Dimensional reduction for sampled priors and application to photometric redshift distributions

Dimensional reduction for sampled priors and application to photometric redshift distributions
Gary Bernstein, William Assignies Doumerg, Michael A. Troxel, Alex Alarcon, Alexandra Amon, Giulia Giannini, Boyan Yin, Sahar Allam, Felipe Andrade-Oliveira, David Brooks, Aurelio Carnero Rosell, Jorge Carretero, Luiz da Costa, Maria Elidaiana da Silva Pereira, Juan De Vicente, Spencer Everett, Josh Frieman, Juan Garcia-Bellido, Daniel Gruen, Samuel Hinton, Devon L. Hollowood, Klaus Honscheid, David James, Sujeong Lee, Jennifer Marshall, Juan Mena-Fern’andez, Ramon Miquel, Andr’es Plazas Malag’on, Eusebio Sanchez, David Sanchez Cid, Ignacio Sevilla, Tae-hyeon Shin, Mathew Smith, Eric Suchyta, Molly Swanson, Noah Weaverdyck, Jochen Weller, Philip Wiseman
arXiv:2506.00758v2 Announce Type: replace
Abstract: 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.arXiv:2506.00758v2 Announce Type: replace
Abstract: 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.