Journal of Data Science

We propose a differentially private Bayesian framework for envelope regression, a technique that improves estimation efficiency by modelling the response as a function of a low-dimensional subspace of the predictors. Our method applies the analytic Gaussian mechanism to privatize sufficient statistics from the data, ensuring formal $(\epsilon ,\delta )$-differential privacy. We develop a tailored Gibbs sampling algorithm that performs valid Bayesian inference using only the noisy sufficient statistics. This approach leverages the envelope structure to isolate the variation in predictors that is relevant to the response, reducing estimation error compared to standard regression under the same privacy constraints. Through simulation studies, we demonstrate improved estimation accuracy and tighter credible intervals relative to a differentially private Bayesian linear regression baseline.

Abstract: Kang (2006) used the log-likelihood function with Lagrangian multipliers for estimation of cell probabilities in two-way incomplete contingency tables. The constraints on cell probabilities can be incorporated through Lagrangian multipliers for the likelihood function. The method can be readily extended to multidimensional tables. Variances of the MLEs are derived from the matrix of second derivatives of the log likelihood with respect to cell probabilities and the Lagrange multiplier. Wald and likelihood ratio tests of independence are derived using the estimates and estimated variances. Simulation results, when data are missing at random, reveal that maximum likelihood estimation (MLE) produces more efficient estimates of population proportions than either multiple imputation (MI) based on data augmentation or complete case (CC) analysis. Neither MLE nor MI, however, leads to an improvement over CC analysis with respect to power of tests for independence in 2×2 tables. Thus, the partially classified marginal information increases precision about proportions, but is not helpful for judging independence.

Abstract: The null distribution of the likelihood ratio test (LRT) of a onecomponent normal model versus two-component normal mixture model is

Cognitive Diagnosis Models (CDMs) are a special family of discrete latent variable models widely used in educational, psychological and social sciences. In many applications of CDMs, certain hierarchical structures among the latent attributes are assumed by researchers to characterize their dependence structure. Specifically, a directed acyclic graph is used to specify hierarchical constraints on the allowable configurations of the discrete latent attributes. In this paper, we consider the important yet unaddressed problem of testing the existence of latent hierarchical structures in CDMs. We first introduce the concept of testability of hierarchical structures in CDMs and present sufficient conditions. Then we study the asymptotic behaviors of the likelihood ratio test (LRT) statistic, which is widely used for testing nested models. Due to the irregularity of the problem, the asymptotic distribution of LRT becomes nonstandard and tends to provide unsatisfactory finite sample performance under practical conditions. We provide statistical insights on such failures, and propose to use parametric bootstrap to perform the testing. We also demonstrate the effectiveness and superiority of parametric bootstrap for testing the latent hierarchies over non-parametric bootstrap and the naïve Chi-squared test through comprehensive simulations and an educational assessment dataset.