Bounding relative entropy by the relative entropy of local specifications in product spaces

Details Bounding relative entropy by the relative entropy of local specifications in product spaces Bounding relative entropy by the relative entropy of local specifications in product spaces

2009-07-26

arxiv.org/abs/0907.4491v1

Mathematics

Probability

31 pages. This is a strengthened version of paper arXiv:math/0605397, titled An explicit bound on the Logarithmic Sobolev cons

Scientific paper

For a class of density functions $q^n(x^n)$ on $\Bbb R^n$ we prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: For any density function $p^n(x^n)$ on $\Bbb R^n$, $D(p^n||q^n)\leq Const. \sum_{i=1}^n \Bbb E D(p_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n) || Q_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n)),$ where $p_i(\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot|x_1,..., x_{i-1},x_{i+1},..., x_n)$ denote the local specifications for $p^n$ resp. $q^n$, i.e., the conditional density functions of the $i$'th coordinate, given the other coordinates. The constant depends on the properties of the local specifications of $q^n$. The above inequality implies a logarithmic Sobolev inequality for $q^n$. We get an explicit lower bound for the logarithmic Sobolev constant of $q^n$ under the assumptions that: (i) the local specifications of $q^n$ satisfy logarithmic Sobolev inequalities with constants $\rho_i$, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian of $q^n$ are not too large relative to the logarithmic Sobolev constants $\rho_i$. Condition (ii) may be weaker than that used in Otto and Reznikoff's recent paper on the estimation of logarithmic Sobolev constants of spin systems.

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