Truncated multivariate normal distribution under linear and nonlinear constraints

Generating truncated multivariate normal distributions is widely used in Bayesian constrained statistical modeling. This technique is applied in various fields, including ecology, economics, physics, computer science, biology, geosciences, and machine learning. In this paper, we extend the approach proposed by Ray et al. (2020) and further developed in Souris et al. (2019). Their main idea is to incorporate a <i>smooth relaxation</i> of the complex constraints appearing in the constrained density function into the likelihood and to employ a highly efficient Markov Chain Monte Carlo (MCMC) sampler. Our main contribution is the extension of this approach to address general linear and nonlinear inequality constraints, thereby enhancing its applicability to a wider range of problems. In light of this, we propose updating the approximation parameter in the likelihood at each MCMC iteration to enhance stability and ensure convergence of the algorithm. The flexibility, efficiency, and accuracy of the proposed approach are demonstrated through several numerical examples and a real-world application.