Dimensional reduction for sampled priors and application to photometric redshift distributions

A typical Bayesian inference on the values of some parameters of interest $\bf q$ from some data $D$ involves running a Markov Chain (MC) to sample from the posterior $p({\bf q},{\bf n} | D) \propto \mathcal{L}(D | {\bf q},{\bf n}) p({\bf q}) p({\bf n}),$ where $\bf n$ are some nuisance parameters with separable prior. In some cases, the nuisance parameters are high-dimensional, and their prior $p({\bf n})$ 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 $p({\bf n})$ at arbitrary values of ${\bf n},$ i.e.\ one needs to provide a density estimator over the full $\bf n$ space from the provided samples. But the high dimensionality of $\bf n$ 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 $\bf n$ space into a much lower-dimensional space $\bf u$ which projects away directions in $\bf n$ space that cannot appreciably alter $\mathcal{L}.$ The algorithm for doing so is a slight modification to principal components analysis, and is less restrictive on $p(\bf n)$ 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 \textit{Dark Energy Survey}, where $\bf n$ is a binned representation of the redshift distribution $n(z)$ of the galaxies.