arXiv subjects

Vadim Sokolov

Publications and source records attributed to Vadim Sokolov.

At least 19 recordsLinked to original sources

arXiv2026

Generative Learner for Distributional Causal Effects

We propose a generative learner for estimating conditional average treatment effects and characterizing the full distribution of these effects. The learner takes the form of a multi-head feed-forward neural network with three jointly estimated subnetworks: propensity score, baseline outcome, and the conditional average treatment effect. Here, the treatment effect subnetwork parameterizes the conditional quantile function via a compositional architecture in which covariate representation and cosine quantile embeddings are combined through element-wise multiplication. We then recover the conditional average treatment effect as an integral over conditional quantile treatment effects. Under the classical causal assumptions within the Neyman--Rubin potential outcomes framework, we find that the proposed generative learner reduces out-of-sample mean squared error relative to the generalized random forest, double machine learning, and generative adversarial networks, with gains ranging from 5.4% to 93.5% on average across experimental designs. In an empirical application, we formalize the Stefan--Boltzmann law within a unidirectional causal model and apply the method to publicly available stellar data. The estimated effects satisfy the restrictions the law implies.

astro-ph.SR↗

arXiv2026

Generative Modeling: A Review

We organize the generative-modeling literature around three classes of generators, corresponding to three distinct inferential tasks: estimating counterfactual outcome distributions in causal inference, recovering posteriors from simulated parameter--outcome pairs, and forming predictive outcome distributions. The unifying representation relies on the noise outsourcing theorem of Kallenberg, which expresses a conditional distribution as a deterministic function of its inputs and an independent noise variable. Within this organization we develop generative Bayesian computation, a method in the parameter--outcome class: a quantile neural network, trained on simulated pairs under the pinball loss, that targets the posterior of the parameter directly, without invertible architectures or density evaluation, and that serves equally as a predictive generator once the roles of parameter and outcome are exchanged. We illustrate the framework on an agent-based Ebola transmission application, where generative Bayesian computation recovers accurate posteriors at substantially lower cost than rejection-based simulation inference, while avoiding the density-evaluation and invertibility constraints of competing generators.

stat.CO↗

arXiv2026

Horseshoe Priors and MDP

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-efficient rate of convergence of the Bayes predictive density to the true sampling density in sparse situations (Theorem~2). The ``Shrink Globally, Act Locally'' paper \citep{polson2010shrink} formalised necessary and sufficient conditions on the prior's behaviour at the origin for sparsity adaptation as $p \to \infty$. We show that these results are not merely descriptive properties of the horseshoe -- they are the finite-sample precursors to the asymptotic moderate deviation principle (MDP) of \citet{datta2026newlook}. The log-pole singularity $\piH(θ) \asymp -\log\absθ$ is precisely the origin integrability boundary that selects the MDP threshold $\tcrit = \sqrt{\log(πn/2)}$; super-efficiency below the threshold and tail robustness above it together produce the ABOS Bayes risk $p_0 \log(p/p_0)/n$; and the Clarke--Barron information-theoretic asymptotics of Bayes methods provide the unifying framework in which all three results are faces of a single logarithmic budget principle.

math.ST↗

arXiv2026

Bell's Inequality, Causal Bounds, and Quantum Bayesian Computation: A Unified Framework

Bell inequalities characterize the boundary of the local-realist correlation polytope -- the set of joint probability distributions achievable by classical hidden-variable models. Quantum mechanics exceeds this boundary through non-commutativity, reaching the Tsirelson bound $2\sqrt{2}$ for CHSH. We show that this polytope structure is not specific to quantum foundations: it appears identically in the causal inference literature, where the instrumental inequality, the Balke--Pearl linear programming bounds, and the Tian--Pearl probabilities of causation all arise as facets of the same marginal compatibility polytope. Fine's theorem -- that CHSH inequalities hold if and only if a joint distribution exists -- is precisely the pivot: the instrumental variable model in causal inference is structurally equivalent to the Bell local hidden-variable model, with the instrument playing the role of the measurement setting and the latent confounder playing the role of the hidden variable $λ$. We develop this correspondence in detail, extending it to algorithmic (Kolmogorov complexity) and entropic formulations of Bell inequalities, the NPA semidefinite programming hierarchy, and the MIP$^*$=RE undecidability result. We further show that the Born-rule / Bayes-rule duality underlying quantum Bayesian computation exploits the same non-commutativity that enables Bell violation, providing polynomial speedups for posterior inference. The framework yields a concrete dictionary between quantum information theory, causal econometrics, and Bayesian computation, and suggests new directions including NPA-based quantum causal inference algorithms and quantum architectures for function approximation.

quant-ph↗

arXiv2026

A New Look at Bayesian Testing

We identify the critical deviation scale governing Bayesian evidence accumulation in regular parametric testing. Under integrated Bayes risk with zero-one loss, the risk-optimal rejection boundary lies in a moderate deviation regime, with a square-root logarithmic inflation relative to the usual local asymptotic normal scale. Under Cramer regularity, local prior smoothness at the null, and symmetric loss, we derive the sharp threshold and show that its leading logarithmic term is universal across regular priors, while lower-order constants depend on the local prior density, Fisher information, and prior model odds. The result extends to one-parameter exponential families through local asymptotic normality and places Jeffreys' testing threshold, the Bayesian information criterion penalty, and Chernoff-Stein type error-exponent arguments within a common asymptotic moderate deviation framework.

math.ST↗

arXiv2026

Bayes, E-values and Testing

E-values and E-processes (nonnegative supermartingales) provide anytime-valid evidence for sequential testing via Ville's inequality, yet their connection to Bayesian reasoning, representational structure, and computational feasibility are often conflated in the literature. We develop a typed framework that separates sequential evidence into three layers: (i) representation (Radon-Nikodym / likelihood-ratio geometry), (ii) validity (supermartingale certificates under optional stopping), and (iii) decision (boundary design and efficiency calibration). Our main results are: (a) under log-loss and Bayes-risk minimization, the likelihood ratio is the unique evidence representation within the coherent predictive subclass; (b) the likelihood-ratio stopping time satisfies E_1[tau_b] = (log b)/mu + O(sqrt(log b)) under Cramer conditions, while validity-only thresholds admit no such growth-rate guarantee; and (c) regret-optimal codes (e.g., NML/MDL) do not in general yield valid E-processes, while prequential codes do. Monte Carlo experiments confirm the theoretical predictions. The framework applies to online model validation, adaptive experimentation, conformal prediction, and sequential changepoint detection.

math.ST↗

arXiv2026

Synthetic Priors

Bayesian inference in generalized linear models requires a prior on the coefficient vector $β$. Practitioners naturally reason about response probabilities at specific covariate values, not about abstract log-odds parameters. We develop synthetic priors: informative Bayesian priors for GLMs grounded in Good's device of imaginary observations -- the principle that every conjugate prior is equivalent to a likelihood on pseudo-data from the same exponential family. The conditional means prior of Bedrick (1996) elicits independent Beta priors on the conditional mean response at $p$ expert-chosen design points; the induced prior on $β$ is a product of binomial likelihoods at synthetic data points. Combined with Pólya-Gamma data augmentation \citep{polson2013}, the posterior admits an exact conjugate Gibbs sampler -- no tuning, no Metropolis step -- by treating the augmented dataset as a standard logistic regression. We show that ridge regression and catalytic priors \citep{huang2020} are instances of Good's device, and identify prediction-powered inference \citep{angelopoulos2023ppi} as a structural analogue in the frequentist setting -- all three mediate a variance-bias tradeoff through a single informativeness parameter. We illustrate the approach on two benchmark problems: the Challenger O-ring data \citep{dalal1989}, where the BCJ prior provides a more moderate posterior predictive at the 31°F launch temperature; and a Phase~II atopic dermatitis dose-finding trial ($n = 300$), where the synthetic prior narrows 95\% credible intervals by 3-6\% and raises decision probabilities by up to 2 percentage points relative to a flat prior.

stat.ME↗

arXiv2026

Generative Bayesian Computation as a Scalable Alternative to Gaussian Process Surrogates

Gaussian process (GP) surrogates are the default tool for emulating expensive computer experiments, but cubic cost, stationarity assumptions, and Gaussian predictive distributions limit their reach. We propose Generative Bayesian Computation (GBC) via Implicit Quantile Networks (IQNs) as a surrogate framework that targets all three limitations. GBC learns the full conditional quantile function from input--output pairs; at test time, a single forward pass per quantile level produces draws from the predictive distribution. Across fourteen benchmarks we compare GBC to four GP-based methods. GBC improves CRPS by 11--26\% on piecewise jump-process benchmarks, by 14\% on a ten-dimensional Friedman function, and scales linearly to 90,000 training points where dense-covariance GPs are infeasible. A boundary-augmented variant matches or outperforms Modular Jump GPs on two-dimensional jump datasets (up to 46\% CRPS improvement). In active learning, a randomized-prior IQN ensemble achieves nearly three times lower RMSE than deep GP active learning on Rocket LGBB. Overall, GBC records a favorable point estimate in 12 of 14 comparisons. GPs retain an edge on smooth surfaces where their smoothness prior provides effective regularization.

cs.LG↗

arXiv2026

Photons = Tokens: The Physics of AI and the Economics of Knowledge

Debates about artificial intelligence capabilities and risks are often conducted without quantitative grounding. This paper applies the methodology of MacKay (2009) -- who reframed energy policy as arithmetic -- to the economy of AI computation. We define the token, the elementary unit of large language model input and output, as a physical quantity with measurable thermodynamic cost. Using Landauer's principle, Shannon's channel capacity, and current infrastructure data, we construct a supply-and-demand balance sheet for global token production. We then derive a finite question budget: the number of meaningful queries humanity can direct at AI systems under physical, information-theoretic, and economic constraints. We apply Coase's theory of the firm and the durable-goods monopoly problem to the AI value chain -- from photon to atom to chip to power to token to question -- to identify where economic value concentrates and where regulatory intervention is warranted. We argue that the expansion of the token budget does not resolve a deeper constraint: under structural uncertainty, the decisive variable is not how many questions can be answered but which questions are worth asking -- a problem of agency and direction that computation alone cannot solve. We connect limits of measurement in the token economy to a structural parallel between Goodhart's law and the Heisenberg uncertainty principle, and to Arrow's impossibility result for efficient information pricing. The framework yields order-of-magnitude estimates that discipline policy discussion: at current efficiency, the projected 2028 US AI energy allocation of 326~TWh could support roughly $6.5 \times 10^{17}$ tokens per year, or 225,000 tokens per person per day -- more than three orders of magnitude above estimated mid-2024 utilization.

physics.soc-ph↗

arXiv2026

Bayes Risk for Goodness of Fit Tests

We develop a unified framework for goodness-of-fit (GOF) testing through the lens of Bayes risk. Classical GOF procedures are commonly calibrated either at fixed significance level (CLT scale) or through exponential error exponents (LDP scale). We establish that Bayes-risk optimal calibration operates on the moderate-deviation (MDP) scale, producing canonical $\sqrt{\log n}$ inflation of rejection thresholds and polynomially decaying Type I error. Our main contributions are: (i) we formalise the Rubin--Sethuraman program for KS-type statistics as a risk-calibration theorem with explicit regularity conditions on priors and empirical-process functionals; (ii) we develop the precise connection between Bayes-risk expansions and Sanov information asymptotics, showing how $\log n$-order truncations arise naturally when risk, rather than pure exponents, is the evaluation criterion; (iii) we provide detailed applications to location testing under Laplace families, shape testing via Bayes factors, and connections to Fisher information geometry. The organizing principle throughout is that sample size enters Bayes-optimal GOF cutoffs through the MDP scale, unifying KS-based and Sanov-based perspectives under a single risk criterion.

math.ST↗

arXiv2026

Fast Compute for ML Optimization

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 latent variables determine observation and parameter weights; these play roles analogous to Adam's second-moment scaling and AdamW's weight decay, but are derived from the model. The resulting Scale Mixture EM (SM-EM) algorithm removes user-specified learning-rate and momentum schedules. On synthetic ill-conditioned logistic regression benchmarks with $p \in \{20, \ldots, 500\}$, SM-EM with Nesterov acceleration attains up to $13\times$ lower final loss than Adam tuned by learning-rate grid search. For a 40-point regularization path, sharing sufficient statistics across penalty values yields a $10\times$ runtime reduction relative to the same tuned-Adam protocol. For the base (non-accelerated) algorithm, EM monotonicity guarantees nonincreasing objective values; adding Nesterov extrapolation trades this guarantee for faster empirical convergence.

stat.CO↗

arXiv2026

Some Bayesian Perspectives on Clinical Trials

We examine three landmark clinical trials -- ECMO, CALGB~49907, and I-SPY~2 -- through a unified Bayesian framework connecting prior specification, sequential adaptation, and decision-theoretic optimisation. For ECMO, the posterior probability of treatment superiority is robust across the range of priors examined. For CALGB, predictive probability monitoring stopped enrolment at 633 instead of 1800 patients. For I-SPY~2, adaptive enrichment graduated nine of 23 arms to Phase~III. These case studies motivate a methodological contribution: exact backward induction for two-arm binary trials, where Beta-Binomial conjugacy yields closed-form transitions on the integer lattice of success counts with no quadrature. A Pólya-Gamma augmentation bridges this to covariate-adjusted logistic regression. Simulation reveals a fundamental tension: the optimal Bayesian design reduces expected sample sizes to 14--26 per arm (versus 42--100 for alternatives) but with substantially lower power. A calibrated variant embedding the declaration threshold in the terminal utility improves power while maintaining sample-size savings; varying the per-stage cost traces a power frontier for selecting the preferred operating point, with suitability highest in patient-sparing contexts such as rare diseases and paediatrics. The Pólya-Gamma Laplace approximation is validated against exact calculations (mean absolute error below 0.01). We discuss implications for the 2026 FDA draft guidance on Bayesian methodology.

stat.ME↗

arXiv2026

E-values for Adaptive Clinical Trials: Anytime-Valid Monitoring in Practice

Adaptive clinical trials rely on interim analyses, flexible stopping, and data-dependent design modifications that complicate statistical guarantees when fixed-horizon test statistics are repeatedly inspected or reused after adaptations. E-values and e-processes provide anytime-valid tests and confidence sequences that remain valid under optional stopping and optional continuation without requiring a prespecified monitoring schedule. This paper is a methodology guide for practitioners. We develop the betting-martingale construction of e-processes for two-arm randomized controlled trials, show how e-values naturally handle composite null hypotheses and support futility monitoring, and provide guidance on when e-values are appropriate, when established alternatives are preferable, and how to integrate e-value monitoring with group sequential and Bayesian adaptive workflows. A numerical study compares five monitoring rules -- naive and calibrated versions of frequentist, Bayesian, and e-value approaches -- in a two-arm binary-endpoint trial. Naive repeated testing and naive posterior thresholds inflate Type I error substantially under frequent interim looks. Among the valid methods, the calibrated group sequential rule achieves the highest power, the e-value rule provides robust anytime-valid control with moderate power, and the calibrated Bayesian rule is the most conservative. Extended simulations show that the power gap between group sequential and e-value methods depends on the monitoring schedule and reverses under continuous monitoring. The methodology, including futility monitoring, platform trial multiplicity control, and hybrid strategies combining e-values with established methods, is implemented in the open-source R package `evalinger` and situated within the regulatory framework of the January 2026 FDA draft guidance on Bayesian methodology.

stat.ME↗

arXiv2026

Bayesian Dynamic Gamma Models for Route-Level Travel Time Reliability

Route-level travel time reliability requires characterizing the distribution of total travel time across correlated segments -- a problem where existing methods either assume independence (fast but miscalibrated) or model dependence via copulas and simulation (accurate but expensive). We propose a conjugate Bayesian dynamic Gamma model with a common random environment that resolves this trade-off. Each segment's travel time follows a Gamma distribution conditional on a shared latent environment process that evolves as a Markov chain, inducing cross-segment dependence while preserving conditional independence. A moment-matching approximation yields a closed-form $F$-distribution for route travel time, from which the Planning Time Index, Buffer Index, and on-time probability are computed instantly -- at the same $O(1)$ cost as independence-based methods. The conjugate structure ensures that Bayesian posterior updates and the full predictive distribution are available in closed form as new sensor data arrives. Applied to 16 sensors spanning 8.26 miles on I-55 in Chicago, the model achieves 95.4% coverage of nominal 90\% predictive intervals versus 34--37% for independence-based convolution, at identical computational cost.

stat.AP↗

arXiv2026

Physics-Informed Diffusion Models for Vehicle Speed Trajectory Generation

Synthetic vehicle speed trajectory generation is essential for evaluating vehicle control algorithms and connected vehicle technologies. Traditional Markov chain approaches suffer from discretization artifacts and limited expressiveness. This paper proposes a physics-informed diffusion framework for conditional micro-trip synthesis, combining a dual-channel speed-acceleration representation with soft physics constraints that resolve optimization conflicts inherent to hard-constraint formulations. We compare a 1D U-Net architecture against a transformer-based Conditional Score-based Diffusion Imputation (CSDI) model using 6,367 GPS-derived micro-trips. CSDI achieves superior distribution matching (Wasserstein distance 0.30 for speed, 0.026 for acceleration), strong indistinguishability from real data (discriminative score 0.49), and validated utility for downstream energy assessment tasks. The methodology enables scalable generation of realistic driving profiles for intelligent transportation systems (ITS) applications without costly field data collection.

stat.AP↗

arXiv2026

Fast Compute via MC Boosting

Modern training and inference pipelines in statistical learning and deep learning repeatedly invoke linear-system solves as inner loops, yet high-accuracy deterministic solvers can be prohibitively expensive when solves must be repeated many times or when only partial information (selected components or linear functionals) is required. We position \emph{Monte Carlo boosting} as a practical alternative in this regime, surveying random-walk estimators and sequential residual correction in a unified notation (Neumann-series representation, forward/adjoint estimators, and Halton-style sequential correction), with extensions to overdetermined/least-squares problems and connections to IRLS-style updates in data augmentation and EM/ECM algorithms. Empirically, we compare Jacobi and Gauss--Seidel iterations with plain Monte Carlo, exact sequential Monte Carlo, and a subsampled sequential variant, illustrating scaling regimes that motivate when Monte Carlo boosting can be an enabling compute primitive for modern statistical learning workflows.

stat.CO↗

arXiv2026

Bayesian Methods for the Navier-Stokes Equations

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--Stokes dynamics as a state-space model and to view numerical solution as posterior computation: priors encode physical structure and modeling error, and the solver outputs a distribution over states and quantities of interest rather than a single trajectory. In two dimensions, stochastic representations (Feynman--Kac and stochastic characteristics for linear advection--diffusion with prescribed drift) motivate Monte Carlo solvers and provide intuition for uncertainty propagation. In three dimensions, we formulate stochastic Navier--Stokes models and describe particle-based and ensemble-based Bayesian workflows for uncertainty propagation in spectral discretizations. A key computational advantage is that parameter learning can be performed stably via particle learning: marginalization and resample--propagate (one-step smoothing) constructions avoid the weight-collapse that plagues naive sequential importance sampling on static parameters. When partial observations are available, the same machinery supports sequential observational updating as an additional capability. We also discuss non-Gaussian (heavy-tailed) error models based on normal variance-mean mixtures, which yield conditionally Gaussian updates via latent scale augmentation.

stat.CO↗

arXiv2026

Horseshoe Mixtures-of-Experts (HS-MoE)

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 adaptive global-local shrinkage with input-dependent gating, yielding data-adaptive sparsity in expert usage. Our primary methodological contribution is a particle learning algorithm for sequential inference, in which the filter is propagated forward in time while tracking only sufficient statistics. We also discuss how HS-MoE relates to modern mixture-of-experts layers in large language models, which are deployed under extreme sparsity constraints (e.g., activating a small number of experts per token out of a large pool).

stat.ML↗