Bayesian Time Series Models: Adaptive Markov chain Monte Carlo: theory and methods

In general, the transition probability P of the Markov chain depends on some tuning parameter θ defined on some space Θ which can be either finite dimensional or infinite dimensional. The success of the MCMC procedure depends crucially upon a proper choice of θ. To illustrate, consider the standard Metropolis-Hastings (MH) algorithm. For simplicity, we assume that π has a density also denoted by π with respect to the Lebesgue measure on X = R endowed with its Borel σ-field X . Given that the chain is at x, a candidate y is sampled from a proposal transition density q(x, ·) and is accepted with probability α(x, y) defined as

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