Hierarchical Inference of the Lensing Convergence from Photometric Catalogs with Bayesian Graph Neural Networks

We present a Bayesian graph neural network (BGNN) that can estimate the weak lensing convergence (κ) from photometric measurements of galaxies along a given line of sight (LOS). The method is of particular interest in strong gravitational time-delay cosmography (TDC), where characterizing the “external convergence” (κ _ext) from the lens environment and LOS is necessary for precise Hubble constant (H ₀) inference. Starting from a large-scale simulation with a κ resolution of ∼1′, we introduce fluctuations on galaxy-galaxy lensing scales of ∼1″ and extract random sight lines to train our BGNN. We then evaluate the model on test sets with varying degrees of overlap with the training distribution. For each test set of 1000 sight lines, the BGNN infers the individual κ posteriors, which we combine in a hierarchical Bayesian model to yield constraints on the hyperparameters governing the population. For a test field well sampled by the training set, the BGNN recovers the population mean of κ precisely and without bias (within the 2σ credible interval), resulting in a contribution to the H ₀ error budget well under 1%. In the tails of the training set with sparse samples, the BGNN, which can ingest all available information about each sight line, extracts a stronger κ signal compared to a simplified version of the traditional method based on matching galaxy number counts, which is limited by sample variance. Our hierarchical inference pipeline using BGNNs promises to improve the κ _ext characterization for precision TDC. The code is available as a public Python package, Node to Joy ⏬.