Conditional Density Estimation Tools in Python and R with Applications to Photometric Redshifts and Likelihood-Free Cosmological Inference

Abstract

It is well known in astronomy that propagating non-Gaussian prediction uncertainty in photometric redshift estimates is key to reducing bias in downstream cosmological analyses. Similarly, likelihood-free inference approaches, which are beginning to emerge as a tool for cosmological analysis, require a characterization of the full uncertainty landscape of the parameters of interest given observed data. However, most machine learning (ML) or training-based methods with open-source software target point prediction or classification, and hence fall short in quantifying uncertainty in complex regression and parameter inference settings. As an alternative to methods that focus on predicting the response (or parameters) y from features x, we provide nonparametric conditional density estimation (CDE) tools for approximating and validating the entire probability density function (PDF) p(y|x) of y given (i.e., conditional on) x. As there is no one-size-fits-all CDE method, the goal of this work is to provide a comprehensive range of statistical tools and open-source software for nonparametric CDE and method assessment which can accommodate different types of settings and be easily fit to the problem at hand. Specifically, we introduce four CDE software packages in 𝙿𝚢𝚝𝚑𝚘𝚗 and 𝚁 based on ML prediction methods adapted and optimized for CDE: 𝙽𝙽𝙺𝙲𝙳𝙴, 𝚁𝙵𝙲𝙳𝙴, 𝙵𝚕𝚎𝚡𝙲𝚘𝚍𝚎, and 𝙳𝚎𝚎𝚙𝙲𝙳𝙴. Furthermore, we present the 𝚌𝚍𝚎𝚝𝚘𝚘𝚕𝚜 package, which includes functions for computing a CDE loss function for tuning and assessing the quality of individual PDFs, along with diagnostic functions. We provide sample code in 𝙿𝚢𝚝𝚑𝚘𝚗 and 𝚁 as well as examples of applications to photometric redshift estimation and likelihood-free cosmological inference via CDE.