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An Exploration of the Role of Principal Inertia Components in Information Theory

The principal inertia components of the joint distribution of two random variables $X$ and $Y$ are inherently connected to how an observation of $Y$ is statistically related to a hidden variable $X$. In this paper, we explore this connection within an information theoretic framework. We show that, under certain symmetry conditions, the principal inertia components play an important role in estimating one-bit functions of $X$, namely $f(X)$, given an observation of $Y$. In particular, the principal inertia components bear an interpretation as filter coefficients in the linear transformation of $p_{f(X)|X}$ into $p_{f(X)|Y}$. This interpretation naturally leads to the conjecture that the mutual information between $f(X)$ and $Y$ is maximized when all the principal inertia components have equal value. We also study the role of the principal inertia components in the Markov chain $B\rightarrow X\rightarrow Y\rightarrow \widehat{B}$, where $B$ and $\widehat{B}$ are binary random variables. We illustrate our results for the setting where $X$ and $Y$ are binary strings and $Y$ is the result of sending $X$ through an additive noise binary channel.