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

Understand

Monte Carlo algorithms often aim to draw from a distribution $\pi$ by simulating a Markov chain with transition kernel $P$ such that $\pi$ is invariant under $P$.

• However, there are many situations for which it is impractical or impossible to draw from the transition kernel $P$.
• For instance, this is the case with massive datasets, where is it prohibitively expensive to calculate the likelihood and is also the case for intractable likelihood models arising from, for example, Gibbs random fields, such as those found in spatial statistics and network analysis.
• A natural approach in these cases is to replace $P$ by an approximation $\hat{P}$.