arXiv subjects

Sucharita Roy

Publications and source records attributed to Sucharita Roy.

6 recordsLinked to original sources

arXiv2026

Function Optimization with Posterior Gaussian Derivative Process

We propose a Bayesian optimization algorithm for differentiable functions with available first and second partial derivatives. The objective function is modeled as a Gaussian process, inducing a Gaussian derivative process. Given initial function evaluations, we obtain the posterior of the derivative process and construct a posterior for stationary points by setting the derivative to zero. A recursive importance-resampling scheme with prior constraints on the gradient norm and Hessian definiteness drives the algorithm toward the true optima. We prove almost sure convergence as the number of stages tends to infinity, relying on novel results establishing almost sure uniform convergence of the Gaussian process and its derivative process posteriors to the true function and its derivatives under fixed-domain infill asymptotics; rates of convergence are also derived. We also give a Bayesian characterization of the number of optima. The method is demonstrated on five diverse problems, including finding maxima, minima, saddle points, and inconclusive cases, ranging from one-dimensional to 100-dimensional nonlinear least-squares. On a real-world Poisson regression (AIDS deaths data), our procedure achieves a substantially smaller gradient norm at the MLE than Fisher scoring, BFGS, simulated annealing, and multi-start quasi-Newton. The posterior simulation nature allows exploration of neighborhoods of solutions from other methods, yielding more accurate results. Code is available at https://github.com/Sourabh-Bhattacharya/FUNCTION_OPT_GDP.

math.OC↗

arXiv2026

The Bayesian Reflex: Online Learning as the Autonomic Nervous System of Modern and Future AI

This chapter introduces the Bayesian reflex -- an analogy with the autonomic nervous system -- as a unifying framework for online learning in AI. Bayesian online algorithms automatically maintain equilibrium in dynamic environments via three mechanisms: belief maintenance through probabilistic representations, sequential updating via Bayes' theorem, and uncertainty-driven action balancing exploration and exploitation. We survey online Bayesian methods, highlighting two computational principles: the look-up table principle for sequential inference in function space, and the ellipsoidal decomposition framework for nearly exact i.i.d. sampling from arbitrary posteriors. These principles are generalized across dynamic emulation, nonparametric state-space models, circular time series, inverse regression for climate model evaluation, and deep architectures via Recursive Gaussian Processes. Decision-making is explored via Thompson sampling and restless bandits. We extend the framework to assess infinite series convergence (applied to climate dynamics and the Riemann Hypothesis), model prime number distributions leading to the discovery of 184 strong Mersenne prime candidates, detect stationarity, and characterize point processes. The Bayesian reflex provides a foundational infrastructure for adaptive AI that continuously learns in a complex world.

stat.ME↗

arXiv2025

Bayes Meets Riemann Again: Large Prime Discovery and Re-emergence of the Bone of Contention

Prime numbers have fascinated mathematicians since antiquity, with ongoing efforts to uncover both their properties and ever-larger examples. While giant primes rarely aid cryptography, they find use in areas such as locally decodable codes. Large prime-hunting, often brute-force in nature, is conceptually linked to the Riemann Hypothesis and the prime number theorem, which portrays prime distribution as essentially random. This motivates a statistical perspective, with Bayesian methodology providing a natural foundation. We show that the prime number theorem suggests a nonhomogeneous Poisson process for prime counts, yielding primes as waiting times. This process agrees with the prime number theorem, asymptotic results, and prime gap properties. Building on it, we develop a recursive Bayesian theory for large prime prediction and Riemann Hypothesis validation. The approach matches traditional but computationally infeasible non-recursive Bayesian formulations in the limit, and it strongly falsifies the Riemann Hypothesis. Finally, we propose a computational method using Transformation-based MCMC to simulate recursive posterior predictives. A simple change of variable enables simulation of Mersenne prime exponents. With modest computing resources, we identified 259 primes over 140 million, including 184 strong Mersenne candidates corresponding to potential primes with 42--242 million digits.

math.GM↗

arXiv2020

Bayesian Appraisal of Random Series Convergence with Application to Climate Change

Roy and Bhattacharya (2020) provided Bayesian characterization of infinite series, and their most important application, namely, to the Dirichlet series characterizing the (in)famous Riemann Hypothesis, revealed insights that are not in support of the most celebrated conjecture for over 150 years. In contrast with deterministic series considered by Roy and Bhattacharya (2020), in this article we take up random infinite series for our investigation. Remarkably, our method does not require any simplifying assumption. Albeit the Bayesian characterization theory for random series is no different from that for the deterministic setup, construction of effective upper bounds for partial sums, required for implementation, turns out to be a challenging undertaking in the random setup. In this article, we construct parametric and nonparametric upper bound forms for the partial sums of random infinite series and demonstrate the generality of the latter in comparison to the former. Simulation studies exhibit high accuracy and efficiency of the nonparametric bound in all the setups that we consider. Finally, exploiting the property that the summands tend to zero in the case of series convergence, we consider application of our nonparametric bound driven Bayesian method to global climate change analysis. Specifically, analyzing the global average temperature record over the years 1850--2016 and Holocene global average temperature reconstruction data 12,000 years before present, we conclude, in spite of the current global warming situation, that global climate dynamics is subject to temporary variability only, the current global warming being an instance, and long term global warming or cooling either in the past or in the future, are highly unlikely.

math.PR↗

arXiv2020

Bayesian Characterizations of Properties of Stochastic Processes with Applications

In this article, we primarily propose a novel Bayesian characterization of stationary and nonstationary stochastic processes. In practice, this theory aims to distinguish between global stationarity and nonstationarity for both parametric and nonparametric stochastic processes. Interestingly, our theory builds on our previous work on Bayesian characterization of infinite series, which was applied to verification of the (in)famous Riemann Hypothesis. Thus, there seems to be interesting and important connections between pure mathematics and Bayesian statistics, with respect to our proposed ideas. We validate our proposed method with simulation and real data experiments associated with different setups. In particular, applications of our method include stationarity and nonstationarity determination in various time series models, spatial and spatio-temporal setups, and convergence diagnostics of Markov Chain Monte Carlo. Our results demonstrate very encouraging performance, even in very subtle situations. Using similar principles, we also provide a novel Bayesian characterization of mutual independence among any number of random variables, using which we characterize the properties of point processes, including characterizations of Poisson point processes, complete spatial randomness, stationarity and nonstationarity. Applications to simulation experiments with ample Poisson and non-Poisson point process models again indicate quite encouraging performance of our proposed ideas. We further propose a novel recursive Bayesian method for determination of frequencies of oscillatory stochastic processes, based on our general principle. Simulation studies and real data experiments with varieties of time series models consisting of single and multiple frequencies bring out the worth of our method.

math.ST↗

arXiv2018

Bayes meet Riemann- Bayesian Characterization of Infinite Series with Application to Riemann Hypothesis

In the classical literature on infinite series there are various tests to determine if a given infinite series converges, diverges, or oscillates. But unfortunately, for very many infinite series all the existing tests can fail to provide definitive answers. In this article we propose a novel Bayesian theory for assessment of convergence properties of any given infinite series. Remarkably, this theory attempts to provide conclusive answers to the question of convergence even where all the existing tests of convergence fail. We apply our ideas to seven different examples, obtaining very encouraging results. Importantly, we also apply our ideas to investigate the Riemann Hypothesis, and obtain results that do not completely support the conjecture. We also extend our ideas to develop a Bayesian theory on oscillating series, where we allow even infinite number of limit points. Analysis of Riemann Hypothesis using Bayesian multiple limit points theory yielded almost identical results as the Bayesian theory of convergence assessment.

math.ST↗