Suppose I have three identically-distributed homogeneous continuous-time discrete state space Markov chains $X_1(t), X_2(t), X_3(t)$, $t\geq 0$. They evolve independently but share a common random variable $X_0$ as an initial condition. I let $$X_1=X_1(t_1), \ \ \ X_2=X_2(t_2), \ \ \ X_3=X_3(t_3)$$ for some times $t_1, t_2, t_3\geq 0$.

I want to show that $$ I(X_1; X_2; X_3) \geq 0 $$ where $I$ is the multivariate mutual information (or information interaction) $$ I(A,B,C)= H(A,B,C) -H(A,B) - H(B,C) - H(A,C) + H(A) + H(B) + H(C) $$ where $H$ is the usual Shannon entropy.

## Background/Motivation

There are well-known situations where $I(A;B;C)<0$, a famous one being if $A$ and $B$ are independent random variables, each $\pm 1$ with probability $1/2$, and $C=AB$. But I conjecture that in the case I have described above $I(X_1;X_2;X_3)\geq 0$. I believe that the Markov chains being continuous-time and homogeneous is essential.

The more general motivation is that I want to find very general situations where multivariate mutual information is non-negative. (One well-known example is if $A,B,C$ form a Markov chain.)