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We present predictions for the statistical error due to finite sampling in the presence of thermal fluctuations in molecular simulation algorithms. Specifically, we establish how these errors depend on Mach number, Knudsen number, number of…
The problem of f-divergence estimation is important in the fields of machine learning, information theory, and statistics. While several nonparametric divergence estimators exist, relatively few have known convergence properties. In…
Machine learning models have emerged as a very effective strategy to sidestep time-consuming electronic-structure calculations, enabling accurate simulations of greater size, time scale and complexity. Given the interpolative nature of…
Power-law distributions are typical macroscopic features occurring in almost all complex systems observable in nature. As a result, researchers in quantitative analyses must often generate random synthetic variates obeying power-law…
Using a one-dimensional macromolecule in aqueous solution as an illustration, we demonstrate that the relative entropy from information theory, $\sum_k p_k\ln(p_k/p_k^*)$, has a natural role in the energetics of equilibrium and…
Analyzing and controlling system entropy is a powerful tool for regulating predictability of control systems. Applications benefiting from such approaches range from reinforcement learning and data security to human-robot collaboration. In…
The Central Limit Theorem provides a foundation for inferential statistics and hypothesis testing. It describes how standardized statistics behave under repeated sampling from large populations. However, if the size of the sample (n)…
In some estimation problems, especially in applications dealing with information theory, signal processing and biology, theory provides us with additional information allowing us to restrict the parameter space to a finite number of points.…
Aleatoric and Epistemic uncertainty have achieved recent attention in the literature as different sources from which uncertainty can emerge in stochastic modeling. Epistemic being intrinsic or model based notions of uncertainty, and…
This paper investigates an expected average error for distributed averaging problems under asynchronous updates. The asynchronism in this context implies no existence of a global clock as well as random characteristics in communication…
All energy measurements of a quantum system are prone to inaccuracies. In particular, if such measurements are carried over a finite period of time the accuracy of the result is affected by the length of that period. Here I show an upper…
Mathematical theory of selection systems is developed for a wide class of dynamical models of inhomogeneous populations with discrete time. The Price equation and its particular case, the Fisher Fundamental theorem of natural selection…
The apparantly irregular (unpredictable) space-time fluctuations in atmospheric flows ranging from climate (thousands of kilometers - years) to turbulence (millimeters - seconds) exhibit the universal symmetry of self-similarity.…
We investigate how to determine whether the states of a set of quantum systems are identical or not. This paper treats both error-free comparison, and comparison where errors in the result are allowed. Error-free comparison means that we…
Entropy, its production, and its change in a dynamical system can be understood from either a fully stochastic dynamic description or from a deterministic dynamics exhibiting chaotic behavior. By taking the former approach based on the…
High dimensional random dynamical systems are ubiquitous, including -- but not limited to -- cyber-physical systems, daily return on different stocks of S&P 1500 and velocity profile of interacting particle systems around McKeanVlasov…
Decision-focused learning (DFL), which differentiates through the KKT conditions, has recently emerged as a powerful approach for predict-then-optimize problems. However, under probabilistic settings, DFL faces three major bottlenecks:…
Switching Kalman Filters (SKF) are well known for their ability to solve the piecewise linear dynamic system estimation problem using the standard Kalman Filter (KF). Practical SKFs are heuristic, approximate filters that are not guaranteed…
The theory of large deviations deals with the probabilities of rare events (or fluctuations) that are exponentially small as a function of some parameter, e.g., the number of random components of a system, the time over which a stochastic…
Fluctuations of dynamical quantities are fundamental and inevitable. For the booming research in nanotechnology, huge relative fluctuation comes with the reduction of system size, leading to large uncertainty for the estimates of dynamical…