Exact Rejection Sampling for Non-Gaussian State Space Models

Abstract

Rejection sampling requires a proposal that dominates the target by a known constant, generally unavailable for non-Gaussian state space models. We construct such a proposal for the latent state path, yielding independent exact smoothing draws and an unbiased likelihood estimator whose relative variance is at most $1/p-1$ per draw at acceptance probability $p$. The method covers scalar states with affine Gaussian dynamics and log-concave observation densities, including multivariate observations. Transition twisting makes the log target-to-proposal ratio separable, and tangent-line twists make each term nonpositive, producing an attained, sharp dominating constant. With a companding node placement, the accumulated envelope error is $O(T/G^2)$ for a sample of length $T$ with $G$ nodes per date, so $G\propto \sqrt{T}$ keeps acceptance bounded away from zero; for stochastic volatility, the required conditions hold almost surely. A simpler mode-centered grid shows the same scaling empirically. At $T=2,000$, acceptance is $75\%$, versus roughly $10^{-16}$ for the Gaussian envelope.