The paper "ENSEMBLE SAMPLERS WITH AFFINE INVARIANCE" by Jonathan Goodman and Jonathan Weare introduces a family of Markov chain Monte Carlo (MCMC) methods that are affine invariant, meaning their performance is not affected by affine transformations of the space. These algorithms are designed to be easy to construct and require minimal additional computational overhead, making them particularly useful for sampling distributions that are badly scaled or highly skewed. The authors motivate their work by noting that standard MCMC methods often require careful tuning of parameters to handle such distributions effectively. They propose a general framework for ensemble MCMC samplers, where multiple walkers move together, and the performance of the sampler is independent of the aspect ratio in highly anisotropic distributions. This property is achieved by using affine transformations that preserve the target density. The paper details two specific affine invariant methods: the stretch move and the walk move. The stretch move involves moving a walker using a proposal that depends on the relative positions of other walkers, while the walk move uses a proposal based on the empirical distribution of a subset of walkers. Both methods are designed to be symmetric and detailed balance preserving. The authors also discuss the evaluation of ensemble sampling methods, comparing them to standard single-particle MCMC methods using integrated autocorrelation times as a criterion. Numerical tests on the Rosenbrock density and a 101-dimensional distribution highlight the advantages of the affine invariant methods, showing significant improvements in performance over standard methods. Finally, the paper provides software documentation and test programs, emphasizing the ease of use and computational efficiency of the proposed methods. The authors conclude that their ensemble MCMC schemes offer uniform effectiveness on problems that can be rescaled by affine transformations to be well-conditioned, with negligible computational overhead compared to standard single-particle schemes.The paper "ENSEMBLE SAMPLERS WITH AFFINE INVARIANCE" by Jonathan Goodman and Jonathan Weare introduces a family of Markov chain Monte Carlo (MCMC) methods that are affine invariant, meaning their performance is not affected by affine transformations of the space. These algorithms are designed to be easy to construct and require minimal additional computational overhead, making them particularly useful for sampling distributions that are badly scaled or highly skewed. The authors motivate their work by noting that standard MCMC methods often require careful tuning of parameters to handle such distributions effectively. They propose a general framework for ensemble MCMC samplers, where multiple walkers move together, and the performance of the sampler is independent of the aspect ratio in highly anisotropic distributions. This property is achieved by using affine transformations that preserve the target density. The paper details two specific affine invariant methods: the stretch move and the walk move. The stretch move involves moving a walker using a proposal that depends on the relative positions of other walkers, while the walk move uses a proposal based on the empirical distribution of a subset of walkers. Both methods are designed to be symmetric and detailed balance preserving. The authors also discuss the evaluation of ensemble sampling methods, comparing them to standard single-particle MCMC methods using integrated autocorrelation times as a criterion. Numerical tests on the Rosenbrock density and a 101-dimensional distribution highlight the advantages of the affine invariant methods, showing significant improvements in performance over standard methods. Finally, the paper provides software documentation and test programs, emphasizing the ease of use and computational efficiency of the proposed methods. The authors conclude that their ensemble MCMC schemes offer uniform effectiveness on problems that can be rescaled by affine transformations to be well-conditioned, with negligible computational overhead compared to standard single-particle schemes.