Appendix of the A&A paper "Interpreting deep learning-based stellar mass estimation via causal analysis and mutual information decomposition"

Zhang et al.: Interpreting deep learning-based stellar mass estimation Appendix C: More results C.1. Detection of external variables In Figs. C.1, C.2, C.3, C.4 and C.5, we show more results on the local correlation distributions supplementary to Fig. 3 for the five photometry-only or image-based models defined in Table 2, respectively. The list of the shown parameters, summarized in Table 1, includes various photometric properties, morphological features, physical properties and other catalog data. Other than the results presented in Fig. 3, we found that there exist clear residual correlations between stellar mass and optical colors for the photometry-only models, illustrated as the inconsistency between the original and reference correlation distributions. This is because optical magnitudes rather than colors are the direct model inputs. The possible misinformation due to noise or data imbalance makes magnitudes and colors not exploited simultaneously. Although not shown, the impact of such residual correlations on stellar mass estimation indicated by the predictive efficiency is insignificant. For image-based models, there are generally no strong residual correlations for either optical magnitudes or colors, probably because the inclusion of morphology covers or synergizes with the color information not fully exploited by the photometry-only models. In addition, for the models not involving infrared photometry, there are correlations between stellar mass and infrared magnitude errors, which is due to the fact that WISE infrared magnitude errors are a strong function of infrared magnitudes. While PSFs may contribute in shaping the dependence relations between measured morphological features and stellar mass, they do not have apparently direct contributions to stellar mass estimation for the SDSS data. Therefore, we did not include PSF FWHMs in the input data for any model. On the contrary, we always used galactic reddening E(BV) as an additional input. Although not shown, the impact of galactic reddening E(BV) on stellar mass estimation cannot be fully covered by photometry or images. The results illustrated in the figures confirm that our models fed with galactic reddening E(BV) have learned to leverage this information for stellar mass estimation. C.2. Analysis of causal structures between external variables and stellar mass To complement the discussions in Sect. 4.2 in which the conditional predictive efficiency was adopted, we applied the correlation metric for the model Mugriz, with both stellar mass and query variables quadratically regressed on conditional variables before estimating correlations. In Fig. C.6, we present the distributions of such conditional correlations between stellar mass and the representative parameters shown in Fig. 5. All the parameters including stellar mass are first quadratically regressed on the grcolor of the nearest neighbors of each test galaxy, then the conditional correlations for every shown parameter are estimated by conditioning on every other parameter, compared to the unconditional correlations in the first column. As shown, there are clear changes in the conditional correlations in contrast to the unconditional ones for some parameters, such as the correlations for W1W2 conditioned on spec-z. These trends are consistent with those shown in Fig. 5. Nonetheless, compared to the conditional predictive efficiency, the correlation is less indicative of the impacts of certain variables on stellar mass estimation. C.3. Decomposition of contributions of different photometric bands In Figs. C.7, C.8, C.9, and C.10, we show the stack plots supplementary to Figs. 7 and 8 for all the cases defined in Table 4, decomposing the contributions of di↵erent photometric bands to stellar mass estimation. In general, other than the synergistic effects exhibited by gband and the behaviors of data imbalance discussed in Sect. 5.2, we found no sharp discrepancy between di↵erent bands. Article number, page 25 of 34 A&A proofs: manuscript no. aa54065-25 Fig. C.1. Distributions of local correlations between stellar mass and the parameters listed in Table 1 for the photometry-only model Mugriz defined in Table 2. The original distributions are separately shown for star-forming, passive and other galaxies from the test sample, illustrated as the colored curves. The distributions shown in grey are used as a contrast, produced by randomly permuting the stellar mass values within the nearest neighbors of each test galaxy. Article number, page 26 of 34 Zhang et al.: Interpreting deep learning-based stellar mass estimation Fig. C.2. Same as Fig. C.1, but for the photometry-only model MugrizW123 defined in Table 2. Article number, page 27 of 34 A&A proofs: manuscript no. aa54065-25 Fig. C.3. Same as Fig. C.1, but for the image-based model Iugriz defined in Table 2. Article number, page 28 of 34 Zhang et al.: Interpreting deep learning-based stellar mass estimation Fig. C.4. Same as Fig. C.1, but for the image-based model Iugriz [MW123 defined in Table 2. Article number, page 29 of 34 A&A proofs: manuscript no. aa54065-25 Fig. C.5. Same as Fig. C.1, but for the image-based model Iugriz [MW123 [zspec defined in Table 2. Article number, page 30 of 34 Zhang et al.: Interpreting deep learning-based stellar mass estimation Fig. C.6. Distributions of conditional local correlations between stellar mass and representative parameters for the photometry-only model Mugriz defined in Table 2. Each row corresponds to a parameter labeled on the first column. Similar to Fig. 5, for each parameter, the first column shows the unconditional correlation distributions to be compared with, and each of the remaining columns shows the conditional correlation distributions with the conditional variable labeled on the bottom. All the parameters including stellar mass are first conditioned on the grcolor of the nearest neighbors of each test galaxy before computing the (conditional) correlations. The original distributions are separately shown for star-forming, passive and other galaxies from the test sample, illustrated as the colored curves. The distributions shown in grey are used as a contrast, produced by randomly permuting the stellar mass values within the nearest neighbors of each test galaxy. Article number, page 31 of 34 A&A proofs: manuscript no. aa54065-25 Fig. C.7. Stack plots of the redundant, unique and synergistic information components (in units of nats) as a function of stellar mass or r-band magnitude for <IX,I\X>defined in Table 4. Each row corresponds to a case in which the images in one band are separated out, shown for starforming, passive and other galaxies from the test sample. The label Xrefers to a single band that is separated out, running over all the optical bands, and \Xrefers to the remaining bands, distinguished for the unique information (shown in red and blue, respectively). Fig. C.8. Same as Fig. C.7, but with two adjacent bands separated out in each case. Article number, page 32 of 34 Zhang et al.: Interpreting deep learning-based stellar mass estimation Fig. C.9. Same as Fig. C.7, but for <MX,M\X>defined in Table 4. Each row corresponds to a case in which the photometry in one band is separated out. The label Xruns over all the optical and infrared bands. Article number, page 33 of 34