Optimal Scaling of the Random Walk Metropolis: General Criteria for the 0.234 Acceptance Rule

The RWM algorithm creates a Markov chain with stationary distribution π(x), and hence (eventually) a dependent sample with distribution ≈ π(x). Given the current value X ∈ R, a new value X∗ = X + Y is proposed by sampling a “jump”, Y, from from a pre-specified Lebesgue density q (y|x) = λ−d r (y/λ) , where r (−y) = r (y); the proposal is then accepted with probability α(x,y) = 1 ∧ (π(x∗)/π(x)). If the proposed value is accepted it becomes the next current value (X′← X∗), otherwise the current value is left unchanged (X′← X).