About

I’m an Assistant Professor in the Department of Philosophy at Carnegie Mellon University. I specialize in logic, formal epistemology, and the philosophy of science—in particular, the philosophy of probability and induction. I obtained my Ph.D. in Philosophy and Symbolic Systems at Stanford University in the summer of 2020. Before that, I completed an M.Sc. in Logic at the Institute for Logic, Language and Computation at the University of Amsterdam, as well as an M.A. in Philosophy at the University of Edinburgh. Most of my work is devoted to showing that the theory of algorithmic randomness—a branch of computability theory—can be fruitfully applied to shed light on the foundations of inductive learning. Algorithmic randomness provides a mathematical analysis of the notion of an object displaying no algorithmically detectable patterns or regularities. I’m particularly interested in the effects of algorithmic randomness, when taken to be a property of data streams, on the learning perfor

Research topics

• Computer Science
• Artificial Intelligence
• Algorithm
• Theoretical computer science
• Mathematics
• Philosophy
• Combinatorics
• Theology
• Discrete mathematics

Selected publications

• Algorithmic randomness and the weak merging of computable probability measures

  Annals of Pure and Applied Logic · 2026-03-09

  articleSenior authorCorresponding
  PublisherDOI

• A New Look at the Modal Logic of Stepwise Removal

  Trends in logic · 2026-01-01

  book-chapterSenior author
  PublisherDOI

• Bayesian Convergence for Computably Bounded Agents

  Philosophy and Phenomenological Research · 2025-12-06

  articleOpen accessSenior author

  ABSTRACT In this article, we pursue two goals. First, we argue that computable probability theory offers a fitting framework for modeling the credences of computably bounded—and, thus, more realistic—Bayesian reasoners. Second, we develop a Bayesian perspective on algorithmic randomness: a branch of computability theory that provides a formal account of what it takes for a sequence of observations (a data stream) to be probabilistically typical in an algorithmically specifiable way. In particular, we argue that adopting such a perspective leads to novel insights for one of the pillars of Bayesian epistemology: Bayesian convergence to the truth. In a companion article, we showed that, for Bayesian agents whose credences are given by computable probability measures, the data streams that guarantee convergence to the truth coincide with the algorithmically random ones. Here, we put these results to use to counter various skeptical arguments which target the philosophical significance of Bayesian convergence‐to‐the‐truth theorems.

  PublisherOA PDFDOI

• Schnorr Randomness and Effective Bayesian Consistency and Inconsistency

  ArXiv.org · 2025-01-20

  preprintOpen accessSenior author

  We study Doob's Consistency Theorem and Freedman's Inconsistency Theorem from the vantage point of computable probability and algorithmic randomness. We show that the Schnorr random elements of the parameter space are computably consistent, when there is a map from the sample space to the parameter space satisfying many of the same properties as limiting relative frequencies. We show that the generic inconsistency in Freedman's Theorem is effectively generic, which implies the existence of computable parameters which are not computably consistent. Taken together, this work provides a computability-theoretic solution to Diaconis and Freedman's problem of ``know[ing] for which [parameters] the rule [Bayes' rule] is consistent'', and it strengthens recent similar results of Takahashi on Martin-Löf randomness in Cantor space.

  PublisherOA PDFDOI

• Algorithmic randomness and the weak merging of computable probability measures

  ArXiv.org · 2025-04-01

  preprintOpen accessSenior author

  We characterize Martin-Löf randomness and Schnorr randomness in terms of the merging of opinions, along the lines of the Blackwell-Dubins Theorem. After setting up a general framework for defining notions of merging randomness, we focus on finite horizon events, that is, on weak merging in the sense of Kalai-Lehrer. In contrast to Blackwell-Dubins and Kalai-Lehrer, we consider not only the total variational distance but also the Hellinger distance and the Kullback-Leibler divergence. Our main result is a characterization of Martin-Löf randomness and Schnorr randomness in terms of weak merging and the summable Kullback-Leibler divergence. The main proof idea is that the Kullback-Leibler divergence between $μ$ and $ν$, at a given stage of the learning process, is exactly the incremental growth, at that stage, of the predictable process of the Doob decomposition of the $ν$-submartingale $L(σ)=-\ln \frac{μ(σ)}{ν(σ)}$. These characterizations of algorithmic randomness notions in terms of the Kullback-Leibler divergence can be viewed as global analogues of Vovk's theorem on what transpires locally with individual Martin-Löf $μ$- and $ν$-random points and the Hellinger distance between $μ,ν$.

  PublisherOA PDFDOI

• Randomness and invariance

  Journal of Logic and Computation · 2024-12-04

  articleOpen accessSenior author

  Abstract Richard von Mises was the first to provide a rigorous definition of randomness for infinite binary sequences, taken to represent indefinitely long sequences of experimental outcomes or indefinitely large (ordered) population samples. According to von Mises, a sequence is random if, within it, the relative frequencies of 0 and 1 each converge to a finite limit, and these limiting relative frequencies are invariant under a class of transformations that he called selection rules. While the notion of randomness defined by von Mises is known to have some serious limitations, his theory was pivotal to the development of algorithmic randomness: a branch of computability theory that provides the now-standard approach to defining randomness for individual mathematical objects. The purpose of this article is to call attention to the fact that, despite its flaws, one of the core ideas behind von Mises’ account of randomness also underlies much of the theory of algorithmic randomness. In particular, we bring together and generalize a number of little-known results, proving that, for a broad class of probability measures, several canonical algorithmic randomness notions are characterizable in terms of invariance: i.e., in terms of the preservation, or the stable satisfaction, of various natural properties under a class of transformations that can be seen as a generalization of von Mises’ selection rules. Many of the properties in question are usually described as ‘minimal randomness properties’: they are not in themselves sufficient for randomness, but they are often taken to be necessary (or at least desirable) for it. From this perspective, our results establish that algorithmic randomness coincides with the stable satisfaction of various minimal randomness properties (including the existence of limiting relative frequencies, as in von Mises’ original account).

  PublisherDOI

• Editorial

  Annals of Pure and Applied Logic · 2024-03-27

  editorialOpen accessSenior authorCorresponding
  PublisherDOI

• Algorithmic Randomness, Effective Disintegrations, and Rates of Convergence to the Truth

  arXiv (Cornell University) · 2024-03-29 · 1 citations

  preprintOpen accessSenior author

  Lévy's Upward Theorem says that the conditional expectation of an integrable random variable converges with probability one to its true value with increasing information. In this paper, we use methods from effective probability theory to characterise the probability one set along which convergence to the truth occurs, and the rate at which the convergence occurs. We work within the setting of computable probability measures defined on computable Polish spaces and introduce a new general theory of effective disintegrations. We use this machinery to prove our main results, which (1) identify the points along which certain classes of effective random variables converge to the truth in terms of certain classes of algorithmically random points, and which further (2) identify when computable rates of convergence exist. Our convergence results significantly generalize earlier results within a unifying novel abstract framework, and there are no precursors of our results on computable rates of convergence. Finally, we make a case for the importance of our work for the foundations of Bayesian probability theory.

  PublisherOA PDFDOI

• From Wald to Schnorr: von Mises’ definition of randomness in the aftermath of Ville’s Theorem

  Studies in History and Philosophy of Science Part A · 2024-07-26 · 3 citations

  articleOpen access1st authorCorresponding

  The first formal definition of randomness, seen as a property of sequences of events or experimental outcomes, dates back to Richard von Mises' work in the foundations of probability and statistics. The randomness notion introduced by von Mises is nowadays widely regarded as being too weak. This is, to a large extent, due to the work of Jean Ville, which is often described as having dealt the death blow to von Mises' approach, and which was integral to the development of algorithmic randomness-the now-standard theory of randomness for elements of a probability space. The main goal of this article is to trace the history and provide an in-depth appraisal of two lesser-known, yet historically and methodologically notable proposals for how to modify von Mises' definition so as to avoid Ville's objection. The first proposal is due to Abraham Wald, while the second one is due to Claus-Peter Schnorr. We show that, once made precise in a natural way using computability theory, Wald's proposal constitutes a much more radical departure from von Mises' framework than intended. Schnorr's proposal, on the other hand, does provide a partial vindication of von Mises' approach: it demonstrates that it is possible to obtain a satisfactory randomness notion-indeed, a canonical algorithmic randomness notion-by characterizing randomness in terms of the invariance of limiting relative frequencies. More generally, we argue that Schnorr's proposal, together with a number of little-known related results, reveals that there is more continuity than typically acknowledged between von Mises' approach and algorithmic randomness. Even though von Mises' exclusive focus on limiting relative frequencies did not survive the passage to the theory of algorithmic randomness, another crucial aspect of his conception of randomness did endure; namely, the idea that randomness amounts to a certain type of stability or invariance under an appropriate class of transformations.

  PublisherDOI

• Pride and Probability

  Philosophy of Science · 2024 · 2 citations

  1st authorCorresponding
  • Philosophy
  • Theology

  Abstract Bayesian agents, argues Belot, are orgulous: they believe in inductive success even when guaranteed to fail on a topologically typical collection of data streams. Here we shed light on how pervasive this phenomenon is. We identify several classes of inductive problems for which Bayesian convergence to the truth is topologically typical. However, we also show that, for all sufficiently complex classes, there are inductive problems for which convergence is topologically atypical. Last, we identify specific topologically typical collections of data streams, observing which guarantees convergence to the truth across all problems from certain natural classes of effective inductive problems.

  PublisherOA PDFDOI

Frequent coauthors

• Johan van Benthem

  3 shared

• Sean Walsh

  2 shared

• Carlos Areces

  Centro Científico Tecnológico - Córdoba

  2 shared

• Simon M. Huttegger

  2 shared

• Krzysztof Mierzewski

  2 shared

Education

• Ph.D. in Philosophy and Symbolic Systems, Philosophy

  Leland Stanford Junior University

  2020