Conditional differential entropy of sum of Gaussians

Question

Is it possible to give an expression for the conditional differential entropy $h(A+B\mid C+D),$ where $A,B,C,D$ are normally distributed with known standard deviations $σ_A,\ldots,σ_D$ and where all but $A$ and $C$ are conditinally independent? For $A$ and $C,$ the Pearson’s correlation value $C$ is known. The resulting expression should depend on $σ_A,\ldots,σ_D$ and $C.$ Thank you very much for your help!

Answer 1

$\newcommand\si\sigma$The question is stated very poorly. In particular, it is unclear

• how you define the conditional differential entropy
• if $A, B, C, D$ are jointly normal
• if they are zero-mean
• what you mean by "all but A and C are conditinally independent"

Assuming that $A, B, C, D$ are jointly normal and zero-mean, $A+B$ and $C+D$ will be jointly normal and zero-mean, and then simple calculations yield $$h(A+B|C+D)=\ln\si_{A+B}+\frac12(1+\ln(2\pi(1-\rho^2))),$$ where $\si_{A+B}$ is the standard deviation of $A+B$, $\rho$ is the correlation coefficient of $A+B$ and $C+D$,
$$h(X|Y):=-\int_{\mathbb R^2} f_{X,Y}(x,y)\ln f_{X|Y}(x|y)\,dx\,dy,$$ $f_{X,Y}$ is the joint pdf of random variables $X$ and $Y$, and $f_{X|Y}$ is the conditional pdf of $X$ given $Y$.

In turn, it is straightforward to express $\si_{A+B}$ and $\rho$ in terms of the variances and covariances of $A, B, C, D$.