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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">q</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo stretchy="false">∣</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∝</mml:mo> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">∣</mml:mo> <mml:mi mathvariant="bold-italic">q</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">L</mml:mi> <mml:mo>.</mml:mo> </mml:math> 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.

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Journal
The Astrophysical Journal
Published
2026-04-28
DOI
https://doi.org/10.3847/1538-4357/ae5baa
Primary Topic
Statistical Mechanics and Entropy
Type
article
Field-Weighted Citation Impact
0.00

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article

Dimensional Reduction for Sampled Priors and Application to Photometric Redshift Distributions

B. Yin, A. Alarcon, J Weller, J Frieman et al.
The Astrophysical Journal
Statistical Mechanics and Entropy
article

Dimensional Reduction for Sampled Priors and Application to Photometric Redshift Distributions

B. Yin, A. Alarcon, J Weller, J Frieman, J García-Bellido, M E C Swanson, M A Troxel, W. d'Assignies, J. Mena-Fernández, A. A. Plazas Malagón, S. S. Allam, M. Smith, F. Andrade-Oliveira, T. Shin, G. Giannini, S. Everett, S. Lee, D. Brooks, P. Wiseman, M. Aguena, J. De Vicente, G. M. Bernstein, D. Sanchez Cid, J. Carretero, I. Sevilla-Noarbe, D. L. Hollowood, M. E. S. Pereira, S. R. Hinton, N. Weaverdyck, D. J. James, E. Sanchez, E. Suchyta, L. N. da Costa, D. Gruen, R. Miquel, J. L. Marshall, A. Amon, A. Carnero Rosell, K. Honscheid
article en

Abstract

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">q</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo stretchy="false">∣</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∝</mml:mo> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">∣</mml:mo> <mml:mi mathvariant="bold-italic">q</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">L</mml:mi> <mml:mo>.</mml:mo> </mml:math> 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.

The Astrophysical JournalVol. 1002(1)
Institució Catalana de Recerca i Estudis Avançats (ES), California Institute of Technology (US), Oak Ridge National Laboratory (US), Centre National de la Recherche Scientifique (FR), Fermi National Accelerator Laboratory (US), Institute of Space Sciences (ES), Jet Propulsion Laboratory (US), Lawrence Berkeley National Laboratory (US), Universidad de La Laguna (ES), Universität Hamburg (DE), The University of Queensland (AU), Duke University (US), Institute for High Energy Physics (ES), Princeton University (US), University of Zurich (CH), Mitchell Institute (US), SLAC National Accelerator Laboratory (US), University of Chicago (US), Max Planck Institute for Extraterrestrial Physics (DE), Kavli Institute for Particle Astrophysics and Cosmology (US), Center for Astrophysics Harvard & Smithsonian (US), Instituto de Astrofísica de Canarias (ES), Institut National de Physique Nucléaire et de Physique des Particules (FR), National Center for Supercomputing Applications (US), Laboratório Interinstitucional de e-Astronomia (BR), Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (ES), University of Southampton (GB), University College London (GB), The Ohio State University (US), Stony Brook University (US), Universidad Autónoma de Madrid (ES), Lancaster University (GB), Ludwig-Maximilians-Universität München (DE), Université Grenoble Alpes (FR), Texas A&M University (US), Philadelphia University (US), University of California, Berkeley (US), Stanford University (US)
U.S. Department of Energy
Openalex Percentile: Top 14%
Statistical Mechanics and Entropy
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