Related papers: On the Use of Approximations in Statistical Physic…
Measurement error is a pervasive issue which renders the results of an analysis unreliable. The measurement error literature contains numerous correction techniques, which can be broadly divided into those which aim to produce exactly…
We present an axiomatic approach to the mean and discuss generalizations of the mean, including one due to Kolmogorov based on the Weak Law of Large Numbers. We offer examples and counterexamples, describe conventional and unconventional…
We consider the discrete three dimensional scan statistics. Viewed as the maximum of an 1-dependent stationary r.v.'s sequence, we provide approximations and error bounds for the probability distribution of the three dimensional scan…
In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein's method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a…
A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of…
Spin, $s$ in quantum theory can assume only half odd integer or integer values. For a given $s$, there exist $n=2s+1$ states $|s,m\rangle$, $m=s,s-1,........,-s$. A statistical assembly of particles (like a beam or target employed in…
In this paper we improve some existing results concerning the approximation of the distribution of extremes of a 1-dependent and stationary sequence of random variables. We enlarge the range of applicability and improve the approximation…
The paper presents a new statistical method that enables the use of systematic errors in the maximum-likelihood regression of integer-count Poisson data to a parametric model. The method is primarily aimed at the characterization of the…
An approximation method is presented for probabilistic inference with continuous random variables. These problems can arise in many practical problems, in particular where there are "second order" probabilities. The approximation, based on…
The basic notions of statistical mechanics (microstates, multiplicities) are quite simple, but understanding how the second law arises from these ideas requires working with cumbersomely large numbers. To avoid getting bogged down in…
Statistical thermodynamics delivers the probability distribution of the equilibrium state of matter through the constrained maximization of a special functional, entropy. Its elegance and enormous success have led to numerous attempts to…
Probably we have observed a new simple phenomena dealing with approximations to two real numbers.
Estimating mutual information (MI) is a fundamental yet challenging task in data science and machine learning. This work proposes a new estimator for mutual information. Our main discovery is that a preliminary estimate of the data…
This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the…
One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and non-stationary regimes is the lack of a suitable limit object. Instead, suitable distributional approximations can be used, where…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
The Stirling approximation formula for $n!$ dates from 1730. Here we give new and instructive proofs of this and related approximation formulae via tools of probability and statistics. There are connections to the Central Limit Theorem and…
A \emph{double extrema form} of the calculus of variations is put forward in which only the smallest one of the finite differences is physically meaningful to represent the variational derivatives defined on the discrete points. The most…
A calculational approach in fluid turbulence is presented. Use is made of the attracting nature of the fluid-dynamic dynamical system. An approximate approach is offerred that effectively propagates the statistics in time. Loss of…
We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$…