Learning Bayesian Networks for Student Modeling

In the last decade, there has been a growing interest in using Bayesian Networks (BN) in the student modelling problem. This increased interest is probably due to the fact that BNs provide a sound methodology for this difficult task. In order to develop a Bayesian student model, it is necessary to define the structure (nodes and links) and the parameters. Usually the structure can be elicited with the help of human experts (teachers), but the difficulty of the problem of parameter specification is widely recognized in this and other domains. In the work presented here we have performed a set of experiments to compare the performance of two Bayesian Student Models, whose parameters have been specified by experts and learnt from data respectively. Results show that both models are able to provide reasonable estimations for knowledge variables in the student model, in spite of the small size of the dataset available for learning the parametersUniversidad de Málaga. Campus de Excelencia Internacional Andalucía Tec

Bayesian psychometric scaling

In educational and psychological studies, psychometric methods are involved in the measurement of constructs, and in constructing and validating measurement instruments. Assessment results are typically used to measure student proficiency levels and test characteristics. Recently, Bayesian item response models received considerable attention to analyze test data and to measure latent variables. Bayesian psychometric modeling allows to include prior information about the assessment in addition to information available in the observed response data. An introduction is given to Bayesian psychometric modeling, and it is shown that this approach is very flexible, provides direct estimates of student proficiencies, and depends less on asymptotic results. Various Bayesian item response models are discussed to provide insight in Bayesian psychometric scaling and the Bayesian way of making psychometric inferences. This is done according to a general multilevel modeling approach, where observations are nested in students and items, and students are nested in schools. Different examples are given to illustrate the influence of prior information, the effects of clustered response data following a PISA study, and Bayesian methods for scale construction

Structure Discovery in Bayesian Networks: Algorithms and Applications

Bayesian networks are a class of probabilistic graphical models that have been widely used in various tasks for probabilistic inference and causal modeling. A Bayesian network provides a compact, flexible, and interpretable representation of a joint probability distribution. When the network structure is unknown but there are observational data at hand, one can try to learn the network structure from the data. This is called structure discovery. Structure discovery in Bayesian networks is a host of several interesting problem variants. In the optimal Bayesian network learning problem (we call this structure learning), one aims to find a Bayesian network that best explains the data and then utilizes this optimal Bayesian network for predictions or inferences. In others, we are interested in finding the local structural features that are highly probable (we call this structure discovery). Both structure learning and structure discovery are considered very hard because existing approaches to these problems require highly intensive computations. In this dissertation, we develop algorithms to achieve more accurate, efficient and scalable structure discovery in Bayesian networks and demonstrate these algorithms in applications of systems biology and educational data mining. Specifically, this study is conducted in five directions. First of all, we propose a novel heuristic algorithm for Bayesian network structure learning that takes advantage of the idea of curriculum learning and learns Bayesian network structures by stages. We prove theoretical advantages of our algorithm and also empirically show that it outperforms the state-of-the-art heuristic approach in learning Bayesian network structures. Secondly, we develop an algorithm to efficiently enumerate the k-best equivalence classes of Bayesian networks where Bayesian networks in the same equivalence class are equally expressive in terms of representing probability distributions. We demonstrate our algorithm in the task of Bayesian model averaging. Our approach goes beyond the maximum-a-posteriori (MAP) model by listing the most likely network structures and their relative likelihood and therefore has important applications in causal structure discovery. Thirdly, we study how parallelism can be used to tackle the exponential time and space complexity in the exact Bayesian structure discovery. We consider the problem of computing the exact posterior probabilities of directed edges in Bayesian networks. We present a parallel algorithm capable of computing the exact posterior probabilities of all possible directed edges with optimal parallel space efficiency and nearly optimal parallel time efficiency. We apply our algorithm to a biological data set for discovering the yeast pheromone response pathways. Fourthly, we develop novel algorithms for computing the exact posterior probabilities of ancestor relations in Bayesian networks. Existing algorithm assumes an order-modular prior over Bayesian networks that does not respect Markov equivalence. Our algorithm allows uniform prior and respects the Markov equivalence. We apply our algorithm to a biological data set for discovering protein signaling pathways. Finally, we introduce Combined student Modeling and prerequisite Discovery (COMMAND), a novel algorithm for jointly inferring a prerequisite graph and a student model from student performance data. COMMAND learns the skill prerequisite relations as a Bayesian network, which is capable of modeling the global prerequisite structure and capturing the conditional independence between skills. Our experiments on simulations and real student data suggest that COMMAND is better than prior methods in the literature. COMMAND is useful for designing intelligent tutoring systems that assess student knowledge or that offer remediation interventions to students

Bayesian Model Estimation and Selection for the Weekly Colombian Exchange Rate

This document reviews and applies recently developed techniques for Bayesian estimation and model selection in the context of Time Series modeling for Stochastic Volatility. After the literature review on Generalized Conditional Autoregressive models, Stochastic Volatility models, and the relevant results on Markov Chain Monte Carlo methods (MCMC), an example applying such techniques is shown. The methodology is used with a series of Weekly Colombian-USA Exchange Rate on seven different mod els. The GARCH model, which uses Type-IV Pearson distribution, is favored for the selecting technique, Reversible Jump MCMC, over other models, including Stochastic Volatility Models with a Student-t distribution.

Dynamic Bayeian Inference Networks and Hidden Markov Models for Modeling Learning Progressions over Multiple Time Points

The current study examines the performance of a Bayesian Inference Network (BIN) for modeling Learning Progressions (LP) as a longitudinal design approach. Recently, Learning Progressions, defined by measurable pathways that a student may follow in building their knowledge and gaining expertise over time (National Research Council, 2007; Shin, Stevens, Short & Krajcik, 2009), have captured attention in mathematics and science education (Learning Progressions in Science Conference, 2009). While substantive, psychological, instructional, and task developmental aspects has been proposed in the LP framework, few assessment design frameworks have been designed to link the theory embodied in a progression, tasks that provide evidence about a student's level on that progression, and psychometric models that can link them. Specially, few psychometric models have been proposed to characterize the relationship between student performance and levels on learning progressions in a longitudinal design approach. This dissertation introduces an approach to modeling LPs over multiple time points using Bayesian Inference Networks, referred to as dynamic Bayesian Inference Networks (DBINs). The DBINs are a framework for modeling LPs over time by integrating the theory embodying LPs, assessment design, and interpretation of student performances. The technical aspects of this dissertation cover the fundamental concepts of the graphical model for constructing a DBIN. It is shown that this modeling strategy for change over multiple time points is equivalent to a hidden Markov model. An expectation-maximization (EM) algorithm is presented for estimating the parameters in the model. Two simulation studies are conducted that focus on the construction of a simple DBIN model and an expanded DBIN model with a covariate. The extension that incorporates a covariate for students is useful for studying the effect of instructional treatments, students' background, and motivation on a student's LP. An application illustrates the ideas with real data from the domain of beginning computer network engineering drawn from work in the Cisco Networking Academy

Optimising ITS behaviour with Bayesian networks and decision theory

We propose and demonstrate a methodology for building tractable normative intelligent tutoring systems (ITSs). A normative ITS uses a Bayesian network for long-term student modelling and decision theory to select the next tutorial action. Because normative theories are a general framework for rational behaviour, they can be used to both define and apply learning theories in a rational, and therefore optimal, way. This contrasts to the more traditional approach of using an ad-hoc scheme to implement the learning theory. A key step of the methodology is the induction and the continual adaptation of the Bayesian network student model from student performance data, a step that is distinct from other recent Bayesian net approaches in which the network structure and probabilities are either chosen beforehand by an expert, or by efficiency considerations. The methodology is demonstrated by a description and evaluation of CAPIT, a normative constraint-based tutor for English capitalisation and punctuation. Our evaluation results show that a class using the full normative version of CAPIT learned the domain rules at a faster rate than the class that used a non-normative version of the same system