Consider a family of distributions $\{π_β\}$ where $X\simπ_β$ means that $\mathbb{P}(X=x)=\exp(-βH(x))/Z(β)$. Here $Z(β)$ is the proper normalizing constant, equal to $\sum_x\exp(-βH(x))$. Then $\{π_β\}$ is known as a Gibbs distribution, and $Z(β)$ is the partition function. This work presents a new method for approximating the partition function to a specified level of relative accuracy using only a number of samples, that is, $O(\ln(Z(β))\ln(\ln(Z(β))))$ when $Z(0)\geq1$. This is a sharp improvement over previous, similar approaches that used a much more complicated algorithm, requiring $O(\ln(Z(β))\ln(\ln(Z(β)))^5)$ samples.
We obtain some approximation results for the weights appearing in the exchangeable partition probability function identifying Gibbs partition models of parameter $α\in (0,1)$, as introduced in Gnedin and Pitman (2006). We rely on approximation results for central and non-central generalized Stirling numbers and on known results for conditional and unconditional $α$ diversity. We provide an application to an approximate Bayesian nonparametric estimation of discovery probability in species sampling problems under normalized inverse Gaussian priors.
This paper deals with the numerical approximation of normalizing constants produced by particle methods, in the general framework of Feynman-Kac sequences of measures. It is well-known that the corresponding estimates satisfy a central limit theorem for a fixed time horizon $n$ as the number of particles $N$ goes to infinity. Here, we study the situation where both $n$ and $N$ go to infinity in such a way that $\lim_{n\rightarrow\infty}% n/N=α>0$. In this context, Pitt et al. \cite{pitt2012} recently conjectured that a lognormal central limit theorem should hold. We formally establish this res…
We discuss a thinning and an embedding procedure to construct finite Gibbs processes with a given Papangelou intensity. Extending the approach in Hofer-Temmel (2019) and Hofer-Temmel and Houdebert (2019) we will use this to couple two finite Gibbs processes with different boundary conditions. As one application we will establish Poisson approximation of point processes derived from certain infinite volume Gibbs processes via dependent thinning. As another application we shall discuss empty space probabilities of certain Gibbs processes.
We study the approximation of action minimizing measures, for a mechanical Lagrangian on the torus, by the probability densities given by quasi-periodic eigenfunctions of the Schrodinger operator.
A Stein operator for the runs is derived as a perturbation of an operator for discrete Gibbs measure. Due to this fact, using perturbation technique, the approximation results for runs arising from identical and non-identical Bernoulli trials are derived via Stein method. The bounds obtained are new and their importance is demonstrated through an interesting application.
In the paper asymptotic properties of functionals of stationary Gibbs particle processes are derived. Two known techniques from the point process theory in the Euclidean space R^d are extended to the space of compact sets on R^d equipped by the Hausdorff metric. First, conditions for the existence of the stationary Gibbs point process with given conditional intensity have been simplified recently. Secondly, the Malliavin-Stein calculus was applied to the estimation of Wasserstein distance between the Gibbs input and standard Gaussian distribution. We transform these theories to the space of co…
We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factor…
In this paper we address the questions of perfectly sampling a Gibbs measure with infinite range interactions and of perfectly sampling the measure together with its finite range approximations. We solve these questions by introducing a perfect simulation algorithm for the measure and for the coupled measures. The algorithm works for general Gibbsian interaction under requirements on the tails of the interaction. As a consequence we obtain an upper bound for the error we make when sampling from a finite range approximation instead of the true infinite range measure.
Stein's method provides a way of bounding the distance of a probability distribution to a target distribution $μ$. Here we develop Stein's method for the class of discrete Gibbs measures with a density $e^V$, where $V$ is the energy function. Using size bias couplings, we treat an example of Gibbs convergence for strongly correlated random variables due to Chayes and Klein [Helv. Phys. Acta 67 (1994) 30--42]. We obtain estimates of the approximation to a grand-canonical Gibbs ensemble. As side results, we slightly improve on the Barbour, Holst and Janson [Poisson Approximation (1992)] bounds f…
Disordered systems such as spin glasses have been used extensively as models for high-dimensional random landscapes and studied from the perspective of optimization algorithms. In a recent paper by L. Addario-Berry and the second author, the continuous random energy model (CREM) was proposed as a simple toy model to study the efficiency of such algorithms. The following question was raised in that paper: what is the threshold $β_G$, at which sampling (approximately) from the Gibbs measure at inverse temperature $β$ becomes algorithmically hard? This paper is a first step towards answering this…
We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $π$ on $\mathbb{R}^d$, based on (overdamped) Langevin diffusions. This method inspired by \cite{mainPPlangevin} and \cite{giles_szpruch_invariant} relies on a multilevel occupation measure, $i.e.$ on an appropriate combination of $R$ occupation measures of (constant-step) Euler schemes with respective steps $γ_r = γ_0 2^{-r}$, $r=0,\ldots,R$. We first state a quantitative result under general assumptions which guarantees an \textit{$\varepsilon$-approximation} (in a $L^2$-sense) with a cost of …
We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariate observations, given a centering and a general normalizing sequence $b(N)$ of invertible square matrices, are approximated by a normal distribution with respect to a metric of regular test functions. Depending on the metric and the normalizing sequence $b(N)$, the conditions imply that the error in the approximation decays either at the rate $O(N^{-1/2})$ or the rate $O(N^{-1/2} \log N)$, under the additional …
We prove an exponential approximation for the law of approximate occurrence of typical patterns for a class of Gibssian sources on the lattice $\mathbb{Z}^d$, $d\ge2$. From this result, we deduce a law of large numbers and a large deviation result for the waiting time of distorted patterns.
We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics like the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity and higher order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.
Let $μ$ be a Gibbs measure of the doubling map $T$ of the circle. For a $μ$-generic point $x$ and a given sequence $\{r_n\} \subset \R^+$, consider the intervals $(T^nx - r_n \pmod 1, T^nx + r_n \pmod 1)$. In analogy to the classical Dvoretzky covering of the circle we study the covering properties of this sequence of intervals. This study is closely related to the local entropy function of the Gibbs measure and to hitting times for moving targets. A mass transference principle is obtained for Gibbs measures which are multifractal. Such a principle was shown by Beresnevich and Velani \cite{BV}…
Boltzmann-Gibbs random fields are defined in terms of the exponential expression exp(-H), where H is a suitably defined energy functional of the field states x(s). This paper presents a new Boltzmann-Gibbs model which features local interactions in the energy functional. The interactions are embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics. A specific model for the interactions based on a second-degree polynomial of the Laplace operator is studied. An explicit, mesh-free…
