Algorithm 1.

The Multiresolution Sampler

Let i = 0. Start from the k-resolution chain. Collect samples from fk (θ, Y{k} | Y) using any combination of local updating algorithms. Discard some initial samples as burn-in, and retain the remaining samples as the empirical distribution of (Y{k}, θ) from fk (θ, Y{k} | Y). Let i ← i + 1. Start the (k+i)-resolution chain. Initialize the chain to a state ( ${\mathit{Y}}_{\mathit{old}}^{\{k+i\}}$, θold). With probability 1 − p, perform a local update step to generate a new sample from fk+i (θ, Y{k+i} | Y), using any combination of local updates. With probability p, perform a cross-resolution move: Randomly select a state ( ${\mathit{Y}}_{\mathit{trial}}^{\{k+i-1\}}$, θtrial) from the empirical distribution of the (k+i-1)-chain. Augment ( ${\mathit{Y}}_{\mathit{trial}}^{\{k+i-1\}}$, θtrial) to ( ${\mathit{Y}}_{\mathit{trial}}^{\{k+i\}}$, θtrial) by generating additional missing data values. With a Metropolis-Hasting type probability r, accept ( ${\mathit{Y}}_{\mathit{trial}}^{\{k+i\}}$, θtrial) as the next sample in the chain; with probability 1 − r, keep the previous values of ( ${\mathit{Y}}_{\mathit{old}}^{\{k+i\}}$, θold) as the next sample in the chain. Rename the most recent draw as ( ${\mathit{Y}}_{\mathit{old}}^{\{k+i\}}$, θold), and repeat from Step 4 until a desired number of samples are achieved (typically determined in part by monitoring the chain for sufficient evidence of convergence). Discard some initial samples of the chain as burn-in, and retain the remaining samples to form an empirical distribution of (Y{k+i}, θ) from fk+i (θ, Y{k+i} | Y). If a finer approximation to the SDE is desired, repeat from Step 3.