PolyChord: nested sampling for cosmology

TL;DR

PolyChord introduces a high-dimensional nested sampling algorithm optimized for cosmology, integrating N-dimensional slice sampling, degeneracy whitening, and cluster-based multimodal exploration. It supports a fast-slow parameter hierarchy (CosmoChord) and is parallelized via OpenMPI. Empirical results show PolyChord scales better with dimension than MultiNest (N_L ~ O(D^3) vs exponential) while preserving accurate evidence estimates; CosmoChord demonstrates practical gains in Planck-scale cosmological analyses. The work enables efficient Bayesian model comparison and robust posterior sampling in very high-dimensional cosmological problems.

Abstract

PolyChord is a novel nested sampling algorithm tailored for high dimensional parameter spaces. In addition, it can fully exploit a hierarchy of parameter speeds such as is found in CosmoMC and CAMB. It utilises slice sampling at each iteration to sample within the hard likelihood constraint of nested sampling. It can identify and evolve separate modes of a posterior semi-independently and is parallelised using openMPI. PolyChord is available for download at: http://ccpforge.cse.rl.ac.uk/gf/project/polychord/

Figures (5)

• Figure 1: Slice sampling in one dimension. Given a likelihood slice $\mathcal{L}_0$, a seed point $x_0$ and an initial parameter $w$, slice sampling generates a new point $x_1$ within the horizontal region defined by $\mathcal{L}>\mathcal{L}_0$. A point $x$ is within the "slice" if $\mathcal{L}(x)>\mathcal{L}_0$. External bounds are first set $\hat{L}<x_0<\hat{R}$ by expanding a random initial bound of width $w$ until they lie outside the slice via the stepping out procedure. $x_1$ is then sampled uniformly within these bounds. If $x_1$ is not in the slice, then the bound is contracted down to $x_1$, and $x_1$ is then drawn again from these new bounds. This procedure is guaranteed to generate a new point $x_1$, and satisfies detailed balance $P(x_0|x_1) = P(x_1|x_0)$. Thus, if $x_0$ is drawn from a uniform distribution within the slice, so is $x_1$.
• Figure 2: Slice Sampling in $N$ dimensions. We begin by "whitening" the unit hypercube by making a linear transformation which turns a degenerate contour into one with dimensions $\sim\mathcal{O}(1)$ in all directions. This is a linear skew transformation defined by the inverse of the Cholesky decomposition of the live points' covariance matrix. We term this whitened space the sampling space. Starting from a randomly chosen live point $x_0$, we pick a random direction and perform one dimensional slice sampling in that direction (Figure \ref{['fig:1d_slice']}), using $w=1$ in the sampling space. This generates a new point $x_1$ in $\sim\mathcal{O}(\text{a few})$ likelihood evaluations. This process is repeated $\sim\mathcal{O}(n_\mathrm{dims})$ times to generate a new uniformly sampled point $x_N$ which is decorrelated from $x_0$.
• Figure 3: Parallelisation of PolyChord. The algorithm parallelises nearly linearly, providing that $n_\mathrm{procs}<n_\mathrm{live}$. For most astronomical applications this is more than sufficient.
• Figure 4: Comparing PolyChord with MultiNest using a Gaussian likelihood for different dimensionalities. PolyChord has at worst $N_\mathcal{L}\sim\mathcal{O}(D^3)$, whereas MultiNest has an exponential scaling that emerges at high dimensions.
• Figure 5: Evidence estimates and errors produced by PolyChord for a Gaussian likelihood as a function of dimensionality. The dashed line indicates the correct analytic evidence value.