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Generate samples from multiple spanning ellipsoids

Source: R/multi_ellipsoid.R
multi_ellipsoid.Rd

[Experimental] Partitions the prior space into a set of ellipsoids whose union bounds the set of live points. Samples are created by randomly selecting an ellipsoid (weighted by their respective volumes), then using it to generate a random point as in unif_ellipsoid. Effective for multimodal posteriors where a single ellipsoid would be inefficient.

Usage

multi_ellipsoid(enlarge = 1.25, min_reduction = 0.7, allow_contact = TRUE)

Arguments

enlarge

Double, greater than or equal to 1. Factor by which to inflate the bounding ellipsoid's volume before sampling (see Details).

min_reduction

Double between 0 and 1. The minimum reduction in total volume required for an ellipsoid to be split in two. Lower values lead to more aggressive splitting.

allow_contact

Logical. Whether to allow ellipsoids to overlap.

Value

A list with class c("multi_ellipsoid", "ernest_lrps"). Use with ernest_sampler() to specify nested sampling behaviour.

Details

Nested likelihood contours for multimodal distributions are poorly represented by a single ellipsoid. This method fits multiple ellipsoids to better capture disconnected or elongated regions.

Ellipsoids are generated using the following procedure:

  1. A single ellipsoid is fit to the set of live points, with volume \(V\).

  2. The live points are clustered into two groups using k-means clustering.

  3. Ellipsoids are fit to each cluster.

  4. The split ellipsoids are accepted if all of the following conditions are met:

    • Both ellipsoids are non-degenerate

    • The combined volume of the split ellipsoids is less than \(min_{red.} * V\)

    • (If allow_contact is FALSE) the ellipsoids do not intersect.

  5. Steps 2–4 are repeated recursively on each new ellipsoid until no further splits are accepted, updating \(V\) to the volume of the currently split ellipsoid.

Ellipsoids

Ellipsoids are stored in the cache environment of the LRPS object. Ellipsoids are defined by their centre \(c\) and shape matrix \(A\). The set of points \(x\) contained within the ellipsoid is given by $$ x \in {\bf{R}}^n | (x-c) A (x-c)' \leq 1 $$

The volume of the ellipsoid is \(V = \mathrm{Vol}(S_n) \sqrt{\det(A^{-1})}\), where \(\mathrm{Vol}(S_n)\) is the volume of the unit hypersphere.

For sampling, we store the matrix \(A^{-1/2}\), the inverse of the positive-semidefinite square root of \(A\). The ellipsoid can equivalently be defined as the set of points $$x = A^{-1/2} y + c,$$ where \(y\) are points from the unit hypersphere.

For more on ellipsoids and their operations, see Algorithms for Ellipsoids by S.B. Pope, Cornell University Report FDA 08-01 (2008).

Status

This LRPS is experimental and has not been extensively validated across different nested sampling problems. You are encouraged to use it, but please exercise caution interpretting results and report any issues or unexpected behaviour.

References

Feroz, F., Hobson, M. P., Bridges, M. (2009) MULTINEST: An Efficient and Robust Bayesian Inference Tool for Cosmology and Particle Physics. Monthly Notices of the Royal Astronomical Society. 398(4), 1601–1614. doi:10.1111/j.1365-2966.2009.14548.x

For implementation, see: https://github.com/kbarbary/nestle/blob/master/runtests.py

See also

Other ernest_lrps: mini_balls(), no_underrun(), rwmh_cube(), slice_rectangle(), unif_cube(), unif_ellipsoid()

Examples

data(example_run)
lrps <- multi_ellipsoid(enlarge = 1.25, min_reduction = 0.5)

ernest_sampler(example_run$log_lik_fn, example_run$prior, sampler = lrps)
#> nested sampling specification <ernest_sampler>
#> • No. Points: 500
#> • LRPS Method: multi_ellipsoid
#> 
#> ernest LRPS method <multi_ellipsoid/ernest_lrps>
#> • Dimensions: 3
#> • No. Log-Lik Calls: 0
#> • No. Ellipsoids: 1
#> • Total Log Volume: 1.001
#> • Min Reduction: 0.5
#> • Allow Contact: TRUE
#> • Enlargement: 1.25