Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance

Abstract

The aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: For a fixed probability measure $q^n$ on ${\mathcal{X}}^n$, ($\mathcal{X}$ is a finite set), and any probability measure $p^n=\mathcal{L}(Y^n)$ on ${\mathcal{X}}^n$, we have \begin{equation}\label{*} D(p^n||q^n)\leq Const. \sum_{i=1}^n \Bbb E_{p^n} D(p_i(\cdot|Y_1,\dots, Y_{i-1},Y_{i+1},\dots, Y_n) || q_i(\cdot|Y_1,\dots, Y_{i-1},Y_{i+1},\dots, Y_n)), \end{equation} where $p_i(\cdot |y_1,\dots ,y_{i-1},y_{i+1},\dots ,y_n)$ and $q_i(\cdot |x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_n)$ denote the local specifications for $p^n$ resp. $q^n$. The constant shall depend on the properties of the local specifications of $q^n$. Inequality (*) is meaningful in product spaces, both in the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for $q^n$, provided uniform logarithmic Sobolev inequalities are available for $q_i(\cdot |x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_n)$, for all fixed $i$ and all fixed $(x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_n)$. Inequality (*) directly implies that the Gibbs sampler associated with $q^n$ is a contraction for relative entropy. We derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance. A logarithmic Sobolev inequality is, roughly speaking, a contractivity property of relative entropy with respect to some Markov semigroup. It is much easier to prove contractivity for a distance between measures than for relative entropy, since distances satisfy the triangle inequality, and for them well known linear tools, like estimates through matrix norms can be applied.

arXiv comment24 pages

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