Snow Zhang

How should rational agents revise their opinions given the opinions of multiple experts? One attractive answer is linear averaging: upon learning multiple experts’ opinions about a proposition A, one’s own probability of A should equal a linear average of the experts’ opinions about A. However, this answer has a well-known problem: it is compatible with Bayesian conditionalization only when the agent is certain that the experts assign the exact same probability to A (Dawid et al. in TEST, 4(2):263–313, 1995, Ranjan and Gneiting in J Royal Stat Soc Ser B Stat Methodol 72(1):71–91, 2010, Bradley in Theory Decis 85(1):5-20, 2018, Gallow in Philos Stud 175(10):2389–2398, 2018). This paper shows that, for priors of finite domains, a similar result holds for the much weaker norm of strict convexity. To this extent, the triviality phenomenon is more widespread than it has been previously appreciated.

A core tenet of Bayesian epistemology is that Bayesian conditionalization is the rule of rational credal revision. But it has been pointed out in the recent literature that if learning can be nontransparent, then Bayesian conditionalization does not universally maximize expected accuracy. This result raises an explanatory challenge for any externalist Bayesian who does not want to give up on a connection between accuracy and epistemic rationality: Why is Bayesian conditionalization the rule of rational credal revision, despite the fact that it does not universally maximize expected accuracy? This article proposes an answer to this challenge. The article argues that an updating rule can be evaluated, not only on the basis of its expected accuracy in particular learning situations, but also on the basis of its total expected accuracy across different learning situations. The article proposes a notion of global evidential constancy, and shows that, out of all globally evidentially constant rules, Bayesian conditionalization has the highest total expected accuracy across all learning situations, including nontransparent ones. Bayes is back!

In ‘The logic of partial supposition’, Eva and Hartmann present a dilemma for a normative account of partial subjunctive supposition: the natural subjunctive analogue of Jeffrey conditionalization is Jeffrey imaging, but this rule violates a natural monotonicity constraint. This paper offers a partial defence of Jeffrey imaging against Eva and Hartmann’s objection. I show that, although Jeffrey imaging is non-monotonic in Eva and Hartmann’s sense, it is what I call status quo monotonic. A status quo monotonic credal revision rule is monotonic in Eva and Hartmann’s sense if it is conservative in the sense of Meehan and Zhang (‘Jeffrey meets Kolmogorov’), but Jeffrey imaging is in general non-conservative. On the other hand, Jeffrey imaging satisfies a different constraint that I call convexity, and the only rule that is both convex and conservative is Jeffrey conditionalization. To this extent, the real dilemma for a normative account of partial subjunctive supposition is not between monotonicity (broadly construed) and Jeffrey imaging, but between convexity and conservativeness.

According to approximate Bayesianism, Bayesian norms are ideal norms worthy of approximation for non-ideal agents. This paper discusses one potential challenge for approximate Bayesianism: in non-transparent learning situations—situations where the agent does not learn what they have or have not learnt—it is unclear that the Bayesian norms are worth satisfying, let alone approximating. I discuss two replies to this challenge and find neither satisfactory. I suggest that what transpires is a general tension between approximate Bayesianism and the possibility of “non-ideal” epistemic situations.

Jeffrey conditionalization is a rule for updating degrees of belief in light of uncertain evidence. It is usually assumed that the partitions involved in Jeffrey conditionalization are finite and only contain positive-credence elements. But there are interesting examples, involving continuous quantities, in which this is not the case. Q1 Can Jeffrey conditionalization be generalized to accommodate continuous cases? Meanwhile, several authors, such as Kenny Easwaran and Michael Rescorla, have been interested in Kolmogorov’s theory of regular conditional distributions as a possible framework for conditional probability which handles probability-zero events. However the theory faces a major shortcoming: it seems messy and ad hoc. Q2 Is there some axiomatic theory which would justify and constrain the use of rcds, thus serving as a possible foundation for conditional probability? These two questions appear unrelated, but they are not, and this paper answers both. We show that when one appropriately generalizes Jeffrey conditionalization as in Q1, one obtains a framework which necessitates the use of rcds. It is then a short step to develop a general theory which addresses Q2, which we call the theory of extensions. The theory is a formal model of conditioning which recovers Bayesian conditionalization, Jeffrey conditionalization, and conditionalization via rcds as special cases.

A vexing question in Bayesian epistemology is how an agent should update on evidence which she assigned zero prior credence. Some theorists have suggested that, in such cases, the agent should update by Kolmogorov conditionalization, a norm based on Kolmogorov’s theory of regular conditional distributions. However, it turns out that in some situations, a Kolmogorov conditionalizer will plan to always assign a posterior credence of zero to the evidence she learns. Intuitively, such a plan is irrational and easily Dutch bookable. In this paper, we propose a revised norm, Kolmogorov-Blackwell conditionalization, which avoids this problem. We prove a Dutch book theorem and converse Dutch book theorem for this revised norm, and relate our results to those of Rescorla (2018).

The Borel-Kolmogorov paradox is often presented as an obscure problem that certain mathematical accounts of conditional probability must face. In this article, we point out that the paradox arises in the physical sciences, for physical probability or chance. By carefully formulating the paradox in this setting, we show that it is a puzzle for everyone, regardless of one’s preferred probability formalism. We propose a treatment that is inspired by the approach that scientists took when confronted with these cases.