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Monte Carlo algorithms often aim to draw from a distribution $\pi$ bysimulating a Markov chain with transition kernel $P$ such that $\pi$ isinvariant under $P$. However, there are many situations for which it isimpractical or impossible to draw from the transition kernel $P$. For instance,this is the case with massive datasets, where is it prohibitively expensive tocalculate the likelihood and is also the case for intractable likelihood modelsarising from, for example, Gibbs random fields, such as those found in spatialstatistics and network analysis. A natural approach in these cases is toreplace $P$ by an approximation $\hat{P}$. Using theory from the stability ofMarkov chains we explore a variety of situations where it is possible toquantify how 'close' the chain given by the transition kernel $\hat{P}$ is tothe chain given by $P$. We apply these results to several examples from spatialstatistics and network analysis.

Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels