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

Understand

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 $q^n$.