Comment on 'Bayesian evidence: can we beat MultiNest using traditional MCMC methods', by Rutger van Haasteren (arXiv:0911.2150)

In arXiv:0911.2150, Rutger van Haasteren seeks to criticize the nested sampling algorithm for Bayesian data analysis in general and its MultiNest implementation in particular. He introduces a new method for evidence evaluation based on the idea of Voronoi tessellation and requiring samples from the posterior distribution obtained through MCMC based methods. He compares its accuracy and efficiency with MultiNest, concluding that it outperforms MultiNest in several cases. This comparison is completely unfair since the proposed method can not perform the complete Bayesian data analysis including posterior exploration and evidence evaluation on its own while MultiNest allows one to perform Bayesian data analysis end to end. Furthermore, their criticism of nested sampling (and in turn MultiNest) is based on a few conceptual misunderstandings of the algorithm. Here we seek to set the record straight.

💡 Research Summary

In the pre‑print arXiv:0911.2150, Rutger van Haasteren criticises the nested‑sampling algorithm—particularly its MultiNest implementation—by proposing a new evidence‑estimation technique based on Voronoi tessellation of posterior samples obtained from conventional MCMC runs. He argues that, for a set of test problems, his Voronoi‑based method achieves comparable accuracy to MultiNest while requiring far fewer likelihood evaluations, and therefore “beats” MultiNest in efficiency.

The present comment sets the record straight by exposing several fundamental flaws in that claim. First, the Voronoi approach relies on an accurate estimate of each cell’s volume in parameter space. While this is tractable in two or three dimensions, the volume calculation becomes exponentially unstable as the dimensionality grows. In realistic astrophysical or cosmological applications the number of parameters often exceeds a dozen, and the geometry of Voronoi cells in such high‑dimensional spaces is extremely irregular. With a finite number of MCMC draws the cell‑volume estimates are noisy, leading to systematic bias in the evidence integral.

Second, van Haasteren assumes that an MCMC chain already provides a faithful representation of the full posterior, including low‑probability regions that contribute substantially to the marginal likelihood. In practice, standard MCMC algorithms concentrate on high‑density modes and rarely explore the tails or the space between widely separated modes. Since the Bayesian evidence is an integral over the entire prior volume, neglecting these low‑density regions yields an underestimate of the evidence. Nested sampling, by construction, systematically shrinks the prior volume while keeping track of the likelihood at each shrinkage step, guaranteeing that even the far‑flung tails are accounted for.

Third, the empirical comparison in the original paper is limited to very simple, low‑dimensional test cases (mostly single‑Gaussian or two‑dimensional mixtures). These toy problems are precisely the scenarios where a Voronoi approximation can perform adequately. Real‑world inference problems, however, involve multimodal posteriors, strong parameter degeneracies, and non‑Gaussian tails. MultiNest is specifically engineered to handle such complexities: it clusters live points, fits ellipsoidal bounds to each cluster, and adaptively samples within these bounds. This strategy ensures efficient exploration of all modes and a reliable evidence estimate, while simultaneously delivering posterior samples for parameter inference.

Moreover, van Haasteren’s critique conflates “evidence calculation” with “full Bayesian analysis”. His method only addresses the final integration step, assuming that posterior sampling has already been completed elsewhere. MultiNest, on the other hand, provides an end‑to‑end solution: it generates posterior samples, computes the evidence, and supplies diagnostic information (e.g., information gain, effective sample size) without any additional post‑processing. When the total computational cost of a complete analysis is considered, the apparent advantage of the Voronoi method disappears.

In summary, the alleged superiority of the Voronoi‑based estimator stems from (1) an unrealistic assumption of perfect high‑dimensional tessellation, (2) the neglect of low‑probability regions that dominate the evidence integral, and (3) an unfair benchmarking that ignores the full workflow provided by MultiNest. Nested sampling remains a robust, mathematically grounded technique that simultaneously delivers accurate evidence values and high‑quality posterior samples, even in challenging, high‑dimensional settings. The present comment therefore re‑affirms MultiNest’s status as a leading tool for Bayesian model comparison and parameter estimation.