Writing

Shorter pieces on how prediction markets and betting work are on Substack.

Prediction Markets and Regulation

  • 2026The Kelly Criterion with an Estimated EdgeResearchers.One, September 2026. Bet sizing as a constrained optimization on stop loss and stopping probability, with a closed-form fractional Kelly stake. PDF
  • 2026Estimating U.S. User Activity in Offshore Prediction MarketsResearch report commissioned by the Coalition for Prediction Markets, June 2026. Cited by the Congressional Research Service in Prediction Markets Legislation in the 119th Congress, R49266, August 2026.
  • 2025Sports Betting Regulation: The States v. the Public InterestFutures and Derivatives Law Report, Vol. 45, No. 8, September 2025.
  • 2024Election Betting Markets Were 2024’s Other Winner. Embrace ThemWith Koleman Strumpf. Barron’s, November 21, 2024. Op-ed.
  • 2024CFTC v. the Public Interest: CFTC Seeks to Restrict Broad Categories of Event ContractsFutures and Derivatives Law Report, Vol. 44, No. 8, September 2024.
  • 2023Models vs. Markets: Forecasting the 2020 U.S. ElectionWith Darrion Vinson. Journal of Prediction Markets, Vol. 17, No. 2. Cited in Rabinovitz and Packin, “All Bets Are On: Addiction, Prediction, Regulation, and the Future of Financial Gambling” (2025).
  • 2022Political Prediction Markets Are an Antidote to Degraded Public DiscourseWith Koleman Strumpf. Chicago Tribune, September 2022. Op-ed.
  • 2022The Cloudy Future of Political FuturesWith Koleman Strumpf. National Review, August 2022. Op-ed.
  • 2020Betting Against the Political PollsLuckbox Magazine, December 2020.
  • 2019Polls, Pundits, or Prediction Markets: An Assessment of Election ForecastingResearchers.One.
  • 2019Rethinking Probabilistic Prediction: Some Insights from the 2016 U.S. Presidential ElectionWith Ryan Martin. Researchers.One.

Probability & Statistics

  • 2024Root and Community Inference on the Latent Growth Process of a NetworkWith Min Xu. Journal of the Royal Statistical Society, Series B, in press. Read Paper, presented and discussed before the Society.
  • 2022Risk Is Random: The Magic of the d’AlembertWith Glenn Shafer. Researchers.One. Featured by Mark Buchanan in Nature Physics.
  • 2021Inference on the History of a Randomly Growing TreeWith Min Xu. Journal of the Royal Statistical Society, Series B, 83(4): 639–668.
  • 2021A Statistical Framework for Modern Network ScienceWith Walter Dempsey. Statistical Science, 36(1): 51–67.
  • 2019Imprecise Probabilities as a Semantics for Intuitive Probabilistic ReasoningProceedings of Machine Learning Research, Vol. 103, ISIPTA 2019.
  • 2019Relational ExchangeabilityWith Walter Dempsey. Journal of Applied Probability.
  • 2019The Logic of TypicalityWith Isaac Wilhelm. In V. Allori (ed.), Statistical Mechanics and Scientific Explanation. World Scientific.
  • 2018Pattern Avoidance for Random Mallows PermutationsWith Stephen DeSalvo and Sergi Elizalde. Random Structures & Algorithms, 53: 417–447.
  • 2018Edge Exchangeable Models for Interaction DataWith Walter Dempsey. Journal of the American Statistical Association, 113(523): 1311–1326.
  • 2018Combinatorial Lévy ProcessesAnnals of Applied Probability, 28(1): 285–339.
  • 2018Relative Exchangeability with Equivalence RelationsWith Henry Towsner. Archive for Mathematical Logic, 57: 533–556.
  • 2018Relatively Exchangeable StructuresWith Henry Towsner. Journal of Symbolic Logic, 83(2): 416–442.
  • 2018Pattern Avoidance for Random PermutationsWith Stephen DeSalvo. Discrete Mathematics and Theoretical Computer Science, 19(2), Paper 13.
  • 2017A Hidden Markov Model for Latent Temporal ClusteringComputational Statistics and Data Analysis, 110: 19–36.
  • 2017Exchangeable Graph-Valued Feller ProcessesProbability Theory and Related Fields, 168: 849–899.
  • 2017Generalized Markov Branching Tree ModelsAdvances in Applied Probability, 49(1).
  • 2016The Ubiquitous Ewens Sampling FormulaStatistical Science, 31(1): 1–39. With discussion and a rejoinder by the author.
  • 2016Dynamic Random Networks and Their Graph LimitsAnnals of Applied Probability, 26(2): 691–721.
  • 2016Exchangeable Markov Processes on [k]ℕ with Càdlàg Sample PathsWith Steven P. Lalley. Journal of Theoretical Probability, 29: 206–230.
  • 2015Clustering from Categorical Data SequencesJournal of the American Statistical Association, 110(510): 810–823.
  • 2015Generalized Ewens–Pitman Model for Bayesian ClusteringBiometrika, 102(1): 231–238.
  • 2015Time-Varying Network ModelsBernoulli, 21(3): 1670–1696.
  • 2015Lipschitz Partition ProcessesBernoulli, 21(3): 1386–1411.
  • 2015Poisson Superposition ProcessesWith Peter McCullagh. Journal of Applied Probability, 52(4): 1013–1027.
  • 2015Reversible Markov Structures on Divisible Set PartitionsWith Peter McCullagh. Journal of Applied Probability, 52(3): 622–635.
  • 2015Left-Right Arrangements, Set Partitions, and Pattern AvoidanceAustralasian Journal of Combinatorics, 61(1): 57–72.
  • 2014The Cut-and-Paste ProcessAnnals of Probability, 42(5): 1952–1979.
  • 2013Permanental Partition Models and Markovian Gibbs StructuresJournal of Statistical Physics, 153: 698–726.
  • 2013Some Algebraic Identities for the α-PermanentLinear Algebra and Its Applications, 439(11): 3445–3459.
  • 2013Consistent Markov Branching Trees with Discrete Edge LengthsElectronic Communications in Probability, 18, Paper 73: 1–14.
  • 2013Convergence Rates of Markov Chains on Spaces of PartitionsWith Steven P. Lalley. Electronic Journal of Probability, 18, Paper 61: 1–22.
  • 2011A Consistent Markov Partition Process Generated from the Paintbox ProcessJournal of Applied Probability, 43(3): 778–791.

Commentary on statistics, science, and risk

  • 2021Naive Probabilism and Covid-19In Covid-19: A Complex Systems Approach (J. Norman and A. J. Morales, eds.), pp. 9–22. STEM Academic Press.
  • 2021Comment on Shafer, “Testing by Betting: A Strategy for Statistical and Scientific Communication”Journal of the Royal Statistical Society, Series A, 184(2): 436–437.
  • 2020Naive ProbabilismResearchers.One. “When gambling, think probability. When hedging, think plausibility. When preparing, think possibility.”
  • 2020A Fiasco in the Making: More Data Is Not the Answer to the Coronavirus PandemicResearchers.One. A response to John Ioannidis. Also published in To Vima and translated into Greek, Finnish, and Spanish.
  • 2019Is Statistics Meeting the Needs of Science?With Ryan Martin. Researchers.One.
  • 2018The Fundamental Principle of ProbabilityResearchers.One. Resolving the replication crisis with skin in the game.
  • 2018The Impact of P-hacking on “Redefine Statistical Significance”Basic and Applied Social Psychology, 40: 219–235.
  • 2017Comment on Caron and Fox, “Sparse Graphs Using Exchangeable Random Measures”Journal of the Royal Statistical Society, Series B, 79, Part 5.
  • 2017Comment on Gelman and Hennig, “Beyond Subjective and Objective in Statistics”Journal of the Royal Statistical Society, Series A, 180, Part 4.

Opinion

With Ryan Martin, I co-founded Researchers.One in 2018: a nonprofit, independent platform for publishing and reviewing research in the open, with authors in control of the process. It has published hundreds of articles, launched four journals, and hosted conferences, and has been covered in Nature, The Atlantic, and The Scientist. The pieces below set out the case for it, along with other one-off comment.

Books

  • 2026Prediction Markets and Event ContractsBook manuscript in progress.
  • 2026Making Sense: The Pursuit of Clarity over TruthIn production.
  • 2019Probabilistic Foundations of Statistical Network AnalysisChapman & Hall/CRC Monographs on Statistics and Applied Probability. Chapter synopses, slides, and a sample chapter on the book page.