The paper introduces a powerful and flexible MCMC algorithm called the pseudo-marginal approach for efficient Monte Carlo computations. This method builds on a pseudo-marginal technique, originally introduced in genetics, to approximate the marginal distribution of a target variable by using importance sampling estimates. The algorithm allows for the use of approximate densities in the acceptance probability of the Metropolis-Hastings (MH) update, while still maintaining the same marginal stationary distribution as the idealized method. Theoretical results are provided to describe the convergence properties of the proposed method, and numerical examples are given to illustrate its empirical characteristics. The paper discusses the importance of choosing an appropriate importance sampling distribution to ensure that the algorithm is uniformly ergodic. It shows that if the importance weights are unbounded for "too many" values of the target variable, the algorithm cannot be geometrically ergodic. The paper also compares the pseudo-marginal approach with other methods, such as the Monte Carlo within Metropolis (MCWM) and grouped independence MH (GIMH) algorithms, and highlights the advantages of the pseudo-marginal approach in terms of computational efficiency and accuracy. Theoretical results are presented to show that under certain conditions, the pseudo-marginal approach inherits the convergence properties of the idealized method. The paper also discusses the uniform and geometric ergodicity of the exact algorithms, showing that the pseudo-marginal approach can be uniformly ergodic under certain conditions. The paper concludes with a discussion of the implications of these results for the design of efficient MCMC algorithms for model selection and other applications.The paper introduces a powerful and flexible MCMC algorithm called the pseudo-marginal approach for efficient Monte Carlo computations. This method builds on a pseudo-marginal technique, originally introduced in genetics, to approximate the marginal distribution of a target variable by using importance sampling estimates. The algorithm allows for the use of approximate densities in the acceptance probability of the Metropolis-Hastings (MH) update, while still maintaining the same marginal stationary distribution as the idealized method. Theoretical results are provided to describe the convergence properties of the proposed method, and numerical examples are given to illustrate its empirical characteristics. The paper discusses the importance of choosing an appropriate importance sampling distribution to ensure that the algorithm is uniformly ergodic. It shows that if the importance weights are unbounded for "too many" values of the target variable, the algorithm cannot be geometrically ergodic. The paper also compares the pseudo-marginal approach with other methods, such as the Monte Carlo within Metropolis (MCWM) and grouped independence MH (GIMH) algorithms, and highlights the advantages of the pseudo-marginal approach in terms of computational efficiency and accuracy. Theoretical results are presented to show that under certain conditions, the pseudo-marginal approach inherits the convergence properties of the idealized method. The paper also discusses the uniform and geometric ergodicity of the exact algorithms, showing that the pseudo-marginal approach can be uniformly ergodic under certain conditions. The paper concludes with a discussion of the implications of these results for the design of efficient MCMC algorithms for model selection and other applications.