# Adding an independent variable does not increase conditional information

Given $$P(X, Y, \hat{Y})$$ discrete with $$\hat{Y}$$ independent of both $$X$$ and $$Y$$, one would thus expect that the following relationship holds $$\max_{f}I(X;Y,\hat{Y} \mid f(Y,\hat{Y})) = \max_{f_1, f_2}I(X;Y,\hat{Y} \mid f_1(Y) f_2(\hat{Y})) = \max_{f_1}I(X;Y \mid f_1(Y))$$ where $$f, f_1, f_2$$ are deterministic, $$f$$ is non injective and either $$f_1$$ or $$f_2$$ is also non-injective functions. Is it the case?

• Can you recall what the notation $I(U ; V | W)$ means? Dec 3, 2022 at 8:53
• Conditional mutual information Dec 3, 2022 at 16:51

Observe that for any $$f(y, \hat{y})$$ we have $$I(X;Y,\hat{Y}|f(Y,\hat{Y})) \leq I(X;Y,\hat{Y},f(Y,\hat{Y})) = I(X;Y,\hat{Y}) = I(X;Y)$$. Equality is achieved by taking $$f$$ to be a constant, hence this is the maximum value.
Similarly, for any $$f_1(y)$$ we have $$I(X;Y|f_1(Y)) \leq I(X;Y)$$. Equality is achieved by taking $$f_1$$ to be a constant, hence this is the maximum value.