Wing Hung Wong

Qiao Liu, Zhongren Chen, Wing Hung Wong

Although understanding and characterizing causal effects have become essential in observational studies, it is challenging when the confounders are high-dimensional. In this article, we develop a general framework $\textit{CausalEGM}$ for estimating causal effects by encoding generative modeling, which can be applied in both binary and continuous treatment settings. Under the potential outcome framework with unconfoundedness, we establish a bidirectional transformation between the high-dimensional confounders space and a low-dimensional latent space where the density is known (e.g., multivariate normal distribution). Through this, CausalEGM simultaneously decouples the dependencies of confounders on both treatment and outcome and maps the confounders to the low-dimensional latent space. By conditioning on the low-dimensional latent features, CausalEGM can estimate the causal effect for each individual or the average causal effect within a population. Our theoretical analysis shows that the excess risk for CausalEGM can be bounded through empirical process theory. Under an assumption on encoder-decoder networks, the consistency of the estimate can be guaranteed. In a series of experiments, CausalEGM demonstrates superior performance over existing methods for both binary and continuous treatments. Specifically, we find CausalEGM to be substantially more powerful than competing methods in the presence of large sample sizes and high dimensional confounders. The software of CausalEGM is freely available at https://github.com/SUwonglab/CausalEGM.

Qiao Liu, Wing Hung Wong

Modern data analysis increasingly requires flexible conditional inference P(X_B | X_A) where (X_A, X_B) is an arbitrary partition of observed variable X. Existing conditional inference methods lack this flexibility as they are tied to a fixed conditioning structure and cannot perform new conditional inference once trained. To solve this, we propose a Bayesian generative modeling (BGM) approach for arbitrary conditional inference without retraining. BGM learns a generative model of X through an iterative Bayesian updating algorithm where model parameters and latent variables are updated until convergence. Once trained, any conditional distribution can be obtained without retraining. Empirically, BGM achieves superior prediction performance with well calibrated predictive intervals, demonstrating that a single learned model can serve as a universal engine for conditional prediction with uncertainty quantification. We provide theoretical guarantees for the convergence of the stochastic iterative algorithm, statistical consistency and conditional-risk bounds. The proposed BGM framework leverages the power of AI to capture complex relationships among variables while adhering to Bayesian principles, emerging as a promising framework for advancing various applications in modern data science. The code for BGM is freely available at https://github.com/liuq-lab/bayesgm.

Wenhui Sophia Lu, Wing Hung Wong

When the likelihood is analytically unavailable and computationally intractable, approximate Bayesian computation (ABC) has emerged as a widely used methodology for approximate posterior inference; however, it suffers from severe computational inefficiency in high-dimensional settings or under diffuse priors. To overcome these limitations, we propose Adaptive Bayesian Inference (ABI), a framework that bypasses traditional data-space discrepancies and instead compares distributions directly in posterior space through nonparametric distribution matching. By leveraging a novel Marginally-augmented Sliced Wasserstein (MSW) distance on posterior measures and exploiting its quantile representation, ABI transforms the challenging problem of measuring divergence between posterior distributions into a tractable sequence of one-dimensional conditional quantile regression tasks. Moreover, we introduce a new adaptive rejection sampling scheme that iteratively refines the posterior approximation by updating the proposal distribution via generative density estimation. Theoretically, we establish parametric convergence rates for the trimmed MSW distance and prove that the ABI posterior converges to the true posterior as the tolerance threshold vanishes. Through extensive empirical evaluation, we demonstrate that ABI significantly outperforms data-based Wasserstein ABC, summary-based ABC, and state-of-the-art likelihood-free simulators, especially in high-dimensional or dependent observation regimes.

Qiao Liu, Wing Hung Wong

Causal inference in observational studies with high-dimensional covariates presents significant challenges. We introduce CausalBGM, an AI-powered Bayesian generative modeling approach that captures the causal relationship among covariates, treatment, and outcome variables. The core innovation of CausalBGM lies in its ability to estimate the individual treatment effect (ITE) by learning individual-specific distributions of a low-dimensional latent feature set (e.g., latent confounders) that drives changes in both treatment and outcome. This approach not only effectively mitigates confounding effects but also provides comprehensive uncertainty quantification, offering reliable and interpretable causal effect estimates at the individual level. CausalBGM adopts a Bayesian model and uses a novel iterative algorithm to update the model parameters and the posterior distribution of latent features until convergence. This framework leverages the power of AI to capture complex dependencies among variables while adhering to the Bayesian principles. Extensive experiments demonstrate that CausalBGM consistently outperforms state-of-the-art methods, particularly in scenarios with high-dimensional covariates and large-scale datasets. Its Bayesian foundation ensures statistical rigor, providing robust and well-calibrated posterior intervals. By addressing key limitations of existing methods, CausalBGM emerges as a robust and promising framework for advancing causal inference in modern applications in fields such as genomics, healthcare, and social sciences. CausalBGM is maintained at the website https://causalbgm.readthedocs.io/.

Wenhui Sophia Lu, Chenyang Zhong, Wing Hung Wong

The generation of synthetic data with distributions that faithfully emulate the underlying data-generating mechanism holds paramount significance. Wasserstein Generative Adversarial Networks (WGANs) have emerged as a prominent tool for this task; however, due to the delicate equilibrium of the minimax formulation and the instability of Wasserstein distance in high dimensions, WGAN often manifests the pathological phenomenon of mode collapse. This results in generated samples that converge to a restricted set of outputs and fail to adequately capture the tail behaviors of the true distribution. Such limitations can lead to serious downstream consequences. To this end, we propose the Penalized Optimal Transport Network (POTNet), a versatile deep generative model based on the marginally-penalized Wasserstein (MPW) distance. Through the MPW distance, POTNet effectively leverages low-dimensional marginal information to guide the overall alignment of joint distributions. Furthermore, our primal-based framework enables direct evaluation of the MPW distance, thus eliminating the need for a critic network. This formulation circumvents training instabilities inherent in adversarial approaches and avoids the need for extensive parameter tuning. We derive a non-asymptotic bound on the generalization error of the MPW loss and establish convergence rates of the generative distribution learned by POTNet. Our theoretical analysis together with extensive empirical evaluations demonstrate the superior performance of POTNet in accurately capturing underlying data structures, including their tail behaviors and minor modalities. Moreover, our model achieves orders of magnitude speedup during the sampling stage compared to state-of-the-art alternatives, which enables computationally efficient large-scale synthetic data generation.

Qiao Liu, Jiaze Xu, Rui Jiang et al.

Density estimation is a fundamental problem in both statistics and machine learning. In this study, we proposed Roundtrip as a general-purpose neural density estimator based on deep generative models. Roundtrip retains the generative power of generative adversarial networks (GANs) but also provides estimates of density values. Unlike previous neural density estimators that put stringent conditions on the transformation from the latent space to the data space, Roundtrip enables the use of much more general mappings. In a series of experiments, Roundtrip achieves state-of-the-art performance in a diverse range of density estimation tasks.

Tung-Yu Wu, Y. X. Rachel Wang, Wing H. Wong

Traditional MCMC algorithms are computationally intensive and do not scale well to large data. In particular, the Metropolis-Hastings (MH) algorithm requires passing over the entire dataset to evaluate the likelihood ratio in each iteration. We propose a general framework for performing MH-MCMC using mini-batches of the whole dataset and show that this gives rise to approximately a tempered stationary distribution. We prove that the algorithm preserves the modes of the original target distribution and derive an error bound on the approximation with mild assumptions on the likelihood. To further extend the utility of the algorithm to high dimensional settings, we construct a proposal with forward and reverse moves using stochastic gradient and show that the construction leads to reasonable acceptance probabilities. We demonstrate the performance of our algorithm in both low dimensional models and high dimensional neural network applications. Particularly in the latter case, compared to popular optimization methods, our method is more robust to the choice of learning rate and improves testing accuracy.

Christopher De Sa, Vincent Chen, Wing Wong

Gibbs sampling is the de facto Markov chain Monte Carlo method used for inference and learning on large scale graphical models. For complicated factor graphs with lots of factors, the performance of Gibbs sampling can be limited by the computational cost of executing a single update step of the Markov chain. This cost is proportional to the degree of the graph, the number of factors adjacent to each variable. In this paper, we show how this cost can be reduced by using minibatching: subsampling the factors to form an estimate of their sum. We introduce several minibatched variants of Gibbs, show that they can be made unbiased, prove bounds on their convergence rates, and show that under some conditions they can result in asymptotic single-update-run-time speedups over plain Gibbs sampling.

Bai Jiang, Tung-yu Wu, Wing H. Wong

In our recent paper, we showed that in exponential family, contrastive divergence (CD) with fixed learning rate will give asymptotically consistent estimates \cite{wu2016convergence}. In this paper, we establish consistency and convergence rate of CD with annealed learning rate $η_t$. Specifically, suppose CD-$m$ generates the sequence of parameters $\{θ_t\}_{t \ge 0}$ using an i.i.d. data sample $\mathbf{X}_1^n \sim p_{θ^*}$ of size $n$, then $δ_n(\mathbf{X}_1^n) = \limsup_{t \to \infty} \Vert \sum_{s=t_0}^t η_s θ_s / \sum_{s=t_0}^t η_s - θ^* \Vert$ converges in probability to 0 at a rate of $1/\sqrt[3]{n}$. The number ($m$) of MCMC transitions in CD only affects the coefficient factor of convergence rate. Our proof is not a simple extension of the one in \cite{wu2016convergence}. which depends critically on the fact that $\{θ_t\}_{t \ge 0}$ is a homogeneous Markov chain conditional on the observed sample $\mathbf{X}_1^n$. Under annealed learning rate, the homogeneous Markov property is not available and we have to develop an alternative approach based on super-martingales. Experiment results of CD on a fully-visible $2\times 2$ Boltzmann Machine are provided to demonstrate our theoretical results.

Bai Jiang, Tung-Yu Wu, Yifan Jin et al.

The Contrastive Divergence (CD) algorithm has achieved notable success in training energy-based models including Restricted Boltzmann Machines and played a key role in the emergence of deep learning. The idea of this algorithm is to approximate the intractable term in the exact gradient of the log-likelihood function by using short Markov chain Monte Carlo (MCMC) runs. The approximate gradient is computationally-cheap but biased. Whether and why the CD algorithm provides an asymptotically consistent estimate are still open questions. This paper studies the asymptotic properties of the CD algorithm in canonical exponential families, which are special cases of the energy-based model. Suppose the CD algorithm runs $m$ MCMC transition steps at each iteration $t$ and iteratively generates a sequence of parameter estimates $\{θ_t\}_{t \ge 0}$ given an i.i.d. data sample $\{X_i\}_{i=1}^n \sim p_{θ_\star}$. Under conditions which are commonly obeyed by the CD algorithm in practice, we prove the existence of some bounded $m$ such that any limit point of the time average $\left. \sum_{s=0}^{t-1} θ_s \right/ t$ as $t \to \infty$ is a consistent estimate for the true parameter $θ_\star$. Our proof is based on the fact that $\{θ_t\}_{t \ge 0}$ is a homogenous Markov chain conditional on the data sample $\{X_i\}_{i=1}^n$. This chain meets the Foster-Lyapunov drift criterion and converges to a random walk around the Maximum Likelihood Estimate. The range of the random walk shrinks to zero at rate $\mathcal{O}(1/\sqrt[3]{n})$ as the sample size $n \to \infty$.

Dangna Li, Kun Yang, Wing Hung Wong

Given $iid$ observations from an unknown absolute continuous distribution defined on some domain $Ω$, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function. Our density estimate is a piecewise constant function defined on a binary partition of $Ω$. The key ingredient of the algorithm is to use discrepancy, a concept originates from Quasi Monte Carlo analysis, to control the partition process. The resulting algorithm is simple, efficient, and has a provable convergence rate. We empirically demonstrate its efficiency as a density estimation method. We present its applications on a wide range of tasks, including finding good initializations for k-means.