Bayesian model comparison in cosmology with population Monte Carlo

The paper on evidence approximation by population Monte Carlo that I mentioned in a previous post is now arXived. (This is the second paper jointly produced by the members of the 2005-2009 ANR Ecosstat.) The full abstract is

“We use Bayesian model selection techniques to test extensions of the standard  flat ACDM paradigm. Dark-energy and curvature scenarios, and primordial perturbation models are considered. To that end, we calculate the Bayesian evidence in favour of each model using Population Monte Carlo (PMC), a new adaptive sampling technique which was recently applied in a cosmological context. In contrast to the case of other sampling-based inference techniques such as Markov chain Monte Carlo (MCMC), the Bayesian evidence is immediately available from the PMC sample used for parameter estimation without further computational eff ort, and it comes with an associated error evaluation. Besides, it provides an unbiased estimator of the evidence after any fixed number of iterations and it is naturally parallelisable, in contrast with MCMC and nested sampling methods. By comparison with analytical predictions for simulated data, we show that our results obtained with PMC are reliable and robust. The variability in the evidence evaluation and the stability for various cases are estimated both from simulations and from data. For the cases we consider, the log-evidence is calculated with a precision of better than 0:08.

Using a combined set of recent CMB, SNIa and BAO data, we fi nd inconclusive evidence between  at ACDM and simple dark-energy models. A curved Universe is moderately to strongly disfavoured with respect to a flat cosmology. Using physically well-motivated priors within the slow-roll approximation of inflation, we find a weak preference for a running spectral index. A Harrison-Zel’dovich spectrum is weakly disfavoured. With the current data, tensor modes are not detected; the large prior volume on the tensor-to-scalar ratio r results in moderate evidence in favour of r = 0.”

and we run an evaluation of the PMC based approximation to the evidence that shows that this particular importance sampling technique allows for an associated evalution of the uncertainty surrounding this approximation. A posteriori, I think we should have included a comparison with nested sampling, since this technique is now very much in use in astronomy, but this will have to wait for another paper.

Note that the paper details the fact that the variance of the evidence approximation

$\dfrac{\mathfrak{E}^2}{N} d(\pi,q)$

is connected with the chi-square distance $d(\pi,q)$, which can indeed be estimated from the simulation output as

$\dfrac{N}{\text{ESS}_N} -1$

if $\text{ESS}_N$ denotes the effective sample size. In the cosmology section, a point worth pointing to non-cosmologists is that the likelihood computation takes ages and therefore that the complete parallelisation offered by importance sampling is quite appreciable.

2 Responses to “Bayesian model comparison in cosmology with population Monte Carlo”

1. […] is missing the recent literature both on the nonparametric aspects of ABC and on the more adaptive [PMC] features of ABC-SMC, as processed in our Biometrika ABC-PMC paper and in Del Moral, Doucet and […]

2. […] model comparison in cosmology” is back! We already have the review back on our paper from Monthly Notices of the Royal Astronomical Society. And it is quite positive! Dear Dr […]