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Consider a finite state, irreducible Markov chain with a rate matrix $Q$ and a stationary distribution $\pi$. Suppose the chain starts with the initial distribution $p$ at time $0$, then at time $t$ the distribution is given by $$p_t = p*e^{Qt}$$ The relative entropy of $p_t$ with respect to the $\pi$ (also known as the Kullback-Leibler divergence $D(p_t || \pi)$) is given by $$D(p_t || \pi) = \sum_i p_t(i) \log \left(\frac{p_t(i)}{\pi(i)}\right)$$ where $i$ ranges over all the states.

It seems well known that $D(p_t || \pi)$ monotonically decreases as $t$ increases. (For instance, section $2.9$ of Cover and Thomas's book here

My question is:

Is $D(p_t || \pi)$ a convex function of $t$?

I simulated for many random matrices using Matlab and did not find a counter example. Any references would be appreciated!

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up vote 4 down vote accepted

Turns out $D(p_t||\pi)$ need not always be convex. The following paper demonstrates a counter-example in section $4.2$-

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