Strong Data Processing Inequalities and <inline-formula> <tex-math notation="LaTeX">$\Phi $ </tex-math> </inline-formula>-Sobolev Inequalities for Discrete Channels

Abstract

The noisiness of a channel can be measured by comparing suitable functionals of the input and output distributions. For instance, the worst case ratio of output relative entropy to input relative entropy for all possible pairs of input distributions is bounded from above by unity, by the data processing theorem. However, for a fixed reference input distribution, this quantity may be strictly smaller than one, giving the so-called strong data processing inequalities (SDPIs). The same considerations apply to an arbitrary $\Phi $ -divergence. This paper presents a systematic study of optimal constants in the SDPIs for discrete channels, including their variational characterizations, upper and lower bounds, structural results for channels on product probability spaces, and the relationship between the SDPIs and the so-called $\Phi $ -Sobolev inequalities (another class of inequalities that can be used to quantify the noisiness of a channel by controlling entropy-like functionals of the input distribution by suitable measures of input–output correlation). Several applications to information theory, discrete probability, and statistical physics are discussed.