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mfouesneau / Nested_Sampling

My playground around nested sampling
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Nested Sampling

This is my playground around Nested Sampling methods

I adapted the C-code of nested sampling from Sivia and Skilling 2006 (p. 188) Attempts to make its usage like emcee. (call it nestee? maybe not...)

D. MacKay Webpage for reference.

Book Data Analysis - A Bayesian Tutorial by Sivia and John Skilling (ISBN 0198568312)

Note that the code is very close to the original C code, which may not be optimized for python usage. Although not python-optimized this works just fine.

Background

Although small, the following main code incorporates the main ideas and should suffice for many applications. It is protected against over/underflow of exponential quantities such as the likelihood L∗ by storing them as logarithms (as in the variable logLstar), adding those values through the PLUS macro, and multiplying them through summation. Rather than attempting greater sophistication, the program uses the simple proclamation of steady compression by log(t) = −1/n each step. The corresponding step-widths h are also stored as logarithms, in logwidth.

The new technique of nested sampling (Skilling 2004) tabulates the sorted likelihood function L(x) in a way that itself uses Monte Carlo methods. The technique uses a collection of n objects x, randomly sampled with respect to the prior p, but also subject to an evolving constraint L(x) > L∗ preventing the likelihood from exceeding the current limiting value L∗.

We define xsi(x) = proportion of prior with likelihood greater than x. In terms of xsi, the objects are uniformly sampled subject to the constraint xsi < xsi∗, where xsi∗ corresponds to L∗;

Example for many objects sampled uniformly in x < x1, or equivalently in L > L∗. Nest (Illustration from Skilling, 2006)

At the outset, sampling is uniform over the entire prior, meaning that xsi∗=1 and L∗=0. The idea is then to iterate inwards in xsi and correspondingly upwards in L, in order to locate and quantify the tiny region of high likelihood where most of the joint distribution is to be found.

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