Introduction

Dependencies

In this example we shall be using the stack that had historically been used to develop supernest.

Sampler

One should firstly install pypolychord.

This project has historically been difficult to install, owing to FORTRAN being an old lanugage with difficult to track dependencies, and poor compiler support from modern GCC or LLVM.

We recommend following the official instructions (yours truly contributed to the install process being tenable on Mac OS X).

Plotting

We would then need a post-processing library. One is welcome to do kernel density estimations by hand, but it would be tiresome to do with something like e.g. gnuplot so, we recommend using eitehr anesthetic or getdist.

TIP

While historically anesthetic has been used to develop supernest, getdist is a more standard tool with a friendlier user interface, and better architectural design decisions.

nestcheck

For additional insight, Edward Higson's nestcheck package is useful, but not required. It can, to some extent diagnose issues within the nested sampling pipeline itself and warn about misbehaviour.

However, all work done with nestcheck is highly technical in nature and only useful to those that want to grasp the concepts more thoroughly, and not required for a quickstart.

Basic theory

In order to perform Bayesian inference, one needs to define two probability functions: the prior and the likelihood. The precise meanings of these terms are better described in specific books, but for the time being, I would recommend one think of them as two functions: pi and logl.

In mathematical terms the prior quantile probability function, pi maps a point in the unit hypercube into the prior coordinate space. We are mainly concerned with computationally simple models, so in this guide we shall never cover anything more complicated than, e.g. mapping to a hyperrectangle and a Gaussian (more on that later).

The log-likelihood function logl is a much more straightforward mapping. It accepts points in the hyperrectangle generated by pi and transforms them into a single number. We conventionally operate in logarithmic space, as almost all realistic distributions are going to contain an exponential element to them, (including the so-called banana distribution).

Example model

Our example model is a single two-dimensional spherical Gaussian peak with a uniform prior located in the hyperrectangle whose 2^nd component is scaled by a factor of two.

This translates to the following prior quantile:

def pi(cube):
	return np.array([cube[0], 2*cube[1]])

TIP

The usage of an explicit list and conversion is done for the readability's benefit, the next examples shall be better optimised, to use numpy operations.

The Gaussian is equally straightforward, but a tad more involved.

DANGER

TODO