Evaluation of multi-parameter likelihoods through iteration of two-dimensional slices

Abstract

In this paper we introduce a method for resolving multi-parameter likelihoods by fixing all parameter values, but two. Evaluation of those two variables is followed by iteratively cycling through each of the parameters in turn until convergence. We test the technique on the temperature power spectrum of the lensed cosmic microwave background (CMB). That demonstration is particularly effective since one of the six parameters that define the power spectra, the power spectrum amplitude, $A_s$, nears linearity at small deviations, reducing computation to incrementation in one-dimension, rather than over a 2D grid. At each iterative step $A_s$ is paired with a different parameter. The iterative process yields parameter values in agreement with those derived by \textit{Planck}, and results are obtained within a few hundred calls for spectra. We further compute parameter values as a function of maximum multipole, ${\ell }_{\text{max}}$, spanning a range from ${\ell }_{\text{max}}$=959 to 2500, and uncover bi-modal behavior at the lower end of that range. In the general case, in which neither variable is linear, we identify moderating factors, such as changing both parameters each iterative step, reducing the number of steps per iteration. Markov chain Monte Carlo (MCMC) computation has been the dominant instrument for evaluating multi-parameter functions. For applications with a quasi-linear variable such as, $A_s$, the 2D iterative method is orders of magnitude more efficient than MCMC.

9 pages, 3 figures

References in corpus (13)

• Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results
• The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes
• A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team
• Planck 2013 results. I. Overview of products and scientific results
• Planck 2018 results. V. CMB power spectra and likelihoods
• The Carnegie-Chicago Hubble Program. VIII. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch
• Planck 2015 results. XI. CMB power spectra, likelihoods, and robustness of parameters
• The Atacama Cosmology Telescope: DR4 Maps and Cosmological Parameters
• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Parameter Estimation Methodology
• The Atacama Cosmology Telescope: A Measurement of the Cosmic Microwave Background Power Spectra at 98 and 150 GHz
• Quantifying discordance in the 2015 Planck CMB spectrum
• Non-Gaussian Error Contribution to Likelihood Analysis of the Matter Power Spectrum
• Likelihood methods for the combined analysis of CMB temperature and polarisation power spectra