# Convex sets **STATS 606:** Computation and Optimization Methods in Statistics University of Michigan
including slides from EE 364Aby Stephen
Boyd
and Lieven
Vandenberghe
## Ex: conditional probability set $P\in\cP\subset[0,1]^{n\times n}$ is the (joint) distribution of $(X,Y)$: $$P_{i,j} = \Pr\\{X=x_i,Y=y_j\\}.$$ Let $Q(P)\in[0,1]^{n\times n}$ be the conditional distribution of $X\mid Y=y$ induced by $P$: $$\big[Q(P)\big]_{i,j} = \Pr\\{X=x_i\mid Y=y_j\\}.$$
## Ex: conditional probability set 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.