Parameter estimation for integer-valued Gibbs distributions
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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).
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