Estimation of cosmological parameters using adaptive importance sampling

We present a Bayesian sampling algorithm called adaptive importance sampling or population Monte Carlo (PMC), whose computational workload is easily parallelizable and thus has the potential to considerably reduce the wall-clock time required for sampling, along with providing other benefits. To assess the performance of the approach for cosmological problems, we use simulated and actual data consisting of CMB anisotropies, supernovae of type Ia, and weak cosmological lensing, and provide a comparison of results to those obtained using state-of-the-art Markov chain Monte Carlo (MCMC). For both types of data sets, we find comparable parameter estimates for PMC and MCMC, with the advantage of a significantly lower wall-clock time for PMC. In the case of WMAP5 data, for example, the wall-clock time scale reduces from days for MCMC to hours using PMC on a cluster of processors. Other benefits of the PMC approach, along with potential difficulties in using the approach, are analyzed and discussed.

Figure 1

Test target density on the $(x_1,x_2)$ plane. Contours represent the 68.3% (blue), 95% (green) and 99% (black) confidence regions.

Figure 2

Evolution of the importance function for the target density (see Fig. 1) over 11 iterations of 10 000 points for $x_1$ (horizontal axis) and $x_2$ (vertical axis), except for the last iteration (11) which is a sample of 100 000 points. Iterations 1 (top left) to 11 (bottom right) from left to right with every second iteration shown (i.e., 1, 3, 5, 7, 9, 11). Colors indicate the mixture components with mean of each component indicated by colored dots and approximate 95% confidence regions for the sample of points from each component by colored ellipses. Every 3rd sample point from the importance function is plotted.

Figure 3

Normalized perplexity (top panel) and normalized effective sample size ($\mathrm{ESS}/N$) (bottom panel) estimates for the first 10 iterations of PMC (represented in Fig. 2) over 500 simulation runs. The distributions are shown as whisker plots: the thick horizontal line represents the median; the box shows the interquartile range (IQR), containing 50% of the points; the whiskers indicate the interval $1.5\times \mathrm{IQR}$ from either $Q1$ (lower) or $Q3$ (upper); points outside the interval $[Q1,Q3]$ (outliers) are represented as circles.

Figure 4

Evolution of $\pi (f_a)$ (top panels) and $\pi (f_b)$ (bottom panels) from 10 000 points to 100 000 points for both PMC (left panels) and MCMC (right panels). See Fig. 3 for details about the whisker plot representation.

Figure 5

Results showing the distributions of the PMC and the MCMC estimates $\pi (f)$ for (top) $f=f_c,f_e,f_h$ and (bottom) $f=f_d,f_g,f_i$ (in this order, left to right). All estimates are based on 500 simulation runs. See Fig. 3 for details about the whisker plot representation.

Figure 6

Perplexity (left panel) and normalized effective sample size $\mathrm{ESS}/N$ (right panel), as a function of the cumulative sample size $N$. The likelihood is WMAP5 for a flat $\Lambda \mathrm{CDM}$ model with six parameters.

Figure 7

Histogram of the normalized importance weights ${\overline{w}}_n^t$ for four iterations $t=0,3,6,9$. The posterior is WMAP5, flat $\Lambda \mathrm{CDM}$ model with six parameters.

Figure 8

Lower left panel: Overlaid to the SNIa confidence contours (68%, 95%, 99.7%) is the movement of the importance function. For each iteration a circle is plotted at the position of the mean of each component, where different colors indicate different components. The circle size indicates the component weight. The starting point [first iteration, at $(0.3,-1.3)$] is marked by a thick circle. The other two panels show the mean positions in projection, fanned out as a function of the iteration.

Figure 9

The sampled points from the final iteration are plotted; the colors indicate the components of the importance function from which the points are drawn (the colors are the same as in Fig. 8). One out of five points is shown. Note that the density of points does not correspond to the posterior density since the former has to be weighted by the importance weights.

Figure 10

Examples of marginalized likelihoods (68%, 95% and 99.7% contours are shown) for PMC (solid blue line) and MCMC (dashed green line) from the SNIa data.

Figure 11

Normalized 1D marginals for ${\Omega }_{\mathrm{b}}$ from weak lensing alone for PMC (solid blue line) and MCMC (dashed green line: adaptive; dotted red line: nonadaptive).