Recent developments in Bayesian inference for time series
The pseudo-marginal algorithm is a variant of the Metropolis Hastings scheme which samples asymptotically from a target probability density when we are only able to estimate unbiasedly an unnormalized version of it. It has found numerous applications in statistics and econometrics as there are many scenarios where the likelihood function is intractable but can be estimated unbiasedly using Monte Carlo samples. Several recent contributions will be discussed which optimise the trade off between computational complexity and statistical efficiency. A modification of the pseudo-marginal algorithm, termed the correlated pseudo-marginal algorithm, is introduced. Guidelines are provided for the optimal settings of this algorithm. The computational gains of the new algorithm are demonstrated by examining large time series, including the estimation of continuous time volatility models