Published
- “A Computable von Neumann-Morgenstern Representation Theorem”, Synthese 205, 182 (2025).
Abstract: Real agents have computational limitations. Various disciplines from bounded rationality to cognitive science represent these limitations mathematically via some computational model, such as a Turing machine. But our commonly accepted foundations for decision theory, such as representation theorems, do not say under what conditions an agent can be represented as having utility and probability functions that are computable via Turing machine. This paper gives such conditions by supplementing von Neumann and Morgenstern’s axioms; I also prove that there is an algorithm which, when given an agent’s preference relation as input, computes a utility function that represents the agent.
- “Cartesian Frames”, with Scott Garrabrant and Daniel Herrmann (2021). arXiv.
Under Review
Abstract: Theories of qualitative probability provide a justification for the use of numerical probabilities to represent an agent’s degrees of belief. If a qualitative probability relation satisfies a set of well-known axioms then there is a probability measure that is compatible with that relation. In the particular case of subjective probability this means that we have sufficient conditions for representing an agent as having probabilistic beliefs. But the classical results are not constructive; there is in no general method for calculating the compatible measure from the qualitative relation. To address this problem this paper introduces the theory of computable qualitative probability. I show that there is an algorithm that computes a probability measure from a qualitative relation in highly general circumstances. Moreover I show that given a natural computability requirement on the qualitative relation the resulting probability measure is also computable. Since computable probability is a growing interest in Bayesian epistemology this result provides a valuable interpretation of that notion.
Abstract: Bayesian epistemology is broadly concerned with providing norms for rational belief and learning using the mathematics of probability theory. But many authors have worried that the theory is too idealized to accurately describe real agents. In this paper I argue that Bayesian epistemology can describe more realistic agents while retaining sufficient generality by introducing ideas from a branch of mathematics called computable analysis. I call this program computable Bayesian epistemology. I situate this program by contrasting it with an ongoing debate about ideal versus bounded rationality. I then present foundational ideas from computable analysis and demonstrate their usefulness by proving the main result: on countably generated spaces there are no computable, finitely additive probability measures. On this basis I argue that bounded agents cannot have finitely additive credences, and so countable additivity is the appropriate norm of rationality. I conclude by discussing prospects for this research program.
Abstract: The Problem of the Priors can be stated as the following question: what norms, if any, motivate our choice of one prior over another? Subjective Bayesians argue that there is no problem: your beliefs are what they are, and there is no rational requirement beyond coherence. Nonetheless I argue that in common problems in Bayesian statistics, we are uncertain which prior best represents our beliefs. Using tools from algorithmic randomness I develop a prior determination method that guides agents to priors in a principled manner. I argue that the method produces priors that can all claim to represent the agent’s credences. I then prove that the method is algorithmically implementable in principle, and the priors thus constructed enjoy nice merging properties.
6. “Countable Dutch Book Arguments”
Abstract: I show that a standard argument against the possibility of a valid Dutch book argument for countable additivity is too quick because it fails to distinguish different notions of convergence for random variables. Making this distinction allows for the precise formulation of such a Dutch book argument.
7. “At the Edge of Putnam’s Program: Limitative
Results For Computable Inductive Logics”, with Antoine Mercier and Elijah Spiegel
Abstract: The task of inductive logic is to develop a formal framework to analyze inductive reasoning. Historically this was accomplished by assigning probabilities to sentences of a logical language. Two natural criteria for such a system are: (i) the underlying language should be rich enough to express scientific hypotheses, and (ii) the probabilities should be, in some sense, accessible. The first criterion suggests that the language should at least contain the language of arithmetic, while the second suggests that probabilities should be computable. We show that these two criteria are in tension with one another. Various natural proposals for an inductive logic result in probabilities that are not arithmetically definable, much less computable. We isolate the assumptions responsible for this result, and search for a weaker inductive logic with more accessible probabilities. The most natural weakening results in probabilities that are arithmetically definable but still are not computable.
In Preparation – Drafts available on request
- “Verification, Falsification, and Merging”, with Elijah Spiegel
Abstract: Among the plethora of contributions to philosophy of science made by the Vienna Circle, perhaps the most widely discussed was their explicit link between linguistic meaning and verification. Among other things, the emphasis on verification was aimed at a standard of intersubjective justifiability or agreement. We show that the intuitive connection between verifiability and intersubjective agreement can be made mathematically precise without adopting a verificationist theory or criterion of meaning. We do so by connecting verifiability to Carnap’s inductive logic, and its successor, Bayesian epistemology. In this context, coming to agreement on verifiable sentences entails coming to total intersubjective agreement.
Josiah Lopez-Wild 