Related papers: Random Series and Discrete Path Integral methods: …
By carefully analyzing the relations between operator methods and the discretized and continuum path integral formulations of quantum-mechanical systems, we have found the correct Feynman rules for one-dimensional path integrals in curved…
Barrier crossing is a widespread phenomenon across natural and engineering systems. While an abundant cross-disciplinary literature on the topic has emerged over the years, the stochastic underpinnings of the process are yet to be linked…
This is a brief review on Brownian functionals in one dimension and their various applications, a contribution to the special issue ``The Legacy of Albert Einstein" of Current Science. After a brief description of Einstein's original…
We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from $O(N^2)$ per time step to…
We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve L\'{e}vy-type…
In this paper, we study efficient approximate sampling for probability distributions known up to normalization constants. We specifically focus on a problem class arising in Bayesian inference for large-scale inverse problems in science and…
Approximate Bayesian computation (ABC) is the most popular approach to inferring parameters in the case where the data model is specified in the form of a simulator. It is not possible to directly implement standard Monte Carlo methods for…
Covariate-adaptive randomization (CAR) procedures are frequently used in comparative studies to increase the covariate balance across treatment groups. However, because randomization inevitably uses the covariate information when forming…
A hierarchical Bayesian approach that permits simultaneous inference for the regression coefficient matrix and the error precision (inverse covariance) matrix in the multivariate linear model is proposed. Assuming a natural ordering of the…
Through a very careful analysis of Dirac's 1932 paper on the Lagrangian in Quantum Mechanics as well as the second and third editions of his classic book {\it The Principles of Quantum Mechanics}, I show that Dirac's contributions to the…
The signature of a path, as a fundamental object in Rough path theory, serves as a generating function for non-commutative monomials on path space. It transforms the path into a grouplike element in the tensor algebra space, summarising the…
Bayesian sampling is an important task in statistics and machine learning. Over the past decade, many ensemble-type sampling methods have been proposed. In contrast to the classical Markov chain Monte Carlo methods, these new methods deploy…
The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially…
In this work we study of the dynamics of large size random neural networks. Different methods have been developed to analyse their behavior, most of them rely on heuristic methods based on Gaussian assumptions regarding the fluctuations in…
The process of data interpretation is always based on the implicit introduction of equivalence relations on the set of walks over the database. Every equivalence relation on the set of walks specifies a Markov chain describing the…
Methods were initiated by Mark Kac and Richard Feynman to evaluate random functionals of the form $\int^t_0V(X_s)ds$ for a nonnegative $V$ and a Markov process $X_t$. Their results evolved into the well known Feynman Kac formula.…
The approach to the theory of a relativistic random process is considered by the path integral method as Brownian motion taking into account the boundedness of speed. An attempt was made to build a relativistic analogue of the Wiener…
We present a sequential sampling methodology for weakly structural Markov laws, arising naturally in a Bayesian structure learning context for decomposable graphical models. As a key component of our suggested approach, we show that the…
Calibrating mathematical models of biological processes is essential for achieving predictive accuracy and gaining mechanistic insight. However, this task remains challenging due to limited and noisy data, significant biological…
The Feynman-Kac Operator Expectation Estimator (FKEE) is an innovative method for estimating the target Mathematical Expectation $\mathbb{E}_{X\sim P}[f(X)]$ without relying on a large number of samples, in contrast to the commonly used…