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While fiducial inference was widely considered a big blunder by R.A. Fisher, the goal he initially set—‘inferring the uncertainty of model parameters on the basis of observations’—has been continually pursued by many statisticians. To this end, we develop a new statistical inference method called extended Fiducial inference (EFI). The new method achieves the goal of fiducial inference by leveraging advanced statistical computing techniques while remaining scalable for big data. Extended Fiducial inference involves jointly imputing random errors realized in observations using stochastic gradient Markov chain Monte Carlo and estimating the inverse function using a sparse deep neural network (DNN). The consistency of the sparse DNN estimator ensures that the uncertainty embedded in observations is properly propagated to model parameters through the estimated inverse function, thereby validating downstream statistical inference. Compared to frequentist and Bayesian methods, EFI offers significant advantages in parameter estimation and hypothesis testing. Specifically, EFI provides higher fidelity in parameter estimation, especially when outliers are present in the observations; and eliminates the need for theoretical reference distributions in hypothesis testing, thereby automating the statistical inference process. Extended Fiducial inference also provides an innovative framework for semisupervised learning.

 

This article presents a novel method for learning time‐varying dynamic Bayesian networks. The proposed method breaks down the dynamic Bayesian network learning problem into a sequence of regression inference problems and tackles each problem using the Markov neighborhood regression technique. Notably, the method demonstrates scalability concerning data dimensionality, accommodates time‐varying network structure, and naturally handles multi‐subject data. The proposed method exhibits consistency and offers superior performance compared to existing methods in terms of estimation accuracy and computational efficiency, as supported by extensive numerical experiments. To showcase its effectiveness, we apply the proposed method to an fMRI study investigating the effective connectivity among various regions of interest (ROIs) during an emotion‐processing task. Our findings reveal the pivotal role of the subcortical‐cerebellum in emotion processing.

 

With the advancement of data science, the collection of increasingly complex datasets has become commonplace. In such datasets, the data dimension can be extremely high, and the underlying data generation process can be unknown and highly nonlinear. As a result, the task of making causal inference with high-dimensional complex data has become a fundamental problem in many disciplines, such as medicine, econometrics, and social science. However, the existing methods for causal inference are frequently developed under the assumption that the data dimension is low or that the underlying data generation process is linear or approximately linear. To address these challenges, this paper proposes a novel causal inference approach for dealing with high-dimensional complex data. The proposed approach is based on deep learning techniques, including sparse deep learning theory and stochastic neural networks, that have been developed in recent literature. By using these techniques, the proposed approach can address both the high dimensionality and unknown data generation process in a coherent way. Furthermore, the proposed approach can also be used when missing values are present in the datasets. Extensive numerical studies indicate that the proposed approach outperforms existing ones. 

Reinforcement learning (RL) tackles sequential decision-making problems by creating agents that interacts with their environment. However, existing algorithms often view these problem as static, focusing on point estimates for model parameters to maximize expected rewards, neglecting the stochastic dynamics of agent-environment interactions and the critical role of uncertainty quantification. Our research leverages the Kalman filtering paradigm to introduce a novel and scalable sampling algorithm called Langevinized Kalman Temporal-Difference (LKTD) for deep reinforcement learning. This algorithm, grounded in Stochastic Gradient Markov Chain Monte Carlo (SGMCMC), efficiently draws samples from the posterior distribution of deep neural network parameters. Under mild conditions, we prove that the posterior samples generated by the LKTD algorithm converge to a stationary distribution. This convergence not only enables us to quantify uncertainties associated with the value function and model parameters but also allows us to monitor these uncertainties during policy updates throughout the training phase. The LKTD algorithm paves the way for more robust and adaptable reinforcement learning approaches. 

Parallel tempering (PT), also known as replica exchange, is the go-to workhorse for simulations of multi-modal distributions. The key to the success of PT is to adopt efficient swap schemes. The popular deterministic even-odd (DEO) scheme exploits the non-reversibility property and has successfully reduced the communication cost from quadratic to linear given the sufficiently many chains. However, such an innovation largely disappears in big data due to the limited chains and few bias-corrected swaps. To handle this issue, we generalize the DEO scheme to promote non-reversibility and propose a few solutions to tackle the underlying bias caused by the geometric stopping time. Notably, in big data scenarios, we obtain a nearly linear communication cost based on the optimal window size. In addition, we also adopt stochastic gradient descent (SGD) with large and constant learning rates as exploration kernels. Such a user-friendly nature enables us to conduct approximation tasks for complex posteriors without much tuning costs.