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Abstract When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how the perturbation of MCMC affects the convergence speed and approximation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has fast convergence speed and high approximation accuracy as well. Our convergence analysis is conducted under either the Wasserstein metric or the$$\chi^2$$metric, both are widely used in the literature. The results can be extended to obtain non-asymptotic error bounds for MCMC estimators. We demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis, Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally, we present some simple numerical examples to verify our theoretical claims. 

Abstract This article develops a conformal prediction method for classification tasks that can adapt to random label contamination in the calibration sample, often leading to more informative prediction sets with stronger coverage guarantees compared to existing approaches. This is obtained through a precise characterization of the coverage inflation (or deflation) suffered by standard conformal inferences in the presence of label contamination, which is then made actionable through a new calibration algorithm. Our solution can leverage different modelling assumptions about the contamination process, while requiring no knowledge of the underlying data distribution or of the inner workings of the classification model. The empirical performance of the proposed method is demonstrated through simulations and an application to object classification with the CIFAR-10H image data set. 

Abstract The inherent qualitative nature of textual data poses significant challenges for direct integration into statistical models. This paper presents a two-stage process for analyzing longitudinal textual data, offering a solution to this inherent challenge. The proposed model comprises (1) initial data preprocessing and sentiment extraction, followed by (2) applying a growth curve model to analyze the extracted sentiments directly. The paper also explores four distinct approaches for extracting sentiment scores in the dialogue, providing versatility to the proposed framework. The practical application of the proposed model is demonstrated through the analysis of an empirical longitudinal textual dataset. This framework offers a valuable contribution to the field by addressing the challenges associated with modeling qualitative textual data, providing a robust methodology for extracting and analyzing sentiments longitudinally. 

Montessori pedagogy is a century-old, whole-school system increasingly used in the public sector. In the United States, public Montessori schools are typically Title I schools that mostly serve children of color. The present secondary, exploratory data analysis examined outcomes of 134 children who entered a lottery for admission to public Montessori schools in the northeastern United States at age 3; half were admitted and enrolled and the rest enrolled at other preschool programs. About half of the children were identified as White, and half were identified as African American, Hispanic, or multiracial. Children were tested in the fall when they enrolled and again in the subsequent three springs (i.e., through the kindergarten year) on a range of measures addressing academic outcomes, executive function, and social cognition. Although the Black, Hispanic, and multiracial group tended to score lower in the beginning of preschool in both conditions, by the end of preschool, the scores of Black, Hispanic, and multiracial students enrolled in Montessori schools were not different from the White children; by contrast, such students in the business-as-usual schools continued to perform less well than White children in academic achievement and social cognition. The study has important limitations that lead us to view these findings as exploratory, but taken together with other findings, the results suggest that Montessori education may create an environment that is more conducive to racial and ethnic parity than other school environments.