Given a matrix $P \in \Re^{n \times d}$, and a column vector $\theta \in \Re^d$. Assume that $\sum\limits_{i=1}^n \ln{(1+e^{P_i\theta})} \leq 1$, where $P_i$ is the $i^{th}$ row in $P$.
What can be said about $\sum\limits_{i=1}^n \ln{(1+e^{P_i(-\theta)})}$??
Please advice and thanks in advance.