Fully Adaptive Gaussian Mixture Metropolis-Hastings Algorithm

Markov Chain Monte Carlo methods are widely used in signal processing and communications for statistical inference and stochastic optimization. In this work, we introduce an efficient adaptive Metropolis-Hastings algorithm to draw samples from generic multi-modal and multi-dimensional target distributions. The proposal density is a mixture of Gaussian densities with all parameters (weights, mean vectors and covariance matrices) updated using all the previously generated samples applying simple recursive rules. Numerical results for the one and two-dimensional cases are provided.

💡 Research Summary

The paper introduces a novel adaptive Metropolis‑Hastings (MH) algorithm called Fully Adaptive Gaussian Mixture Metropolis‑Hastings (FA‑GMM‑MH) designed to efficiently sample from generic multimodal and high‑dimensional target distributions, which are common in signal processing and communications applications such as Bayesian inference, channel estimation, and stochastic optimization. Traditional MH methods rely on a fixed proposal distribution; if the proposal poorly matches the target, the Markov chain mixes slowly, especially when the target exhibits several separated modes. Existing adaptive MH schemes typically adjust a single covariance matrix or a scaling factor, but they still struggle to traverse between distant modes.

FA‑GMM‑MH addresses this limitation by employing a mixture of K Gaussian components as the proposal density:

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