Maximum Likelihood Through a Noisy Channel
I'm inclined to think this is true, but so far I've only gotten a counterexample:

Consider the channel:

\begin{bmatrix}0.5 &amp; 0.5 &amp; 0 &amp; 0\\ 0 &amp; 0.5 &amp; 0.5 &amp; 0 \\ 0 &amp; 0 &amp; 0.5 &amp; 0.5 \\ 0.5 &amp; 0 &amp; 0 &amp; 0.5\end{bmatrix}

This a typewriter channel. This matrix is not invertible (it has zero determinant, and rank 2). So the ML estimate of $\Theta$ cannot be gotten from this channel. 

However, for $p \neq 0.5$ I think it could well be true.

EDIT: In fact I think the noisy channel aspect of this may well be a red-herring. The ML estimator of $\Theta$ is as you describe, simply because you have a linear model. You don't need any conditions of $C$ other than it should be invertible (i.e. all rows are linearly independent).