Dimensional Reduction for Sampled Priors and Application to Photometric Redshift Distributions

Abstract A typical Bayesian inference on the values of some parameters of interest q from some data D involves running a Markov Chain (MC) to sample from the posterior p ( q , n ∣ D ) ∝ L ( D ∣ q , n ) p ( q ) p ( n ) , where n are some nuisance parameters with a separable prior. In some cases, the nuisance parameters are high-dimensional, and their prior p ( n ) is itself defined only by a set of samples that have been drawn from some other MC. The MC for the posterior will typically require evaluation of p ( n ) at arbitrary values of n , i.e., one needs to provide a density estimator over the full n space from the provided samples. But the high dimensionality of n hinders both the density estimation and the efficiency of the MC for the posterior. We describe a solution to this problem: a linear compression of the n space into a much lower-dimensional space u , which projects away directions in n space that cannot appreciably alter L . The algorithm for doing so is a slight modification to principal components analysis, and is less restrictive on p ( n ) than other proposed solutions to this issue. We demonstrate this “mode projection” technique using the analysis of 2-point correlation functions of weak lensing fields and galaxy density in the Dark Energy Survey, where n is a binned representation of the redshift distribution n ( z ) of the galaxies.