Alexander Meehan

A core tenet of Bayesian epistemology is that rational agents update by conditionalization. Accuracy arguments in favour of this norm are well known. Meanwhile, scholars working in quantum probability and quantum state estimation have proposed multiple updating rules, all of which look prima facie like analogues of Bayesian conditionalization. The most common are Lüders conditionalization and Bayesian mean estimation (BME). Some authors also endorse a lesser-known alternative that we call retrodiction. We show how one can view Lüders and BME as complementary rules, and we give expected accuracy and accuracy dominance arguments for both. By contrast, we find that retrodiction is accuracy-dominated, at least on many measures of accuracy.

In this article I raise a new problem for quantum mechanics, which I call the control problem. Like the measurement problem, the control problem places a fundamental constraint on quantum theories. The characteristic feature of the problem is its focus on state preparation. In particular, whereas the measurement problem turns on a premise about the completeness of the quantum state (‘no hidden variables’), the control problem turns on a premise about our ability to prepare or control quantum states. After raising the problem, I discuss some applications. I suggest that it provides a useful new lens through which to view existing theories or interpretations, in part because it draws attention to aspects of those theories that the measurement problem does not (such as the role of conditional and relative states). I suggest that it also helps clarify the physical significance of the well-known no-go result—the no-cloning theorem—on which it is based.

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.

Branden Fitelson and Alan Hájek have suggested that it is finally time for a “revolution” in which we jettison Kolmogorov’s axiomatization of probability, and move to an alternative like Popper’s. According to these authors, not only did Kolmogorov fail to give an adequate analysis of conditional probability, he also failed to give an adequate account of another central notion in probability theory: probabilistic independence. This paper defends Kolmogorov, with a focus on this independence charge. I show that Kolmogorov’s sophisticated theory of conditional distributions, advocated by authors such as Kenny Easwaran but often overlooked by philosophers, has the resources to define the notion of independence Fitelson and Hájek seek. I then argue that Kolmogorov’s original definition of independence, once suitably reinterpreted as capturing a slight logical weakening of this notion, is not so problematic after all. Indeed, rather than discarding Kolmogorov’s definition, we should make room for two distinct notions of probabilistic independence, one stronger and one weaker, which play different theoretical roles.

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).

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.

In this article I raise a new problem for quantum mechanics, which I call the control problem. Like the measurement problem, the control problem places a fundamental constraint on quantum theories. The characteristic feature of the problem is its focus on state preparation. In particular, whereas the measurement problem turns on a premise about the completeness of the quantum state ('no hidden variables'), the control problem turns on a premise about our ability to prepare or control quantum states. After raising the problem, I discuss some applications. I suggest that it provides a useful new lens through which to view existing theories or interpretations, in part because it draws attention to aspects of those theories that the measurement problem does not. I suggest that it also helps clarify the physical significance of the well-known no-go result—the no-cloning theorem—on which it is based.