This exercise explores the stationary distribution for Gibbs sampling methods. a. The convex composition [, q 1

This exercise explores the stationary distribution for Gibbs sampling methods.

a. The convex composition [α, q₁; 1 − α, q₂] of q₁ and q₂ is a transition probability distribution that first chooses one of q₁ and q₂ with probabilities α and 1 − α, respectively, and then applies whichever is chosen. Prove that if q₁ and q₂ are in detailed balance with π, then their convex composition is also in detailed balance with π. This result justifies a variant of GIBBS-ASK in which variables are chosen at random rather than sampled in a fixed sequence.)

b. Prove that if each of q₁ and q₂ has π as its stationary distribution, then the sequential composition q = q₁ ◦ q₂ also has π as its stationary distribution.