$(2$pt$)$ The relative entropy between two probability distributions $x,yin{R}^{n_{++}}$is defined as

${\displaystyle \underset{k=1}{\overset{n}{}}}{x}_{k}log(\frac{x_k}{y_k})$

which is a convex function, jointly in $x$ and $y.$

Given a probability distribution $y=(y_1,$dots,$y_n),$ we want to find a distribution $x=(x_1,$dots,$x_n)$ that minimizes the relative entropy with $y,$ subject to equality constraints on $x$ :

$\backslash$table$[[mi{n}_{xin{R}^n},{\displaystyle \underset{k=1}{\overset{n}{}}}{x}_{k}log(\frac{x_k}{y_k})$