As sample sizes grow, scalability has become a central concern in the development of Markov chain Monte Carlo (MCMC) methods. One general approach to this problem, exemplified by the popular stochastic gradient Langevin dynamics (SGLD) algorithm, is to use a small random subsample of the data at every time step. This paper shows that this approach often fails: while decreasing the sample size increases the speed of each MCMC step, for typical datasets this is balanced by a matching decrease in accuracy. This result complements recent work on lower bounds for subsampling MCMC by focusing on actual errors for SGLD, including non-reversible algorithms, and by giving assumptions that are comparatively easy to verify.

Multiple response (MR) items—such as multiple true-false, multiple-select, and select-N items—are increasingly used in assessments to identify partial knowledge and differentiate latent abilities more accurately. Allowing multiple selections, MR items provide richer information and reduce guessing effects compared to single-answer multiple-choice items. However, traditional scoring methods (e.g., Dichotomous, Ripkey, Partial scoring) compress response combination (RC) data, losing valuable information and ignoring issues like local dependence and incompatibility across item types. To address these challenges, we introduce a novel psychometric model framework: the Multiple Response Model with Inter-option Local Dependencies (MRM-LD), and its simplified version, the Multiple Response Model (MRM). These models preserve RC data across MR item types, offering a more comprehensive understanding for MR assessment. Parameters for MRM-LD and MRM were estimated using Markov chain Monte Carlo algorithms in Stan and R. Empirical data from an eighth-grade physics test showed that MRM-LD and MRM outperform Graded Response Model and Nominal Response Model combined with three scoring methods, by retaining more test information, improving reliability and validity, and providing more detailed analysis of item characteristics. Simulation studies confirmed the proposed models perform robustly under various conditions, including small samples and few items, demonstrating their applicability across diverse testing scenarios.

Let $\pi$ be a probability distribution in $\mathbb{R}^d$ and f a test function, and consider the problem of variance reduction in estimating $\mathbb{E}_\pi(f)$. We first construct a sequence of estimators for $\mathbb{E}_\pi (f)$, say $({1}/{k})\sum_{i=0}^{k-1} g_n(X_i)$, where the $X_i$ are samples from $\pi$ generated by the Metropolized Hamiltonian Monte Carlo algorithm and $g_n$ is the approximate solution of the Poisson equation through the weak approximate scheme recently invented by Mijatović and Vogrinc (2018). Then we prove under some regularity assumptions that the estimation error variance $\sigma_\pi^2(g_n)$ can be as arbitrarily small as the approximation order parameter $n\rightarrow\infty$. To illustrate, we confirm that the assumptions are satisfied by two typical concrete models, a Bayesian linear inverse problem and a two-component mixture of Gaussian distributions.

We study the Markov chain Monte Carlo estimator for numerical integration for functions that do not need to be square integrable with respect to the invariant distribution. For chains with a spectral gap we show that the absolute mean error for $L^p$ functions, with $p \in (1,2)$, decreases like $n^{({1}/{p}) -1}$, which is known to be the optimal rate. This improves currently known results where an additional parameter $\delta \gt 0$ appears and the convergence is of order $n^{(({1+\delta})/{p})-1}$.

In this paper, we propose a novel and highly effective variational Bayesian expectation maximization-maximization (VBEM-M) inference method for log-linear cognitive diagnostic model (CDM). In the implementation of the variational Bayesian approach for the saturated log-linear CDM, the conditional variational posteriors of the parameters that need to be derived are in the same distributional family as the priors, the VBEM-M algorithm overcomes this problem. Our algorithm can directly estimate the item parameters and the latent attribute-mastery pattern simultaneously. In contrast, Yamaguchi and Okada’s (2020a) variational Bayesian algorithm requires a transformation step to obtain the item parameters for the log-linear cognitive diagnostic model (LCDM). We conducted multiple simulation studies to assess the performance of the VBEM-M algorithm in terms of parameter recovery, execution time, and convergence rate. Furthermore, we conducted a series of comparative studies on the accuracy of parameter estimation for the DINA model and the saturated LCDM, focusing on the VBEM-M, VB, expectation-maximization, and Markov chain Monte Carlo algorithms. The results indicated that our method can obtain more stable and accurate estimates, especially for the small sample sizes. Finally, we demonstrated the utility of the proposed algorithm using two real datasets.

We present an hierarchical Bayes approach to modeling parameter heterogeneity in generalized linear models. The model assumes that there are relevant subpopulations and that within each subpopulation the individual-level regression coefficients have a multivariate normal distribution. However, class membership is not known a priori, so the heterogeneity in the regression coefficients becomes a finite mixture of normal distributions. This approach combines the flexibility of semiparametric, latent class models that assume common parameters for each sub-population and the parsimony of random effects models that assume normal distributions for the regression parameters. The number of subpopulations is selected to maximize the posterior probability of the model being true. Simulations are presented which document the performance of the methodology for synthetic data with known heterogeneity and number of sub-populations. An application is presented concerning preferences for various aspects of personal computers.

We propose a nonparametric item response theory model for dichotomously-scored items in a Bayesian framework. The model is based on a latent class (LC) formulation, and it is multidimensional, with dimensions corresponding to a partition of the items in homogenous groups that are specified on the basis of inequality constraints among the conditional success probabilities given the latent class. Moreover, an innovative system of prior distributions is proposed following the encompassing approach, in which the largest model is the unconstrained LC model. A reversible-jump type algorithm is described for sampling from the joint posterior distribution of the model parameters of the encompassing model. By suitably post-processing its output, we then make inference on the number of dimensions (i.e., number of groups of items measuring the same latent trait) and we cluster items according to the dimensions when unidimensionality is violated. The approach is illustrated by two examples on simulated data and two applications based on educational and quality-of-life data.

This article devises a Bayesian multivariate formulation for analysis of ordinal data that records teacher classroom performance along multiple dimensions to assess aspects characterizing good instruction. Study designs for scoring teachers seek to measure instructional performance over multiple classroom measurement event sessions at varied occasions using disjoint intervals within each session and employment of multiple ratings on intervals scored by different raters; a design which instantiates a nesting structure with each level contributing a source of variation in recorded scores. We generally possess little a priori knowledge of the existence or form of a sparse generating structure for the multivariate dimensions at any level in the nesting that would permit collapsing over dimensions as is done under univariate modeling. Our approach composes a Bayesian data augmentation scheme that introduces a latent continuous multivariate response linked to the observed ordinal scores with the latent response mean constructed as an additive multivariate decomposition of nested level means that permits the extraction of de-noised continuous teacher-level scores and the associated correlation matrix. A semi-parametric extension facilitates inference for teacher-level dependence among the dimensions of classroom performance under multi-modality induced by sub-groupings of rater perspectives. We next replace an inverse Wishart prior specified for the teacher covariance matrix over dimensions of instruction with a factor analytic structure to allow the simultaneous assessment of an underlying sparse generating structure. Our formulation for Bayesian factor analysis employs parameter expansion with an accompanying post-processing sign re-labeling step of factor loadings that together reduce posterior correlations among sampled parameters to improve parameter mixing in our Markov chain Monte Carlo (MCMC) scheme. We evaluate the performance of our formulation on simulated data and make an application for the assessment of the teacher covariance structure with a dataset derived from a study of middle and high school algebra teachers.

It is shown that measurement error in predictor variables can be modeled using item response theory (IRT). The predictor variables, that may be defined at any level of an hierarchical regression model, are treated as latent variables. The normal ogive model is used to describe the relation between the latent variables and dichotomous observed variables, which may be responses to tests or questionnaires. It will be shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a Bayesian framework using Gibbs sampling. In this article, handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model. Examples using real data are given.

Higher-order latent traits are proposed for specifying the joint distribution of binary attributes in models for cognitive diagnosis. This approach results in a parsimonious model for the joint distribution of a high-dimensional attribute vector that is natural in many situations when specific cognitive information is sought but a less informative item response model would be a reasonable alternative. This approach stems from viewing the attributes as the specific knowledge required for examination performance, and modeling these attributes as arising from a broadly-defined latent trait resembling the ϑ of item response models. In this way a relatively simple model for the joint distribution of the attributes results, which is based on a plausible model for the relationship between general aptitude and specific knowledge. Markov chain Monte Carlo algorithms for parameter estimation are given for selected response distributions, and simulation results are presented to examine the performance of the algorithm as well as the sensitivity of classification to model misspecification. An analysis of fraction subtraction data is provided as an example.

A Bayesian procedure to estimate the three-parameter normal ogive model and a generalization of the procedure to a model with multidimensional ability parameters are presented. The procedure is a generalization of a procedure by Albert (1992) for estimating the two-parameter normal ogive model. The procedure supports analyzing data from multiple populations and incomplete designs. It is shown that restrictions can be imposed on the factor matrix for testing specific hypotheses about the ability structure. The technique is illustrated using simulated and real data.

The analysis of variance, and mixed models in general, are popular tools for analyzing experimental data in psychology. Bayesian inference for these models is gaining popularity as it allows to easily handle complex experimental designs and data dependence structures. When working on the log of the response variable, the use of standard priors for the variance parameters can create inferential problems and namely the non-existence of posterior moments of parameters and predictive distributions in the original scale of the data. The use of the generalized inverse Gaussian distributions with a careful choice of the hyper-parameters is proposed as a general purpose option for priors on variance parameters. Theoretical and simulations results motivate the proposal. A software package that implements the analysis is also discussed. As the log-transformation of the response variable is often applied when modelling response times, an empirical data analysis in this field is reported.

A Metropolis–Hastings Robbins–Monro (MH-RM) algorithm for high-dimensional maximum marginal likelihood exploratory item factor analysis is proposed. The sequence of estimates from the MH-RM algorithm converges with probability one to the maximum likelihood solution. Details on the computer implementation of this algorithm are provided. The accuracy of the proposed algorithm is demonstrated with simulations. As an illustration, the proposed algorithm is applied to explore the factor structure underlying a new quality of life scale for children. It is shown that when the dimensionality is high, MH-RM has advantages over existing methods such as numerical quadrature based EM algorithm. Extensions of the algorithm to other modeling frameworks are discussed.