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An Old Look at Empirical Bayes
Nicholas G. Polson, Vadim O. Sokolov, Daniel Zantedeschi
Dennis Lindley once said that there is only one thing worse than a frequentist, and that is an empirical Bayesian. The quip has the air of caricature, but its technical content is…
Horseshoe Priors and MDP
Nick Polson, Vadim Sokolov, Daniel Zantedeschi
Carvalho (2010) established two foundational theorems for the horseshoe prior: tight two-sided logarithmic bounds on the marginal density near the origin (Theorem~1.1), and a super…
Bell's Inequality, Causal Bounds, and Quantum Bayesian Computation: A Unified Framework
Nick Polson, Vadim Sokolov, Daniel Zantedeschi
Bell inequalities characterize the boundary of the local-realist correlation polytope -- the set of joint probability distributions achievable by classical hidden-variable models.…
Bayes, E-values and Testing
Nicholas G. Polson, Vadim Sokolov, Daniel Zantedeschi
E-values and E-processes (nonnegative supermartingales) provide anytime-valid evidence for sequential testing via Ville's inequality, yet their connection to Bayesian reasoning, re…
Martingale Posterior Predictive Coherence: Hausdorff Moment Hierarchy
Nicholas G. Polson, Daniel Zantedeschi
For an exchangeable Bernoulli sequence with de Finetti mixing measure Pi, the k-step predictive probability P(X_{n+1}=...=X_{n+k}=0 | F_n) equals the posterior expectation E[(1-the…
Synthetic Priors
Nick Polson, Vadim Sokolov
Bayesian inference in generalized linear models requires a prior on the coefficient vector . Practitioners naturally reason about response probabilities at specific covariate va…
Generative Bayesian Computation as a Scalable Alternative to Gaussian Process Surrogates
Nick Polson, Vadim Sokolov
Gaussian process (GP) surrogates are the default tool for emulating expensive computer experiments, but cubic cost, stationarity assumptions, and Gaussian predictive distributions…
Photons = Tokens: The Physics of AI and the Economics of Knowledge
Alec Litowitz, Nick Polson, Vadim Sokolov
Debates about artificial intelligence capabilities and risks are often conducted without quantitative grounding. This paper applies the methodology of MacKay (2009) -- who reframed…
Fast Compute for ML Optimization
Nick Polson, Vadim Sokolov
We study optimization for losses that admit a variance-mean scale-mixture representation. Under this representation, each EM iteration is a weighted least squares update in which l…
Some Bayesian Perspectives on Clinical Trials
Alexandra Sokolova, Vadim Sokolov, Nick Polson
We examine three landmark clinical trials -- ECMO, CALGB~49907, and I-SPY~2 -- through a unified Bayesian framework connecting prior specification, sequential adaptation, and decis…
Bayesian Methods for the Navier-Stokes Equations
Nicholas Polson, Vadim Sokolov
We develop a Bayesian methodology for numerical solution of the incompressible Navier--Stokes equations with quantified uncertainty. The central idea is to treat discretized Navier…
Horseshoe Mixtures-of-Experts (HS-MoE)
Nick Polson, Vadim Sokolov
Horseshoe mixtures-of-experts (HS-MoE) models provide a Bayesian framework for sparse expert selection in mixture-of-experts architectures. We combine the horseshoe prior's adaptiv…