Bayesian Latent Space Models for Graphs Are Misspecified: Toward Robust Inference via Generalized Posteriors

Bayesian causal discovery is widely used for its ability to quantify epistemic uncertainty over directed acyclic graphs (DAGs) through posterior inference. However, its behaviour under latent confounding remains poorly understood, as existing work typically notes that confounding breaks identifiability without characterising how the posterior distribution over DAGs responds. In this work, we analyse posterior behaviour under latent confounding in linear Gaussian causal models, focusing on additive latent confounding between exactly two observed variables. We derive a critical correlation threshold above which the score function favours graphs with a spurious edge between the confounded variables, and show that this threshold decreases with sample size -- more data lowers the correlation required for the spurious edge to be favoured. Beyond this threshold, we characterize two distinct posterior failure regimes determined by the local structure around the confounded variables. Our findings are supported by exact posterior computations on multiple graph structures, demonstrating both the predicted failure regimes.

Hierarchical data is ubiquitous in the empirical sciences and is most commonly analyzed with generalized linear mixed-effects models (GLMMs). Bayesian inference for GLMMs yields calibrated uncertainty but requires MCMC; the No-U-Turn Sampler (NUTS) is the gold standard but is slow and must restart from scratch for every new dataset, model and prior. We introduce metabeta, a pretrained neural network for prior-amortized in-context Bayesian inference over GLMMs. Unlike previous neural posterior estimators that fix the prior at training time, metabeta accepts prior families and hyperparameters as inputs at test time, enabling zero-shot generalization. Two set transformers and conditional normalizing flows mirror the posterior's two-level structure (global parameters shared across groups, local parameters per group). The model is trained on millions of realistic simulated datasets spanning continuous, binary, and count outcomes. By default, the flow posterior is refined by Independence Metropolis-Hastings against the unnormalized posterior, so its correctness rests on the sampler rather than the network; this yields tuning-free inference two to three orders of magnitude faster than NUTS. Alternatively, the flow can warm-start NUTS, giving nearly identical inference with substantially increased speed and stability. On controlled benchmarks with ground-truth parameters, metabeta matches NUTS in parameter recovery, calibration and out-of-sample prediction. On out-of-distribution real datasets, its posteriors closely match those of NUTS across all parameter types, and they remain faithful under misspecified likelihoods and priors, out-of-distribution predictors, collinear designs, and data-poor regimes. The model is open-source and open-weights and thus immediately deployable.

Latent Gaussian models (LGMs) are a popular class of Bayesian hierarchical models that include Gaussian processes, as well as certain spatial models and mixed-effect models. Efficient Bayesian inference of LGMs often requires marginalizing out the latent variables. For LGMs with a non-Gaussian likelihood, exact marginalization is not possible and a popular approach is to do approximate marginalization with an integrated Laplace approximation (ILA). Using ILA produces an approximate posterior which, in some settings, can differ significantly from the correct posterior, which impacts downstream applications. We propose an importance sampling scheme to correct the error introduced by ILA. By increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior. This idea is realized with various techniques, including pseudo-marginalization, quasi-Monte Carlo and randomized quasi-Monte Carlo. We implement our methods in an automatic differentiation framework to support gradient-based algorithms when doing inference on the hyperparameters. For the latter, we specifically consider the use of Hamiltonian Monte Carlo. We demonstrate the benefits of reduced error in various applied models.