The noisiness of a discrete channel can be measured by comparing suitable 
functionals of the input and output distributions. For instance, if we fix a
reference input distribution, then the worst-case ratio of output relative
entropy to input relative entropy for any other input distribution is bounded by
one, 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). In this talk, I will show that the
problem of determining both the best constant in an SDPI and any input
distributions that achieve it can be addressed using so-called logarithmic
Sobolev inequalities, which relate input relative entropy to certain measures of
input-output correlation.