Parameter estimation for integer-valued Gibbs distributions

by   David G. Harris, et al.

We consider the family of Gibbs distributions, which are probability distributions over a discrete space Ω given by μ^Ω_β(x)=e^β H(x)/Z(β). Here H:Ω→{0,1,...,n} is a fixed function (called a Hamiltonian), β is the parameter of the distribution, and Z(β)=∑_x∈Ωe^β H(x) = ∑_k=0^n c_k e^β k is the normalization factor called the partition function. We study how function Z(·) can be estimated using an oracle that produces samples x∼μ_β(·) for a value β in a given interval [β_, β_]. Recently, it has been shown how to estimate quantity q=Z(β_)/Z(β_) with additive error ε using Õ(q/ε^2) samples in expectation. We improve this result to Õ({q,n^2}/ε^2), matching a lower bound of Kolmogorov (2018) up to logarithmic factors. We also consider the problem of estimating the normalized coefficients c_k for indices k∈{0,1,...,n} that satisfy _βμ_β^Ω({x | H(x)=k}) >μ_∗, where μ_∗∈(0,1) is a given parameter. We solve this problem using Õ({ q + √(q)/μ_∗, n^2 + n/μ_∗}/ε^2) expected samples, and we show that this complexity is optimal up to logarithmic factors. This is improved to roughly Õ( 1/μ_∗+{q + n,n^2}/ε^2) for applications in which the coefficients are known to be log-concave (e.g. for connected subgraphs of a given graph).


Parameter estimation for Gibbs distributions

We consider Gibbs distributions, which are families of probability distr...

Characterizations of non-normalized discrete probability distributions and their application in statistics

From the distributional characterizations that lie at the heart of Stein...

Dualizing Le Cam's method, with applications to estimating the unseens

One of the most commonly used techniques for proving statistical lower b...

Sample-efficient learning of quantum many-body systems

We study the problem of learning the Hamiltonian of a quantum many-body ...

A Concentration Result of Estimating Phi-Divergence using Data Dependent Partition

Estimation of the ϕ-divergence between two unknown probability distribut...

Order Optimal One-Shot Distributed Learning

We consider distributed statistical optimization in one-shot setting, wh...

Deterministic Finite-Memory Bias Estimation

In this paper we consider the problem of estimating a Bernoulli paramete...