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

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Volume 703 (November 2025)

A&A, 703 (2025) A276

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Table 3

Summary of the cases with different data modalities for the mutual information decomposition. Galactic reddening E(B – V) is included in all the datasets.

$< M_{u g r i z}, I_{u g r i z}^{n o r m} >$ $< {M_{ugriz}},\,I_{ugriz}^{norm} >$	The set M_u𝑔riz contains Petrosian magnitudes and errors in the five optical bands u, 𝑔, r, i, ɀ. M_u𝑔riz corresponds to the photometry-only model defined in Table. 2. The notations for models and datasets are used interchangeably. The set $I_{u g r i z}^{n o r m}$ $I_{ugriz}^{norm}$ contains the five-band images normalized by the corresponding Petrosian fluxes before rescaling the pixel intensities using Eq. (1). The use of $I_{u g r i z}^{n o r m}$ $I_{ugriz}^{norm}$ is intended to investigate the contribution of images in the absence of magnitude information. The overall morphological information is in principle encompassed in $I_{u g r i z}^{n o r m}$ $I_{ugriz}^{norm}$ . Before normalizing the images, clipping is applied to the Petrosian fluxes in the five bands at the 2nd, 0.1th, 0.1th, 0.1th, and 0.1th percentiles of the per-band flux distributions for the whole sample, respectively, in order to avoid near-zero or negative fluxes in the devidend. This clipping operation is only concerned to a tiny fraction of the sample and does not affect our conclusion.

<M_u𝑔riz, Morph>	The set M_u𝑔riz remains unchanged. The set Morph contains a few representative morphological parameters including Sérsic indices n, inclinations b/a, and Petrosian radii R₅₀ and R₉₀ in the five optical bands u, 𝑔, r, i, ɀ (20 parameters in total). These parameters (or their combinations) have been found to contribute to the stellar mass estimation in addition to integrated photometry by the causal analysis.

<M_u𝑔riz, M_W123>	The set M_u𝑔riz remains unchanged. The set M_W123 contains magnitudes and errors in the three infrared bands W 1, W 2, W 3 (except errors in the W 3 band).

<I_u𝑔riz, M_W123>	The set I_u𝑔riz contains the unnormalized images in the five optical bands. The set M_W123 remains unchanged.

<I_u𝑔riz ∪ M_W123, {ɀ_Spec}>	The set I_u𝑔riz ∪ M_W123 contains the union of I_u𝑔riz and M_W123. The set {ɀ_spec} contains spec-ɀ.

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