A normalizing flow approach for the inference of star cluster properties from unresolved broadband photometry I: Comparison to spectral energy distribution fitting

TL;DR

This work tackles the challenge of inferring star-cluster properties from unresolved broadband photometry, where grid-based SED fitting becomes computationally prohibitive with nuisance parameters. It introduces a normalizing-flow framework using a conditional invertible neural network (cINN) trained on $5 imes 10^6$ CIGALE-generated photometries to learn $p(oldsymbol{ heta} obreak obreak| obreak obreak oldsymbol{x})$ for $(oldsymbol{ heta} = (\log_{10} t, \, \log_{10} m, \, E_{B-V}))$ conditioned on five-band photometry and ancillary inputs. The approach yields reasonable posterior approximations on both synthetic tests and PHANGS DR3 data, enabling efficient density estimation and sampling, with results broadly agreeing with PHANGS MLEs except where mode selection and noise-model choices drive differences. The method is particularly advantageous when forward-model nuisance parameters render the likelihood intractable or when large inference catalogs demand amortized, likelihood-free density estimation, and it sets the stage for extending to more parameters in follow-up work.

Abstract

Estimating properties of star clusters from unresolved broadband photometry is a challenging problem that is classically tackled by spectral energy distribution (SED) fitting methods that are based on simple stellar population models. However, because of their exponential scaling, grid-based methods suffer from computational limitations. In addition, nuisance parameters in the model can make the computation of the likelihood function intractable. These limitations can be overcome by modern generative deep learning methods that offer flexible and powerful tools for modeling high-dimensional posterior distributions and fast inference from learned data. We present a normalizing flow approach for the inference of cluster age, mass, and reddening from Hubble Space Telescope broadband photometry. In particular, we explore our network's behavior on an inference problem that has been analyzed in previous works. We used the SED modeling code CIGALE to create a dataset of synthetic photometric observations for $5 \times 10^6$ mock star clusters. Subsequently, this data set was used to train a coupling-based flow in the form of a conditional invertible neural network (cINN) to predict posterior probability distributions for cluster age, mass, and reddening from photometric observations. We predicted cluster parameters for the 'Physics at High Angular resolution in Nearby GalaxieS' (PHANGS) Data Release 3 catalog. To evaluate the capabilities of the network, we compared our results to the publicly available PHANGS estimates and found that the estimates agree reasonably well. We demonstrate that normalizing flow methods can be a viable tool for the inference of cluster parameters, and argue that this approach is especially useful when nuisance parameters make the computation of the likelihood intractable and in scenarios that require efficient density estimation.

Figures (13)

• Figure 1: Flowchart of single coupling block (representing Eqs. \ref{['eq:coupling_equation_1']}, \ref{['eq:coupling_equation_2']}, and \ref{['eq:coupling_equation_3']}) taken from MasterThesis. The input vector, $\boldsymbol{\theta}_j$, is split into two sub-vectors, $\boldsymbol{\theta_j}^{(1)}$ and $\boldsymbol{\theta}_j^{(2)}$. The conditioning input, $\mathbf{x,}$ is used as an additional input to the subnetworks $s_1$, $s_2$, $t_1,$ and $t_2$.
• Figure 2: Top row: Histograms of relative flux uncertainties in the PHANGS cluster catalog. The dotted vertical lines denote the median relative flux uncertainties, and the dashed lines denote the maximum relative flux uncertainties. Middle row: Comparison of the normal noise model and the log-normal noise model for median relative flux uncertainties. Bottom row: Comparison of the normal noise model and the log-normal noise model for maximum relative flux uncertainties.
• Figure 3: Results of SBC analysis over the entire test set ($\approx 10^5$ parameter--observation pairs). We generated a hundred posterior samples for every element in the test set. Left column: Rank histograms for the proposed statistics. Horizontal dashed lines correspond to the $\pm1\sigma$ range under the assumption of uniformity. Right column: Difference between rank ECDF and uniform cumulative distribution function. The gray area highlights the expected 0.01-0.99 quantile range of the ECDF difference under the assumption of uniformity.
• Figure 4: Results of PPC analysis. Posterior predictive distributions with 500 samples each were generated for all clusters in the PHANGS catalog. Left column: Comparison of the mean predictive photometry and the initial photometry. Right column: Rank distribution of the initial photometry within the posterior predictive distribution. Horizontal dashed lines correspond to the $\pm1\sigma$ range under the assumption of uniformity.
• Figure 5: 2D and 1D density plots of example posteriors. Top rows: Marginal 1D density for the cINN posterior (gray histogram) and ground truth (dashed black line). Middle rows: 2D density of the cINN generated posterior ($10^4$ samples per posterior). Bottom rows: Contour plot of ground truth densities calculated over a parameter grid. Note that we use the same color maps in the second and third rows (although the colorbars show different ranges). The yellow markers indicate positions of cINN MAP estimates (triangle) and PHANGS MLEs (cross).