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

Understand

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 $q^n$.