In generative models with obscured likelihood, Approximate Bayesian Computation (ABC) is often the tool of last resort for inference. However, ABC demands many prior parameter trials to keep only a small fraction that passes an acceptance test. To accelerate ABC rejection sampling, this paper develops a self-aware framework that learns from past trials and errors. We apply recursive partitioning classifiers on the ABC lookup table to sequentially refine high-likelihood regions into boxes. Each box is regarded as an arm in a binary bandit problem treating ABC acceptance as a reward. Each arm has a proclivity for being chosen for the next ABC evaluation, depending on the prior distribution and past rejections. The method places more splits in those areas where the likelihood resides, shying away from low-probability regions destined for ABC rejections. We provide two versions: (1) ABC-Tree for posterior sampling, and (2) ABC-MAP for maximum a posteriori estimation. We demonstrate accurate ABC approximability at much lower simulation cost. We justify the use of our tree-based bandit algorithms with nearly optimal regret bounds. Finally, we successfully apply our approach to the problem of masked image classification using deep generative models.

Clinical data informs the personalization of health care with a potential for more effective disease management. In practice, this is achieved by subgrouping, whereby clusters with similar patient characteristics are identified and then receive customized treatment plans with the goal of targeting subgroup-specific disease dynamics. In this paper, we propose a novel mixture hidden Markov model for subgrouping patient trajectories from chronic diseases. Our model is probabilistic and carefully designed to capture different trajectory phases of chronic diseases (i.e., "severe", "moderate", and "mild") through tailored latent states. We demonstrate our subgrouping framework based on a longitudinal study across 847 patients with non-specific low back pain. Here, our subgrouping framework identifies 8 subgroups. Further, we show that our subgrouping framework outperforms common baselines in terms of cluster validity indices. Finally, we discuss the applicability of the model to other chronic and long-lasting diseases.

Learning from human feedback plays an important role in aligning generative models, such as large language models (LLM). However, the effectiveness of this approach can be influenced by adversaries, who may intentionally provide misleading preferences to manipulate the output in an undesirable or harmful direction. To tackle this challenge, we study a specific model within this problem domain--contextual dueling bandits with adversarial feedback, where the true preference label can be flipped by an adversary. We propose an algorithm namely robust contextual dueling bandit (\algo), which is based on uncertainty-weighted maximum likelihood estimation. Our algorithm achieves an $\tilde O(d\sqrt{T}+dC)$ regret bound, where $T$ is the number of rounds, $d$ is the dimension of the context, and $ 0 \le C \le T$ is the total number of adversarial feedback. We also prove a lower bound to show that our regret bound is nearly optimal, both in scenarios with and without ($C=0$) adversarial feedback. Additionally, we conduct experiments to evaluate our proposed algorithm against various types of adversarial feedback. Experimental results demonstrate its superiority over the state-of-the-art dueling bandit algorithms in the presence of adversarial feedback.

Classic Bayesian methods with complex models are frequently infeasible due to an intractable likelihood. Simulation-based inference methods, such as Approximate Bayesian Computing (ABC), calculate posteriors without accessing a likelihood function by leveraging the fact that data can be quickly simulated from the model, but converge slowly and/or poorly in high-dimensional settings. In this paper, we propose a framework for Bayesian posterior estimation by mapping data to posteriors of parameters using a neural network trained on data simulated from the complex model. Posterior distributions of model parameters are efficiently obtained by feeding observed data into the trained neural network. We show theoretically that our posteriors converge to the true posteriors in Kullback-Leibler divergence. Our approach yields computationally efficient and theoretically justified uncertainty quantification, which is lacking in existing simulation-based neural network approaches. Comprehensive simulation studies highlight our method's robustness and accuracy.

We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

We propose an analytical solution for approximating the gradient of the Evidence Lower Bound (ELBO) in variational inference problems where the statistical model is a Bayesian network consisting of observations drawn from a mixture of a Gaussian distribution embedded in unrelated clutter, known as the clutter problem. The method employs the reparameterization trick to move the gradient operator inside the expectation and relies on the assumption that, because the likelihood factorizes over the observed data, the variational distribution is generally more compactly supported than the Gaussian distribution in the likelihood factors. This allows efficient local approximation of the individual likelihood factors, which leads to an analytical solution for the integral defining the gradient expectation. We integrate the proposed gradient approximation as the expectation step in an EM (Expectation Maximization) algorithm for maximizing ELBO and test against classical deterministic approaches in Bayesian inference, such as the Laplace approximation, Expectation Propagation and Mean-Field Variational Inference. The proposed method demonstrates good accuracy and rate of convergence together with linear computational complexity.

With the evolution of single-cell RNA sequencing techniques into a standard approach in genomics, it has become possible to conduct cohort-level causal inferences based on single-cell-level measurements. However, the individual gene expression levels of interest are not directly observable; instead, only repeated proxy measurements from each individual's cells are available, providing a derived outcome to estimate the underlying outcome for each of many genes. In this paper, we propose a generic semiparametric inference framework for doubly robust estimation with multiple derived outcomes, which also encompasses the usual setting of multiple outcomes when the response of each unit is available. To reliably quantify the causal effects of heterogeneous outcomes, we specialize the analysis to standardized average treatment effects and quantile treatment effects. Through this, we demonstrate the use of the semiparametric inferential results for doubly robust estimators derived from both Von Mises expansions and estimating equations. A multiple testing procedure based on Gaussian multiplier bootstrap is tailored for doubly robust estimators to control the false discovery exceedance rate. Applications in single-cell CRISPR perturbation analysis and individual-level differential expression analysis demonstrate the utility of the proposed methods and offer insights into the usage of different estimands for causal inference in genomics.

Variational inference (VI) has emerged as a popular method for approximate inference for high-dimensional Bayesian models. In this paper, we propose a novel VI method that extends the naive mean field via entropic regularization, referred to as $\Xi$-variational inference ($\Xi$-VI). $\Xi$-VI has a close connection to the entropic optimal transport problem and benefits from the computationally efficient Sinkhorn algorithm. We show that $\Xi$-variational posteriors effectively recover the true posterior dependency, where the dependence is downweighted by the regularization parameter. We analyze the role of dimensionality of the parameter space on the accuracy of $\Xi$-variational approximation and how it affects computational considerations, providing a rough characterization of the statistical-computational trade-off in $\Xi$-VI. We also investigate the frequentist properties of $\Xi$-VI and establish results on consistency, asymptotic normality, high-dimensional asymptotics, and algorithmic stability. We provide sufficient criteria for achieving polynomial-time approximate inference using the method. Finally, we demonstrate the practical advantage of $\Xi$-VI over mean-field variational inference on simulated and real data.

Spectral Graph Neural Networks (GNNs) have attracted great attention due to their capacity to capture patterns in the frequency domains with essential graph filters. Polynomial-based ones (namely poly-GNNs), which approximately construct graph filters with conventional or rational polynomials, are routinely adopted in practice for their substantial performances on graph learning tasks. However, previous poly-GNNs aim at achieving overall lower approximation error on different types of filters, e.g., low-pass and high-pass, but ignore a key question: \textit{which type of filter warrants greater attention for poly-GNNs?} In this paper, we first show that poly-GNN with a better approximation for band-pass graph filters performs better on graph learning tasks. This insight further sheds light on critical issues of existing poly-GNNs, i.e., those poly-GNNs achieve trivial performance in approximating band-pass graph filters, hindering the great potential of poly-GNNs. To tackle the issues, we propose a novel poly-GNN named TrigoNet. TrigoNet constructs different graph filters with novel trigonometric polynomial, and achieves leading performance in approximating band-pass graph filters against other polynomials. By applying Taylor expansion and deserting nonlinearity, TrigoNet achieves noticeable efficiency among baselines. Extensive experiments show the advantages of TrigoNet in both accuracy performances and efficiency.

Epistemic uncertainty quantification (UQ) identifies where models lack knowledge. Traditional UQ methods, often based on Bayesian neural networks, are not suitable for pre-trained non-Bayesian models. Our study addresses quantifying epistemic uncertainty for any pre-trained model, which does not need the original training data or model modifications and can ensure broad applicability regardless of network architectures or training techniques. Specifically, we propose a gradient-based approach to assess epistemic uncertainty, analyzing the gradients of outputs relative to model parameters, and thereby indicating necessary model adjustments to accurately represent the inputs. We first explore theoretical guarantees of gradient-based methods for epistemic UQ, questioning the view that this uncertainty is only calculable through differences between multiple models. We further improve gradient-driven UQ by using class-specific weights for integrating gradients and emphasizing distinct contributions from neural network layers. Additionally, we enhance UQ accuracy by combining gradient and perturbation methods to refine the gradients. We evaluate our approach on out-of-distribution detection, uncertainty calibration, and active learning, demonstrating its superiority over current state-of-the-art UQ methods for pre-trained models.