The Riemann Sampler: A fast, accurate, and efficient alternative to MCMC

Music City Center 

Many Bayesian analyses depend on posterior samples obtained via Markov Chain Monte Carlo (MCMC) methods such as Metropolis, Slice, and Hamiltonian samplers. While effective, these samplers require assessing autocorrelation, burn in, and convergence, all of which can affect accuracy, efficiency, and computation time. We propose a novel Riemann sampler that discretizes the posterior kernel into equally spaced regions (rectangle in 1D, hyperrectangles in higher dimensions), weights them proportionally, and draws independent samples within each region. Unlike MCMC, this method does not generate a Markov chain. Instead, it produces an independent sample that converges to the true posterior as the region width approaches zero. This approach to sampling is fully efficient, requires no burn-in or convergence checks, is parallelizable, and can be computationally faster than traditional samplers. We compare its accuracy, efficiency, and run time to Metropolis, Slice, Hamiltonian, and No-U-Turn for low dimensional settings.

MCMC Algorithm

Bayesian Methods

Sampling Algorithm

Prior Distribution and Likelihood

Posterior Distribution

Kernel Density Estimation 

Section on Bayesian Statistical Science