[Paper Review] Nested sampling on non-trivial geometries

Kamran Javid|arXiv (Cornell University)|May 22, 2019

Pulsars and Gravitational Waves Research13 references4 citations

TL;DR

This paper introduces a geometric nested sampling algorithm that enhances sampling efficiency on manifolds like circles, tori, and spheres by wrapping proposal distributions or transforming to Cartesian coordinates, preventing automatic rejection of points outside the domain. It outperforms MultiNest in recovering multimodal posteriors on high-dimensional toroidal and spherical geometries, especially with low livepoint counts.

ABSTRACT

Metropolis nested sampling evolves a Markov chain from a current livepoint and accepts new points along the chain according to a version of the Metropolis acceptance ratio modified to satisfy the likelihood constraint, characteristic of nested sampling algorithms. The geometric nested sampling algorithm we present here is a based on the Metropolis method, but treats parameters as though they represent points on certain geometric objects, namely circles, tori and spheres. For parameters which represent points on a circle or torus, the trial distribution is `wrapped' around the domain of the posterior distribution such that samples cannot be rejected automatically when evaluating the Metropolis ratio due to being outside the sampling domain. Furthermore, this enhances the mobility of the sampler. For parameters which represent coordinates on the surface of a sphere, the algorithm transforms the parameters into a Cartesian coordinate system before sampling which again makes sure no samples are automatically rejected, and provides a physically intutive way of the sampling the parameter space. We apply the geometric nested sampler to two types of toy model which include circular, toroidal and spherical parameters. We find that the geometric nested sampler generally outperforms extsc{MultiNest} in both cases. \\ %We also apply the algorithm to a gravitational wave detection model which includes circular and spherical parameters, and find that the geometric nested sampler and extsc{MultiNest} appear to perform equally well as one another. Our implementation of the algorithm can be found at \url{https://github.com/SuperKam91/nested_sampling}.

Motivation & Objective

• To address the inefficiency of standard nested sampling in sampling distributions defined on non-trivial manifolds such as circles, tori, and spheres.
• To improve sampler mobility and reduce automatic rejection of proposals due to domain boundary effects in constrained parameter spaces.
• To develop a geometrically informed variant of the Metropolis nested sampling algorithm that respects the intrinsic topology of circular, toroidal, and spherical parameters.
• To evaluate performance on toy models with circular, toroidal, and spherical symmetries, particularly in high-dimensional multimodal posteriors.
• To demonstrate that geometric nested sampling can outperform standard algorithms like MultiNest in recovering complex, symmetric modes on manifolds.

Proposed method

• For circular and toroidal parameters, the algorithm wraps the proposal distribution around the periodic domain, ensuring all proposed points remain within the valid sampling space.
• This wrapping prevents automatic rejection of proposals due to domain boundaries, enhancing mobility, especially near edges or discontinuities.
• For spherical parameters, the algorithm maps angular coordinates (e.g., polar and azimuthal angles) to Cartesian coordinates in Euclidean space for sampling.
• The transformation to Cartesian space ensures no proposals are rejected due to being outside the spherical surface domain.
• The algorithm uses a Metropolis-style acceptance ratio modified to respect the likelihood constraint, while maintaining geometric consistency in the parameter space.
• The method is implemented in Python 2.7 and applied to toy models including von Mises distributions on circles, 6-tori, and multiple Kent-distributed flowers on spheres.

Experimental results

Research questions

• RQ1 Can a nested sampling algorithm that respects the geometric structure of circular, toroidal, and spherical parameters improve sampling efficiency and mobility?
• RQ2 How does geometric nested sampling perform compared to MultiNest in recovering multimodal posteriors on high-dimensional toroidal and spherical manifolds?
• RQ3 Does the use of wrapped proposals or Cartesian transformation reduce the number of rejected samples and improve convergence in constrained parameter spaces?
• RQ4 To what extent does geometric nested sampling outperform MultiNest in low livepoint regimes on complex, symmetric distributions?
• RQ5 Can geometric nested sampling better preserve the symmetry and shape of multimodal features (e.g., 'petals' on a sphere) than standard nested sampling methods?

Key findings

• On a circular toy model (von Mises distribution), geometric nested sampling (MG) and MultiNest (MN) perform comparably due to low dimensionality.
• On a 6-torus model with 60 quarter-peaks, MG with 50 livepoints recovers all peaks well, while MN misses three entirely and poorly recovers 14.
• With 500 livepoints, MN improves but still underperforms MG, which maintains consistent performance, indicating MG’s robustness to low livepoint counts.
• On a 12-dimensional problem with six spheres each having eight symmetric petals, both algorithms fail to recover the true distribution perfectly, but MG better preserves the symmetry and shape of the modes.
• The geometric nested sampler’s performance advantage arises from treating the problem as unimodal in the geometric space, avoiding the clustering pitfalls of MultiNest in high-dimensional, symmetric manifolds.
• The implementation is available at https://github.com/SuperKam91/nested_sampling, though it is not optimized for performance compared to FORTRAN-based tools like MultiNest or POLYCHORD.