Analytic Convergence Rates and Parameterization Issues for the Gibbs Sampler Applied to State Space Models

In this paper we obtain a closed form expression for the convergence rate of the Gibbs sampler applied to the unobserved states of a first‐order autoregression plus noise model. The rate is expressed in terms of the parameters of the model, which are regarded as fixed. For the case where the unconditional mean of the states is a parameter of interest we provide evidence that a ‘centred’ parameterization of a state space model is preferable for the performance of the Gibbs sampler. These two results provide guidance when the Gaussianity or linearity of the state space form is lost. We illustrate this by examining the performance of a Markov chain Monte Carlo sampler for the stochastic volatility model.