This exercise explores the stationary
distribution for Gibbs sampling methods.
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The convex composition
$[\alpha, q_1; 1-\alpha, q_2]$ of$q_1$ and$q_2$ is a transition probability distribution that first chooses one of$q_1$ and$q_2$ with probabilities$\alpha$ and$1-\alpha$ , respectively, and then applies whichever is chosen. Prove that if$q_1$ and$q_2$ are in detailed balance with$\pi$ , then their convex composition is also in detailed balance with$\pi$ . (Note: this result justifies a variant of GIBBS-ASK in which variables are chosen at random rather than sampled in a fixed sequence.) -
Prove that if each of
$q_1$ and$q_2$ has$\pi$ as its stationary distribution, then the sequential composition$q {{,=,}}q_1 \circ q_2$ also has$\pi$ as its stationary distribution.