This article presents a novel method for causal discovery with generalized structural equation models suited for analyzing diverse types of outcomes, including discrete, continuous, and mixed data. Causal discovery often faces challenges due to unmeasured confounders that hinder the identification of causal relationships. The proposed approach addresses this issue by developing two peeling algorithms (bottom-up and top-down) to ascertain causal relationships and valid instruments. This approach first reconstructs a super-graph to represent ancestral relationships between variables, using a peeling algorithm based on nodewise GLM regressions that exploit relationships between primary and instrumental variables. Then, it estimates parent-child effects from the ancestral relationships using another peeling algorithm while deconfounding a child's model with information borrowed from its parents' models. The article offers a theoretical analysis of the proposed approach, which establishes conditions for model identifiability and provides statistical guarantees for accurately discovering parent-child relationships via the peeling algorithms. Furthermore, the article presents numerical experiments showcasing the effectiveness of our approach in comparison to state-of-the-art structure learning methods without confounders. Lastly, it demonstrates an application to Alzheimer's disease (AD), highlighting the utility of the method in constructing gene-to-gene and gene-to-disease regulatory networks involving Single Nucleotide Polymorphisms (SNPs) for healthy and AD subjects.

Both approaches enable sampling and density estimation of conditional probability distributions, which are core tasks in Bayesian inference. Our methods represent the target conditional distributions as transformations of a tractable reference distribution and, therefore, fall into the framework of measure transport. COT maps are a canonical choice within this framework, with desirable properties such as uniqueness and monotonicity. However, the associated COT problems are computationally challenging, even in moderate dimensions. To improve the scalability, our numerical algorithms leverage neural networks to parameterize COT maps. Our methods exploit the structure of the static and dynamic formulations of the COT problem. PCP-Map models conditional transport maps as the gradient of a partially input convex neural network (PICNN) and uses a novel numerical implementation to increase computational efficiency compared to state-of-the-art alternatives. COT-Flow models conditional transports via the flow of a regularized neural ODE; it is slower to train but offers faster sampling. We demonstrate their effectiveness and efficiency by comparing them with state-of-the-art approaches using benchmark datasets and Bayesian inverse problems.

We study empirical Bayes estimation in high-dimensional linear regression. To facilitate computationally efficient estimation of the underlying prior, we adopt a variational empirical Bayes approach, introduced originally in Carbonetto and Stephens (2012) and Kim et al. (2022). We establish asymptotic consistency of the nonparametric maximum likelihood estimator (NPMLE) and its (computable) naive mean field variational surrogate under mild assumptions on the design and the prior. Assuming, in addition, that the naive mean field approximation has a dominant optimizer, we develop a computationally efficient approximation to the oracle posterior distribution, and establish its accuracy under the 1-Wasserstein metric. This enables computationally feasible Bayesian inference; e.g., construction of posterior credible intervals with an average coverage guarantee, Bayes optimal estimation for the regression coefficients, estimation of the proportion of non-nulls, etc. Our analysis covers both deterministic and random designs, and accommodates correlations among the features. To the best of our knowledge, this provides the first rigorous nonparametric empirical Bayes method in a high-dimensional regression setting without sparsity.

A fundamental problem of causal discovery is cause-effect inference, learning the correct causal direction between two random variables. Significant progress has been made through modelling the effect as a function of its cause and a noise term, which allows us to leverage assumptions about the generating function class. The recently introduced heteroscedastic location-scale noise functional models (LSNMs) combine expressive power with identifiability guarantees. LSNM model selection based on maximizing likelihood achieves state-of-the-art accuracy, when the noise distributions are correctly specified. However, through an extensive empirical evaluation, we demonstrate that the accuracy deteriorates sharply when the form of the noise distribution is misspecified by the user. Our analysis shows that the failure occurs mainly when the conditional variance in the anti-causal direction is smaller than that in the causal direction. As an alternative, we find that causal model selection through residual independence testing is much more robust to noise misspecification and misleading conditional variance.

Adaptive importance sampling (AIS) algorithms are widely used to approximate expectations with respect to complicated target probability distributions. When the target has heavy tails, existing AIS algorithms can provide inconsistent estimators or exhibit slow convergence, as they often neglect the target's tail behaviour. To avoid this pitfall, we propose an AIS algorithm that approximates the target by Student-t proposal distributions. We adapt location and scale parameters by matching the escort moments - which are defined even for heavy-tailed distributions - of the target and the proposal. These updates minimize the $\alpha$-divergence between the target and the proposal, thereby connecting with variational inference. We then show that the $\alpha$-divergence can be approximated by a generalized notion of effective sample size and leverage this new perspective to adapt the tail parameter with Bayesian optimization. We demonstrate the efficacy of our approach through applications to synthetic targets and a Bayesian Student-t regression task on a real example with clinical trial data.

Normalizing Flows are generative models that directly maximize the likelihood. Previously, the design of normalizing flows was largely constrained by the need for analytical invertibility. We overcome this constraint by a training procedure that uses an efficient estimator for the gradient of the change of variables formula. This enables any dimension-preserving neural network to serve as a generative model through maximum likelihood training. Our approach allows placing the emphasis on tailoring inductive biases precisely to the task at hand. Specifically, we achieve excellent results in molecule generation benchmarks utilizing $E(n)$-equivariant networks. Moreover, our method is competitive in an inverse problem benchmark, while employing off-the-shelf ResNet architectures.

The Laplacian-constrained Gaussian Markov Random Field (LGMRF) is a common multivariate statistical model for learning a weighted sparse dependency graph from given data. This graph learning problem can be formulated as a maximum likelihood estimation (MLE) of the precision matrix, subject to Laplacian structural constraints, with a sparsity-inducing penalty term. This paper aims to solve this learning problem accurately and efficiently. First, since the commonly used $\ell_1$-norm penalty is inappropriate in this setting and may lead to a complete graph, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to existing first-order methods for this problem, we develop a second-order proximal Newton approach to obtain an efficient solver, utilizing several algorithmic features, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of both computational complexity and graph learning accuracy compared to existing methods.

Bayesian inference with deep generative prior has received considerable interest for solving imaging inverse problems in many scientific and engineering fields. The selection of the prior distribution is learned from, and therefore an important representation learning of, available prior measurements. The SA-Roundtrip, a novel deep generative prior, is introduced to enable controlled sampling generation and identify the data's intrinsic dimension. This prior incorporates a self-attention structure within a bidirectional generative adversarial network. Subsequently, Bayesian inference is applied to the posterior distribution in the low-dimensional latent space using the Hamiltonian Monte Carlo with preconditioned Crank-Nicolson (HMC-pCN) algorithm, which is proven to be ergodic under specific conditions. Experiments conducted on computed tomography (CT) reconstruction with the MNIST and TomoPhantom datasets reveal that the proposed method outperforms state-of-the-art comparisons, consistently yielding a robust and superior point estimator along with precise uncertainty quantification.

In many machine learning applications, well-calibrated posterior predictive distributions are required for a number of closely-related datasets. Given similarity between datasets, it is natural to wish to develop meta-learning algorithms that utilise other datasets to reduce the computational complexity and / or improve predictive performance when deploying models on newly-seen datasets at test time. There have been a number of significant recent developments in meta-learning for predictive distributions, most notably that of the neural process (NP) family (Garnelo et al., 2018a,b; Foong et al., 2020; Gordon et al., 2018, 2019). Despite the utility of these methods on large-scale meta-datasets, they perform poorly in settings where the number of datasets and the total number of datapoints is small. We argue that this is a result of the large number of shared model parameters overfitting to the meta-dataset.

A topic of great current interest is Causal Representation Learning (CRL), whose goal is to learn a causal model for hidden features in a data-driven manner. Unfortunately, CRL is severely ill-posed since it is a combination of the two notoriously ill-posed problems of representation learning and causal discovery. Yet, finding practical identifiability conditions that guarantee a unique solution is crucial for its practical applicability. Most approaches so far have been based on assumptions on the latent causal mechanisms, such as temporal causality, or existence of supervision or interventions; these can be too restrictive in actual applications. Here, we show identifiability based on novel, weak constraints, which requires no temporal structure, intervention, nor weak supervision. The approach is based assuming the observational mixing exhibits a suitable grouping of the observational variables. We also propose a novel self-supervised estimation framework consistent with the model, prove its statistical consistency, and experimentally show its superior CRL performances compared to the state-of-the-art baselines. We further demonstrate its robustness against latent confounders and causal cycles.