  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 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 SDPIs for discrete channels, including their variational characterizations,
upper and lower bounds, structural results for channels on product probability
spaces, and the relationship between SDPIs and 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.
