On Dependence Measures Based on $f$-Divergence and $\Phi$-Entropy

arXiv:2503.03754v3 Announce Type: replace Abstract: We study dependence measures induced by $\Phi$-divergence (also known as $ f$-divergence in the literature) for finite random variables. We define conditional $\Phi$-mutual information through an exact chain rule and develop its structural properties and variational representations. But in general it lacks conditional symmetry and deterministic self-reduction. To tackle this, we introduce max-$\Phi$-mutual information $\wIPhi$, a variational envelope obtained by optimizing over $\IPhi$ quantities containing auxiliary random variables, which restores both properties and satisfies a subadditive chain rule. Both quantities reduce to Shannon's mutual information when $\Phi(t)=t\log t$. For smooth convex generators with $\Phi(0)