Parameter Estimation of Mutual Information Maximized Channels

We study the problem of estimating a parametric discrete memoryless channel $p(y \mid x; \boldsymbol{\theta})$ when the transmitter selects its input distribution $\pi$ to maximize mutual information under the true parameter $\boldsymbol{\theta}^*$. Using only i.i.d.\ observations of the channel output, we aim to jointly estimate the capacity-achieving input distribution $\boldsymbol{\pi}^*$ and the true channel parameter $\boldsymbol{\theta}^*$. In general, recovery of $\boldsymbol{\pi}^*$ and $\boldsymbol{\theta}^*$ 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 $\boldsymbol{\theta}^*$ and $\boldsymbol{\pi}^*$, whereas a naive maximum-likelihood approach that ignores the mutual-information maximization constraint fails to do so.