Interpreting deep learning-based stellar mass estimation via causal analysis and mutual information decomposition

¹ Pengcheng Laboratory, Nanshan District, Shenzhen, Guangdong 518000, PR China
² Harbin Institute of Technology, Nanshan District, Shenzhen, Guangdong 518000, PR China
³ Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA
⁴ Center for Cosmology and AstroParticle Physics (CCAPP), The Ohio State University, Columbus, OH 43210, USA

^★ Corresponding author: linqf@pcl.ac.cn

Received: 7 February 2025
Accepted: 22 August 2025

Abstract

End-to-end deep learning models fed with multi-band galaxy images are powerful data-driven tools used to estimate galaxy physical properties in the absence of spectroscopy. However, due to a lack of interpretability and the associational nature of such models, it is difficult to understand how the information that is included in addition to integrated photometry (e.g., morphology) contributes to the estimation task. Improving our understanding in this field would enable further advances into unraveling the physical connections among galaxy properties and optimizing data exploitation. Therefore, our work is aimed at interpreting the deep learning-based estimation of stellar mass via two interpretability techniques: causal analysis and mutual information decomposition. The former reveals the causal paths between multiple variables beyond nondirectional statistical associations, while the latter quantifies the multicomponent contributions (i.e., redundant, unique, and synergistic) of different input data to the stellar mass estimation. We leveraged data from the Sloan Digital Sky Survey (SDSS) and the Wide-field Infrared Survey Explorer (WISE). With the causal analysis, meaningful causal structures were found between stellar mass, photometry, redshift, and various intra- and cross-band morphological features. The causal relations between stellar mass and morphological features not covered by photometry indicate contributions coming from images that are complementary to the photometry. With respect to the mutual information decomposition, we found that the total information provided by the SDSS optical images is effectively more than what can be obtained via a simple concatenation of photometry and morphology, since having the images separated into these two parts would dilute the intrinsic synergistic information. A considerable degree of synergy also exists between the 𝑔 band and other bands. In addition, the use of the SDSS optical images may essentially obviate the incremental contribution of the WISE infrared photometry, even if infrared information is not fully covered by the optical bands available. Taken altogether, these results provide physical interpretations for image-based models. Our work demonstrates the gains from combining deep learning with interpretability techniques, and holds promise in promoting more data-driven astrophysical research (e.g., astrophysical parameter estimations and investigations on complex multivariate physical processes).