mathbb{P}[\cdot |F_n]$ and ${\mathbb{Q}}_{F_n}=\mathbb{Q}[\cdot |F_n]$.Footnote ⁵

With ℚ-probability 1, ℙ merges to ℚ when $\mathbb{Q}\ll \mathbb{P}$. If $\mathbb{P}\ll \mathbb{Q}$ also, we have merging from the perspectives of both ℙ and ℚ. The almost surely qualifications are needed in the statement of theorem 1 because the conditional probabilities ${\mathbb{P}}_{F_n}$ and ${\mathbb{Q}}_{F_n}$ in this general setting are random objects; that is, they depend on $\omega \in \Omega$ (see n. 2 for details).

4. Merging of Opinions for Sets of Probabilities

Schervish and Seidenfeld (Reference Schervish and Seidenfeld1990) provide sufficient conditions to secure merging of opinions for sets of probability functions. A set of probability functions C is called convex if ℙ, $\mathbb{Q}\in C$ implies $\alpha \mathbb{P}+\left(1-\alpha \right)\mathbb{Q}\in C$, where $\alpha \in \left[0,1\right]$. That is, for any two elements of C, C includes all convex combinations of those elements. We call $\mathbb{P}\in C$ an extreme point of C if $\mathbb{P}=\alpha \mathbb{Q}+\left(1-\alpha \right)ℝ$ with ℚ, $ℝ\in C$, and $\alpha \in \left[0,1\right]$ implies $\mathbb{Q}=\mathbb{P}$ or $ℝ=\mathbb{P}$. The following result is a consequence of theorem 1.

Another way of stating the conclusion of theorem 2 is that for almost all ω and all $\epsilon >0$ there is some m such that for all $n\ge m$ and all ℙ, $\mathbb{Q}\in C$, $d\left({\mathbb{P}}_{F_n}\left(\omega \right),{\mathbb{Q}}_{F_n}\left(\omega \right)\right)<\epsilon$. That the same m works for all ℙ and ℚ is what it means for merging to be uniform.

Here is the gist of their proof. Since C is generated by finitely many extreme points, ℙ¹, …, ℙ^k, the (total variation) distance between the conditional probabilities of any of the extreme points of C is bounded by the maximum distance for pairs of conditional probabilities of extreme points. Moreover, that maximum is a bound for the distance between the conditional probabilities of any points in the whole set. To see this, note two things. First, d(ℙ, ℚ) bounds the distances d(ℙ, ℝ) and d(ℝ, ℚ), where ℝ is a convex combination of ℙ and ℚ. Second, at any stage n, the conditional probabilities of points in C can be written as convex combinations of the conditional probabilities of the extreme points; that is, there exist α₁, …, α_k with ${\alpha }_i\in \left[0,1\right]$ and ${\sum }_{i=1}^k{\alpha }_i=1$ such that ${ℝ}_{F_n}={\sum }_{i=1}^k{\alpha }_i{\mathbb{P}}_{F_n}^i$ for all $ℝ\in C$ and all n. Theorem 2 is stronger than it may at first appear. Notice that any set of probabilities that is included in a closed, convex set generated by finitely many extreme points merges uniformly by this result.

It is not easy to relax the assumption in theorem 2 that C is convex and generated by finitely many extreme points. Schervish and Seidenfeld (Reference Schervish and Seidenfeld1990) demonstrate this with several other results and examples. For instance, if C is d-compact, an assumption that is weaker than the assumptions of theorem 2, the conditional probabilities of elements of C may not exhibit almost sure uniform merging. However, Schervish and Seidenfeld show that weaker modes of merging can be achieved under weaker assumptions about C, such as d-compactness (see their corollaries 2 and 3). It is worth noting that any negative result for Bayesian conditionalization and merging carries over to Jeffrey conditioning, a generalization of conditionalization that we introduce and study below. An immediate consequence of Schervish and Seidenfeld’s results, then, is that Jeffrey updates of a d-compact set of probability measures may not a.s. merge uniformly.

There are a number of uses to which results such as theorem 2 might be put. We delay explaining some of the significance of theorem 2 and propositions 1 and 2 (to come) until section 9.

5. Jeffrey Conditioning

Bayesian conditionalization is a putative diachronic norm that specifies how probabilistic learning takes place.Footnote ⁶ A basic assumption of conditionalization is that it is propositions (or events) that are learned. In other words, learning experiences can always be represented by some $E\in F$. When an agent learns E, conditionalization says that her posterior probability ℙ_E[A] for an event $A\in F$ should be equal to her prior conditional probability of A given E (provided, of course, that this is well defined):

${\mathbb{P}}_E\left[A\right]=\mathbb{P}\left[A\right|E]\text{.}$

It is clear from (1) that conditionalization requires that E have posterior probability 1. Jeffrey conditioning relaxes both of the fundamental features of conditionalization: it does not assume that learning experiences are always represented by propositions, and it does not require that learning involves assigning propositions probability 1.

Proponents of Jeffrey conditioning want to allow for “uncertain learning.” Uncertain learning induces a change in the probabilities P_n assigned to the members of the partition 𝔈_n. Jeffrey conditioning applies just in case the change over 𝔈_n is rigid:

$\begin{array}{}\text{(4)}& {\mathbb{P}}_n\left[A\left|E\left]={\mathbb{P}}_{n-1}\right[A\right|E\right],\text{for all }E\in E_n\text{ and all }A\in F\text{.}\end{array}$

The rigidity condition (4) says that the update from ${\mathbb{P}}_{n-1}$ to ℙ_n does not change conditional probabilities given members of the partition 𝔈_n. This is equivalent to the requirement that for all members E of the partition 𝔈_n and all subsets A, B of E, the update from ${\mathbb{P}}_{n-1}$ to ℙ_n does not change the ratio of the probabilities of A and B. The law of total probability and (4) yield the familiar Jeffrey conditioning equation, which extends P_n from 𝔈_n to the entire algebra 𝔉:

$\begin{array}{}\text{(5)}& {\mathbb{P}}_n\left[A\right]=\sum _{E\in E_n}{\mathbb{P}}_{n-1}\left[A\right|E]P_n\left[E\right],\text{ for all }E\in E_n\text{ and all }A\in F\text{.}\end{array}$

If $E_n=\left\{E,E^c\right\}$ and $P_n\left[E\right]=1$, then Jeffrey conditioning reduces to standard conditionalization; that is, ${\mathbb{P}}_n\left[A\right]={\mathbb{P}}_{n-1}\left[A\right|E]$ for all $A\in F$.Footnote ⁷ Equation (5) simplifies in another way under our assumption that the sequence of partitions 𝔈₁, …, 𝔈_n is such that $E_{i+1}$ refines 𝔈_i for all $1\le i\le n-1$. In this case, after $n-1$ applications of equation (4) in equation (5), we have

$\begin{array}{}\text{(6)}& {\mathbb{P}}_n\left[A\right]=\sum _{E\in E_n}\mathbb{P}\left[A\right|E]P_n\left[E\right],\text{ for all }E\in E_n\text{ and all }A\in F\text{.}\end{array}$

Equation (6) shows that, in our framework, posterior probabilities ℙ_n are determined by their values on members of the partition 𝔈_n and prior (ℙ) conditional probabilities given members of 𝔈_n. Unless otherwise stated, we will be assuming that the events $E\in E_n$ have positive prior probability so that the right-hand side of (6) is well defined.

It is worth pointing out that, in certain situations, Jeffrey conditioning can be represented as standard Bayesian conditionalization in a richer algebra. The cases in which such so-called superconditioning is possible are characterized by Diaconis and Zabell’s superconditioning criterion (Reference Diaconis and Zabell1982, theorem 2.1). But the superconditioning criterion is not trivial; it fails to hold in many cases. Hence, Huttegger’s merging result for Jeffrey conditioning in the following section and our merging result for sets of Jeffrey updates in section 7 genuinely generalize Blackwell and Dubins’s and Schervish and Seidenfeld’s results, respectively. (For other reservations about reducing Jeffrey conditioning to Bayesian conditionalization in the setting of merging of opinions, see Huttegger’s [Reference Huttegger2015b, 630–31] discussion.)

6. Merging of Opinions for Jeffrey Conditioning

Huttegger proves an analogue of Blackwell and Dubins’s merging result for Jeffrey conditioning. Just as we considered random conditional probabilities in the case of conditionalization, we now treat posteriors over the learning partition as random. By random probability, we mean P_n[⋅] is not a determinate number but rather a (measurable) function on Ω such that P_n[⋅](ω) is a probability measure on 𝔈_n for each $\omega \in \Omega$. Let $p_n^E=P_n\left[E\right]={\mathbb{P}}_n\left[E\right]$ be a random probability assigned to $E\in E_n$. We treat future probabilities as random quantities because, from an agent’s present point of view concerning $E\in E_n$, the precise value $p_n^E$ is unknown.

Because Jeffrey conditioning is less constrained than Bayesian conditionalization, some additional assumptions figure into Huttegger’s theorem. The first additional assumption is that the sequence {Q_n} is uniformly absolutely continuous with respect to ℚ. This condition consists of two requirements: (i) $Q_n\ll \mathbb{Q}{|}_{F_n}$ for all n, and (ii) for every $\epsilon >0$ there is a $\delta >0$ such that for all n and all $A\in F_n$,

$\mathbb{Q}\left[A\right]<\delta \Rightarrow Q_n\left[A\right]<\epsilon \text{.}$

Requirement i is the usual constraint on Bayesian learning that events that are null for the prior ℚ remain null for the posterior ℚ_n. The additional uniformity requirement ii ensures that this relation holds as we pass to the limit (Huttegger Reference Huttegger2015b, 623).Footnote ⁸

The other assumption demands a certain sort of stability of probability judgments as the agent updates over increasingly fine partitions. While we treat posterior probabilities as random, condition (M′) places significant constraints on sequences of probability judgments. For all n and all $E\in E_n$,

$\begin{array}{}\text{(M′)}& {\int }_G{p}_{m+1}^{E}d\mathbb{P}={\int }_G{p}_m^{E}d\mathbb{P}\text{ for all }m\ge n\text{ and all }G\in F_m{M}^{\prime }\end{array}$

Property (M′) is the martingale condition, and it requires that future probabilities are equal to present probabilities on average. For all n and all $E\in E_n$, the sequence $p_n^E$, $p_{n+1}^E$, … must form a martingale. The martingale condition has been defended as an essential feature of rational learning experiences (Huttegger Reference Huttegger2013, Reference Huttegger2015b). In the case of Bayesian conditionalization, future probabilities are fixed at 1 for those events learned. This, of course, is not generally the case for Jeffrey conditioning.

Theorem 3 is Huttegger’s merging result for Jeffrey conditioning.

7. Merging of Opinions for Jeffrey Conditioning and Sets of Probabilities

We are now in a position to turn to the contributions of the current article. Our first result extends Huttegger’s theorem for merging of opinions via Jeffrey conditioning to sets of probabilities, an analogue of theorem 2.

A proof is included in the appendix. As Schervish and Seidenfeld (Reference Schervish and Seidenfeld1990) indicate, theorem 2 is a simple corollary of theorem 1 given that point-by-point conditionalization preserves the convexity of a set. We find proposition 1 especially interesting in light of the fact that, unlike Bayesian conditionalization, Jeffrey conditioning does not generally preserve the convexity of the initial set—even for convex sets of priors generated by finitely many extreme points. We provide a demonstration in the appendix (proposition 3).

8. Maximally Informed Opinions

Another consequence of Doob’s martingale convergence theorem is the so-called convergence to truth or convergence to certainty for conditional probabilities, captured by the following statement:

$\text{For all }A\in F,\underset{n\to \infty }{\lim }{\mathbb{P}}_{F_n}\left[A\right]=1_A\mathrm{a.s.}\left(\mathbb{P}\right)\text{.}$

Put another way, coherence requires assigning probability 1 to converging to “the truth” for any event $A\in F$ whose truth value is determined by observations. One way of interpreting such convergence is that “it shows that evidence triumphs over prior opinions under the appropriate circumstances” (Huttegger Reference Huttegger2015a, 590). Two mutually absolutely continuous priors assign probability 1 not just to merging but to being certain of the same observational hypotheses in the limit.Footnote ⁹

Convergence to certainty does not obtain in general for Jeffrey conditioning. However, Huttegger shows that Jeffrey conditioning does lead probabilities to settle down to some limit if we assume (M′), just not necessarily to 0 or 1.Footnote ¹⁰ In particular, Huttegger’s (Reference Huttegger2015b) theorem 9.1 states that sequences of a.s. uniformly absolutely continuous Jeffrey updates that satisfy (M′) converge setwise a.s.; that is, for all $A\in F$, ${\mathbb{P}}_{\infty }\left[A\right]≔{\lim }_{n\to \infty }{\mathbb{P}}_n\left[A\right]$ exists a.s. Skyrms calls this convergence to a “maximally informed opinion” (Reference Skyrms1996).

Convergence to maximally informed opinions can be extended to sets of probabilities. For all $A\in F$, all probabilities in the set C converge to the same limit ℙ_∞[A] a.s. To see this, consider that for all $A\in F$, by the result of Huttegger just mentioned, all elements in C a.s. are converging to a limit. Also, all the elements of C are merging (theorem 3). If two elements ℙ, $\mathbb{Q}\in C$ converged to different limits (with positive probability) for some $A\in F$, then there would be some ε and some m such that $d\left({\mathbb{P}}_n,{\mathbb{Q}}_n\right)>\epsilon$ for any $n\ge m$, which is inconsistent with merging. In fact, a slightly stronger observation can be demonstrated using our proposition 1. As indicated in the introduction, dynamic coherence can be relaxed. For convergence to a maximally informed opinion, it suffices that probabilities take their values in the set of Jeffrey updates of elements of C.

We omit the proof. Proposition 2 is the Jeffrey conditioning analogue of Schervish and Seidenfeld’s corollary 4 (Reference Schervish and Seidenfeld1990, 410).

9. Philosophical Significance

Standard Bayesian learning proceeds by conditionalization, which requires that the event or proposition learned be assigned posterior probability 1. As fans of Jeffrey conditioning sometimes put the point, “learning experiences need not be like that at all” (Huttegger Reference Huttegger2015b, 613). Some learning experiences do not lead to certainty. Here, the standard Bayesian framework cannot help. If there are “uncertain learning experiences,” Jeffrey conditioning, and the extension of learning rules to accommodate “uncertain evidence” more generally, are topics of considerable philosophical interest. Similarly, as fans of IP often point out, not all uncertainty is reducible to numerically precise probabilities. Here, too, the standard Bayesian framework cannot help. In the presence of “deep uncertainty,” sets of probability measures, and generalizations of precise probability models more generally, warrant philosophical attention and scrutiny.

We find it somewhat surprising, then, that with very few exceptions (e.g., Škulj Reference Škulj2006; Stewart and Ojea Quintana Reference Stewart and Quintana2018a) studies that treat “uncertain learning” do not do so in the context of uncertainty that is not reducible to a numerically precise probability, nor do studies of IP treat uncertain evidence. A very general motivation for our current study is to redress this lacuna in the literature. We turn now to three more specific points of philosophical interest.

9.1. Relaxing Dynamic Coherence

Earlier, we indicated some consequences of merging results for sets of probabilities. First, Schervish and Seidenfeld (Reference Schervish and Seidenfeld1990) claim that results such as theorem 2 and proposition 1 have interesting implications for issues related to dynamic coherence. Precisely what they have in mind about the connection between their results and dynamic coherence remains a bit obscure to us (more on this below). The upshot of their discussion, however, seems to be that to achieve asymptotic consensus, agents need not be dynamically coherent—they need not conditionalize or update by Jeffrey conditioning. As Schervish and Seidenfeld put the point with respect to conditionalizing, “We do not impose a constraint of (full) dynamic coherence. For consensus, it suffices that the agents use conditional probabilities arbitrarily chosen from a class C enveloped by finitely many (mutually absolutely continuous) distributions. Under the conditions of [theorem 2], asymptotic certainty follows from static coherence” (402). Interpreting merging results as claims about agents’ unconditional probabilities through time, it is sufficient for achieving asymptotic consensus that each posterior be a conditionalization of some element of C.

By the results of this article, this point can be extended to Jeffrey conditioning. For agents with priors contained within a convex polytope of mutually absolutely continuous probabilities, C, as long as they take posterior probabilities in the set that results from Jeffrey updating the elements of C under the conditions of theorem 3, they will achieve consensus. That is, each agent is free to choose her posterior arbitrarily from the set of Jeffrey updates of elements of C without surrendering the assurance of long-run consensus. For the purpose of achieving consensus with elements of C, an agent’s posterior need not result from Jeffrey updating her prior; it suffices that it be obtained as the result of Jeffrey updating some prior in C.

Analogous points hold for convergence. As Schervish and Seidenfeld (Reference Schervish and Seidenfeld1990) point out, a consequence of their results is that convergence to the truth does not require conditionalization. Similarly, it follows from proposition 2 that convergence to a maximally informed opinion does not demand Jeffrey conditioning. Once again, it suffices that an agent’s posterior be in the set of Jeffrey updates of elements of C.

While Schervish and Seidenfeld’s point that dynamic coherence is not necessary for convergence is certainly correct, and while the results discussed in this article certainly suffice to make this point, the same moral can be gleaned in an even more straightforward way. For example, let { $f_n:n\in \mathbb{N}$} be any sequence of “updates” of ℚ that satisfies

$f_n\left[A\right]\left(\omega \right)\in \{x\in ℝ:\left|x-{\mathbb{Q}}_{F_n}\left[A\right]\left(\omega \right)\right|<1/n\}$

for all $n\in \mathbb{N}$, all $A\in F$, and ℚ-almost every $\omega \in \Omega$. Then, f_n[A] converges to the truth for all $A\in ℱ$ a.s. But { $f_n:n\in \mathbb{N}$} need not be a dynamically coherent sequence of updates. In view of simple examples of this kind, it is not clear to us that there is an especially interesting or significant connection between dynamic coherence and the asymptotics of sets of probabilities.

Whether or not one finds such a connection, it is clear that we need not impose full dynamic coherence in order to achieve the desirable epistemic ends of convergence and merging. Implicit in Schervish and Seidenfeld’s focus on these sorts of asymptotic properties, however, is the idea that we may not be able to relax dynamic coherence in this way and retain just any good-making property typically associated with it. For example, if one is compelled by dynamic Dutch books, taking arbitrary values in the set as above will not cut it. And Skyrms (Reference Skyrms1987) has shown that Dutch book arguments can be extended to Jeffrey conditioning. It bears repeating, though, that the status of dynamic Dutch book arguments is a subject of considerable controversy (e.g., Levi Reference Levi1987; Maher Reference Maher1992; Skyrms Reference Skyrms1993).