STATS 606: Computation and Optimization Methods in Statistics
University of Michigan
including slides by Stephen Boyd and Lieven Vandenberghe
$P\in\cP\subset[0,1]^{n\times n}$ encode the (joint) distribution of $(X,Y)$:
$$P_{i,j} = \Pr\{X=x_i,Y=y_j\}.$$Let $Q(P)\in[0,1]^{n\times n}$ encode the conditional distribution of $X\mid Y$ (induced by $P$):
$$\big[Q(P)\big]_{i,j} = \Pr\{X=x_i\mid Y=y_j\}.$$The cols of $Q(P)$ are linear fractional functions of the cols of $P$:
$$\big[Q(P)\big]_{\cdot,j} = \frac{P_{\cdot,j}}{P_{\cdot,j}^\top 1_n}.$$Thus $Q(\cP)$ is convex whenever $\cP$ is convex!