Parameter Estimation of Mutual Information Maximized Channels

We study the problem of estimating a parametric discrete memoryless channel $p(y\mid x;\mathit{\theta })$ when the transmitter selects its input distribution $\pi$ to maximize mutual information under the true parameter ${\mathit{\theta }}^{\ast }$. Using only i.i.d.\ observations of the channel output, we aim to jointly estimate the capacity-achieving input distribution ${\mathit{\pi }}^{\ast }$ and the true channel parameter ${\mathit{\theta }}^{\ast }$. In general, recovery of ${\mathit{\pi }}^{\ast }$ and ${\mathit{\theta }}^{\ast }$ can be challenging. To that end, we propose two efficient algorithms based on the Blahut–Arimoto (BA) optimality conditions: (i) a bilevel fixed-point method and (ii) an augmented Lagrangian method. Empirical results demonstrate that both proposed algorithms successfully recover the true ${\mathit{\theta }}^{\ast }$ and ${\mathit{\pi }}^{\ast }$, whereas a naive maximum-likelihood approach that ignores the mutual-information maximization constraint fails to do so.