The paper introduces a powerful and flexible MCMC algorithm for stochastic simulation, building on a pseudo-marginal method originally proposed in genetics. The method approximates the marginal distribution of a target density by using importance sampling estimates of the joint density. The authors provide theoretical results on the convergence properties of the proposed method and illustrate its empirical characteristics through numerical examples. They also compare it with a more straightforward Monte Carlo approximation to the marginal algorithm, highlighting the advantages of their approach. The paper discusses two variants of the pseudo-marginal approach: the Monte Carlo within Metropolis (MCWM) algorithm and the grouped independence Metropolis-Hastings (GIMH) algorithm. The MCWM algorithm independently updates auxiliary variables at each iteration, while the GIMH algorithm recycles these variables, leading to a Markov chain that targets the marginal distribution. The authors prove that under certain conditions, the GIMH algorithm converges to the marginal distribution, while the MCWM algorithm may not. They also establish finite horizon convergence properties and geometric and uniform convergence results for the GIMH algorithm under mild assumptions. The paper concludes with a discussion on the design of efficient reversible jump MCMC algorithms for model selection using the pseudo-marminal approach.The paper introduces a powerful and flexible MCMC algorithm for stochastic simulation, building on a pseudo-marginal method originally proposed in genetics. The method approximates the marginal distribution of a target density by using importance sampling estimates of the joint density. The authors provide theoretical results on the convergence properties of the proposed method and illustrate its empirical characteristics through numerical examples. They also compare it with a more straightforward Monte Carlo approximation to the marginal algorithm, highlighting the advantages of their approach. The paper discusses two variants of the pseudo-marginal approach: the Monte Carlo within Metropolis (MCWM) algorithm and the grouped independence Metropolis-Hastings (GIMH) algorithm. The MCWM algorithm independently updates auxiliary variables at each iteration, while the GIMH algorithm recycles these variables, leading to a Markov chain that targets the marginal distribution. The authors prove that under certain conditions, the GIMH algorithm converges to the marginal distribution, while the MCWM algorithm may not. They also establish finite horizon convergence properties and geometric and uniform convergence results for the GIMH algorithm under mild assumptions. The paper concludes with a discussion on the design of efficient reversible jump MCMC algorithms for model selection using the pseudo-marminal approach.