Large deviations for Independent Metropolis Hastings and Metropolis-adjusted Langevin algorithm

In this paper, we prove large deviation principles for the empirical measures associated with the Independent Metropolis Hastings (IMH) sampler and the Metropolis-adjusted Langevin Algorithm (MALA). These are the first large deviation results for empirical measures of Markov chains arising from specific Metropolis-Hastings methods on a continuous state space. Moreover, we show that the existing large deviation framework, that we developed in a previous work (Milinanni and Nyquist, 2024) does not cover the Random Walk Metropolis sampler, even in cases when the underlying Markov chain is geometrically ergodic.