Song Xi Chen

Xinwen Liu, Lei Qian, Song Xi Chen et al.

Causal inference in settings involving complex spatio-temporal dependencies, such as environmental epidemiology, is challenging due to the presence of unmeasured confounding. However, a significant gap persists in existing methods: current diffusion-based causal models rely on restrictive assumptions of causal sufficiency or static confounding. To address this limitation, we introduce the Partially Functional Dynamic Backdoor Diffusion-based Causal Model (PFD-BDCM), a generative framework designed to bridge this gap. Our approach uniquely incorporates valid backdoor adjustments into the diffusion sampling mechanism to mitigate bias from unmeasured confounders. Specifically, it captures their intricate dynamics through region-specific structural equations and conditional autoregressive processes, and accommodates multi-resolution variables via functional data techniques. Furthermore, we provide theoretical guarantees by establishing error bounds for counterfactual estimates. Extensive experiments on synthetic data and a real-world air pollution case study confirm that PFD-BDCM outperforms current state-of-the-art methods.

Lei Qian, Wu Su, Yanqi Huang et al.

We propose a Likelihood Matching approach for training diffusion models by first establishing an equivalence between the likelihood of the target data distribution and a likelihood along the sample path of the reverse diffusion. To efficiently compute the reverse sample likelihood, a quasi-likelihood is considered to approximate each reverse transition density by a Gaussian distribution with matched conditional mean and covariance, respectively. The score and Hessian functions for the diffusion generation are estimated by maximizing the quasi-likelihood, ensuring a consistent matching of both the first two transitional moments between every two time points. A stochastic sampler is introduced to facilitate computation that leverages on both the estimated score and Hessian information. We establish consistency of the quasi-maximum likelihood estimation, and provide non-asymptotic convergence guarantees for the proposed sampler, quantifying the rates of the approximation errors due to the score and Hessian estimation, dimensionality, and the number of diffusion steps. Empirical and simulation evaluations demonstrate the effectiveness of the proposed Likelihood Matching and validate the theoretical results.

Zelin Xiao, Jia Gu, Song Xi Chen

We study methods for identifying heterogeneous parameter components in distributed M-estimation with minimal data transmission. One is based on a re-normalized Wald test, which is shown to be consistent as long as the number of distributed data blocks $K$ is of a smaller order of the minimum block sample size and the level of heterogeneity is dense. The second one is an extreme contrast test (ECT) based on the difference between the largest and smallest component-wise estimated parameters among data blocks. By introducing a sample splitting procedure, the ECT can avoid the bias accumulation arising from the M-estimation procedures, and exhibits consistency for $K$ being much larger than the sample size while the heterogeneity is sparse. The ECT procedure is easy to operate and communication-efficient. A combination of the Wald and the extreme contrast tests is formulated to attain more robust power under varying levels of sparsity of the heterogeneity. We also conduct intensive numerical experiments to compare the family-wise error rate (FWER) and the power of the proposed methods. Additionally, we conduct a case study to present the implementation and validity of the proposed methods.

Huiming Zhang, Song Xi Chen

This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses of mathematical statistics in a wide range of settings, from distribution-free to distribution-dependent, from sub-Gaussian to sub-exponential, sub-Gamma, and sub-Weibull random variables, and from the mean to the maximum concentration. This review provides results in these settings with some fresh new results. Given the increasing popularity of high-dimensional data and inference, results in the context of high-dimensional linear and Poisson regressions are also provided. We aim to illustrate the concentration inequalities with known constants and to improve existing bounds with sharper constants.

Xiaojun Mao, Raymond K. W. Wong, Song Xi Chen

Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion methods often assume a simple uniform missing mechanism. In this work, we study matrix completion from corrupted data under a novel low-rank missing mechanism. The probability matrix of observation is estimated via a high dimensional low-rank matrix estimation procedure, and further used to complete the target matrix via inverse probabilities weighting. Due to both high dimensional and extreme (i.e., very small) nature of the true probability matrix, the effect of inverse probability weighting requires careful study. We derive optimal asymptotic convergence rates of the proposed estimators for both the observation probabilities and the target matrix.