Detailed Proofs of Lemmas, Theorems, and Corollaries

Here, η(b, f) = 2αb = 2βf is the total probability of cross-space transition, φ = min{Φ∗(QBF ),Φ∗(QFB)}. Corollary 1. The joint chain P+ is ergodic when the collapsed chains (QBF and QFB) are both ergodic. Lemma 3. Let P be a reversible transition matrix over X, such that P(x, x) ≥ ξ > 0 for each x ∈ X then its smallest eigenvalue λn has λn ≥ 2ξ − 1. Theorem 4. The hierarchically bridging Markov chain with bk < 1 for k = 0, . . . ,K−1, and fk < 1 for k = 1, . . . ,K is ergodic. If we write the equilibrium distribution in form of (αμ0, β1μ1, . . . , βKμK), then (S1) μ0 equals the target distribution μ; (S2) for each k ≥ 1, and y ∈ Yk, μk(y) is proportional to the total probability of its descendant target states (the target states derived by filling all its placeholders); (S3) α, the probability of being at the target level, is given by α−1 = 1 + ∑K k=1(b0 · · · bk−1)/(f1 · · · fk). Corollary 2. If bk/fk+1 ≤ κ < 1 for each k = 1, . . . ,K, then α > 1− κ.