Sampling from manifold-restricted distributions using tangent bundle projections

Abstract

A common problem in Bayesian inference is the sampling of target probability distributions at sufficient resolution and accuracy to estimate the probability density, and to compute credible regions. Often by construction, many target distributions can be expressed as some higher-dimensional closed-form distribution with parametrically constrained variables, i.e., one that is restricted to a smooth submanifold of Euclidean space. I propose a derivative-based importance sampling framework for such distributions. A base set of n samples from the target distribution is used to map out the tangent bundle of the manifold, and to seed nm additional points that are projected onto the tangent bundle and weighted appropriately. The method essentially acts as an upsampling complement to any standard algorithm. It is designed for the efficient production of approximate high-resolution histograms from manifold-restricted Gaussian distributions, and can provide large computational savings when sampling directly from the target distribution is expensive.

Reduced-Order Modeling with Artificial Neurons for Gravitational-Wave Inference

“…5. Using the derivative-based sampling method of [52], a "base chain" (left; red points) is formed from the first 5,000 samples of the 10 5 -sample chain in Fig. 4.…”

“…A different type of derivative-based method for local sampling and density estimation exploits the constrained-Gaussian form of a target density such as Eq. (4) to produce approximate samples efficiently [52]. Applying this scheme to the above example gives the estimated histogram in Fig.…”

An Efficient Scheme for Sampling in Constrained Domains

“…In [9] and [16], Hamiltonian Monte Carlo (HMC) samplers were developed for implementation on manifolds embedded within Euclidean spaces with demonstrations on unit hyperspeheres. Additionally, a derivative based importance sampler for smooth Euclidean submanifolds was presented in [10].…”