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.