Fast and reliable MCMC for cosmological parameter estimation

TL;DR

The paper develops a power-spectrum–based convergence diagnostic for MCMC in cosmology, enabling reliable termination of a single chain by fitting a parametric template to the chain’s $P(k)$ and extracting $P_0$, $k^*$, and $\alpha$ to assess convergence via $j^*>20$ and $r < 0.01$. It then analyzes how to maximize sampling efficiency through optimal step-size tuning and accurate estimation of the trial-covariance, providing scaling relations that depend on dimension $D$ and the underlying distribution. The authors validate the approach on Gaussian toy models and apply it to real cosmological data, including pure adiabatic and mixed adiabatic/isocurvature models, highlighting both the gains in efficiency and the limitations in challenging posterior geometries (e.g., multi-modality). The work offers practical guidance for robust cosmological parameter inference from CMB and LSS datasets, balancing convergence diagnostics, computational cost, and the dangers of non-Gaussian posteriors.

Abstract

Markov Chain Monte Carlo (MCMC) techniques are now widely used for cosmological parameter estimation. Chains are generated to sample the posterior probability distribution obtained following the Bayesian approach. An important issue is how to optimize the efficiency of such sampling and how to diagnose whether a finite-length chain has adequately sampled the underlying posterior probability distribution. We show how the power spectrum of a single such finite chain may be used as a convergence diagnostic by means of a fitting function, and discuss strategies for optimizing the distribution for the proposed steps. The methods developed are applied to current CMB and LSS data interpreted using both a pure adiabatic cosmological model and a mixed adiabatic/isocurvature cosmological model including possible correlations between modes. For the latter application, because of the increased dimensionality and the presence of degeneracies, the need for tuning MCMC methods for maximum efficiency becomes particularly acute.

Figures (17)

• Figure 1: (Top) The discrete power spectrum of an 'ideal,' uncorrelated chain formed by drawing points at random from a Gaussian distribution of unit variance. (Bottom) The discrete power spectrum from an MCMC chain of length $N=3000$ sampling the same five-dimensional Gaussian.
• Figure 2: The exact power spectrum of one of the variables of an MCMC chain sampling the same 5-D Gaussian distribution as in the lower panel of Fig. \ref{['ideal']}, measured by averaging over a large number of chains.
• Figure 3: In the top panel, the chain of the lower panel in Fig. (\ref{['ideal']}) is fitted to the template defined in Eqn. (\ref{['template:e']}). This fit is indicated by the dashed curve. The lower panel compares the obtained fit to the measured exact power spectrum (shown as the solid curve).
• Figure 4: (Left) Distribution for the quality of fit $P_0^{fit}/P_0$, obtaining $P_0^{fit}$ by fitting 5000 individual chains, compared to the true value $P_0$ obtained by averaging. (Right) Quality of fit $P_0^{fit}/P_0$ at the lower $1\sigma$ level as a function of $j^*$, the number of Fourier modes in the white noise regime. The dashed line indicates the mean value for $j^*>20,$ with very short chains tending to overestimate $P_0.$
• Figure 5: True (solid) and fitted (dashed) power spectra of chains sampling the distribution as in Fig. \ref{['model']}, using trial distributions of widths $\sigma_T/\sigma_0 =0.2$ (left), $0.5$ (middle) and $2$ (right).