# A mutual information equality

$$x$$ and $$y$$ are two random variables and $$I(u;v)$$ is the mutual information between random variables $$u$$ and $$v$$. Does the following equality hold? $$\text{argmax}_a I(y;ax)=\text{argmin}_a I(y;y-ax).$$

It can be seen that for any $$a\neq 0$$, $$I(y;ax)=I(y;x)$$ because $$y\rightarrow ax\rightarrow x$$ and $$y\rightarrow x\rightarrow ax$$ are both Markov chain hence by the data processing inequality both $$I(y;x)\leq I(y;ax)$$ and $$I(y;x)\geq I(y;ax)$$ respectively.
Since any $$a\neq 0$$ is a maximizer of $$I(y;ax)$$, your question is ill defined.
• Could there be a misunderstanding, say, as shown in the expression $I(y;x)>0=\mathrm{argmin}_a I(y;y-ax)$? I am asking about argmin which the argument $a$ for which a minimization is achieved, not min. – Hans Jan 8 at 16:48
• Yes, Sorry about that, but then you have to define $\mathrm{argmax}$ more carefully since any $a\neq 0$ is a maximizer of $I(y;ax)$ hence yes, it could be true depending on how you do this definition – P. Quinton Jan 8 at 17:07
• You are right. Indeed, from the definition $I(x;y)$ is invariant under any injective one variable transformation, i.e. $I(x;y)=I(f(x),g(y))$ where $f$ and $g$ measurable injective functions. More importantly, you may be interested in taking a look at the question of my main concern stats.stackexchange.com/q/386101/44368. – Hans Jan 8 at 23:00
• well, what you say is true if and only if $f$ and $g$ are almost surely one to one. About your main question, I don't understand what "z distributed normally conditioned on x" means, I would say you may want $x$ and $z$ independ and $z\sim \mathcal N(0,\sigma^2)$ or something of the sort. I would say without proof that if $x$ is also gaussian then your minimization is exactly the same as the usual regression so it may be of interest in other cases. I'm pretty sure this has been done before but I do not have any references. – P. Quinton Jan 9 at 6:28