![[Experimental]](figures/lifecycle-experimental.svg)
Uses the bounding ellipsoid of the live points to define the region of prior
space that contains new points. Effective for unimodal and roughly-Gaussian
posteriors.
Value
A list with class c("unif_ellipsoid", "ernest_lrps"). Use with
ernest_sampler() to specify nested sampling behaviour.
Details
Nested likelihood contours rarely form perfect ellipses, so sampling
from the spanning ellipsoid without enlargement may exclude valid regions.
This can bias proposals towards the ellipsoid centre and overestimate
evidence. Setting enlarge = 1 will produce a warning.
The covariance matrix of the points is used to estimate the ellipsoid's shape. In exceptional cases (e.g., perfect collinearity), this matrix may be singular. Should this occur, the covariance matrix is reconditioned by adjusting its eigenvalues. Should this also fail, the algorithm falls back to sampling from the circumscribed sphere bounding the unit hypercube.
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
Mukherjee, P., Parkinson, D., & Liddle, A. R. (2006). A Nested Sampling Algorithm for Cosmological Model Selection. The Astrophysical Journal, 638(2), L51. doi:10.1086/501068
See also
Other ernest_lrps:
mini_balls(),
multi_ellipsoid(),
no_underrun(),
rwmh_cube(),
slice_rectangle(),
unif_cube()
Examples
data(example_run)
lrps <- unif_ellipsoid(enlarge = 1.25)
ernest_sampler(example_run$log_lik_fn, example_run$prior, sampler = lrps)
#> nested sampling specification <ernest_sampler>
#> • No. Points: 500
#> • LRPS Method: unif_ellipsoid
#>
#> ernest LRPS method <unif_ellipsoid/ernest_lrps>
#> • Dimensions: 3
#> • No. Log-Lik Calls: 0
#> • Center: 0.5000, 0.5000, 0.5000
#> • Log Volume: 1.001
#> • Enlargement: 1.25