Weighted information and entropy rates

Abstract

The weighted entropy $H^{\rm w}_\phi (X)=H^{\rm w}_\phi (f)$ of a random variable $X$ with values $x$ and a probability-mass/density function $f$ is defined as the mean value ${\mathbb E} I^{\rm w}_\phi(X)$ of the weighted information $I^{\rm w}_\phi (x)=-\phi (x)\log\,f(x)$. Here $x\mapsto\phi (x)\in{\mathbb R}$ is a given weight function (WF) indicating a 'value' of outcome $x$. For an $n$-component random vector ${\mathbf{X}}_0^{n-1}=(X_0,\ldots ,X_{n-1})$ produced by a random process ${\mathbf{X}}=(X_i,i\in{\mathbb Z})$, the weighted information $I^{\rm w}_{\phi_n}({\mathbf x}_0^{n-1})$ and weighted entropy $H^{\rm w}_{\phi_n}({\mathbf{X}}_0^{n-1})$ are defined similarly, with an WF $\phi_n({\mathbf x}_0^{n-1})$. Two types of WFs $\phi_n$ are considered, based on additive and a multiplicative forms ($\phi_n({\mathbf x}_0^{n-1})=\sum\limits_{i=0}^{n-1}{\varphi} (x_i)$ and $\phi_n({\mathbf x}_0^{n-1})=\prod\limits_{i=0}^{n-1}{\varphi} (x_i)$, respectively). The focus is upon ${\it rates}$ of the weighted entropy and information, regarded as parameters related to ${\mathbf{X}}$. We show that, in the context of ergodicity, a natural scale for an asymptotically additive/multiplicative WF is $\frac{1}{n^2}H^{\rm w}_{\phi_n}({\mathbf{X}}_0^{n-1})$ and $\frac{1}{n}\log\;H^{\rm w}_{\phi_n}({\mathbf{X}}_0^{n-1})$, respectively. This gives rise to ${\it primary}$ ${\it rates}$. The next-order terms can also be identified, leading to ${\it secondary}$ ${\it rates}$. We also consider emerging generalisations of the Shannon-McMillan-Breiman theorem.