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E-raamat: Bayesian Networks in Educational Assessment

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Bayesian Networks in Educational AssessmentSuurem pilt
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Bayesian inference networks, a synthesis of statistics and expert systems, have advanced reasoning under uncertainty in medicine, business, and social sciences. This innovative volume is the first comprehensive treatment exploring how they can be applied to design and analyze innovative educational assessments.

Part I develops Bayes nets’ foundations in assessment, statistics, and graph theory, and works through the real-time updating algorithm. Part II addresses parametric forms for use with assessment, model-checking techniques, and estimation with the EM algorithm and Markov chain Monte Carlo (MCMC). A unique feature is the volume’s grounding in Evidence-Centered Design (ECD) framework for assessment design. This “design forward” approach enables designers to take full advantage of Bayes nets’ modularity and ability to model complex evidentiary relationships that arise from performance in interactive, technology-rich assessments such as simulations. Part III describes ECD, situates Bayes nets as an integral component of a principled design process, and illustrates the ideas with an in-depth look at the BioMass project: An interactive, standards-based, web-delivered demonstration assessment of science inquiry in genetics.

This book is both a resource for professionals interested in assessment and advanced students. Its clear exposition, worked-through numerical examples, and demonstrations from real and didactic applications provide invaluable illustrations of how to use Bayes nets in educational assessment. Exercises follow each chapter, and the online companion site provides a glossary, data sets and problem setups, and links to computational resources.

Arvustused

The three parts of the book give an excellent overview of current developments and the status of a still developing and emerging field. In this way, it helps researchers in psychometrics and educational assessment to familiarize themselves with Bayesian networks, and their applications. ... The targeted audience of the book seems to focus on students at the graduate level. In this way, it is written for a large audience, and it suits the needs of many researchers. (Nikky van Buuren, Sebastiaan de Klerk and Bernard P. Veldkamp, Psychometrika, Vol. 82, 2017)

This book will provide valuable information on using data-mining along with graphical models in educational assessment. It is one of the initial works that well explain the operative procedures of designing, validating, and implementing the data-driven, competency-oriented diagnostic assessment. The book should be a good reference for both scholars and practitioners in the areas of educational assessment, learning environments and curriculum design, and school improvement. (Fengfeng Ke, Technology, Knowledge and Learning, Vol. 24, 2019)

Part I Building Blocks for Bayesian Networks
1 Introduction
3(16)
1.1 An Example Bayes Network
4(3)
1.2 Cognitively Diagnostic Assessment
7(4)
1.3 Cognitive and Psychometric Science
11(3)
1.4 Ten Reasons for Considering Bayesian Networks
14(2)
1.5 What Is in This Book
16(3)
2 An Introduction to Evidence-Centered Design
19(22)
2.1 Overview
20(1)
2.2 Assessment as Evidentiary Argument
21(2)
2.3 The Process of Design
23(3)
2.4 Basic ECD Structures
26(13)
2.4.1 The Conceptual Assessment Framework
27(7)
2.4.2 Four-Process Architecture for Assessment Delivery
34(4)
2.4.3 Pretesting and Calibration
38(1)
2.5 Conclusion
39(2)
3 Bayesian Probability and Statistics: a Review
41(40)
3.1 Probability: Objective and Subjective
41(5)
3.1.1 Objective Notions of Probability
42(1)
3.1.2 Subjective Notions of Probability
43(2)
3.1.3 Subjective-Objective Probability
45(1)
3.2 Conditional Probability
46(5)
3.3 Independence and Conditional Independence
51(6)
3.3.1 Conditional Independence
53(1)
3.3.2 Common Variable Dependence
54(1)
3.3.3 Competing Explanations
55(2)
3.4 Random Variables
57(5)
3.4.1 The Probability Mass and Density Functions
57(3)
3.4.2 Expectation and Variance
60(2)
3.5 Bayesian Inference
62(19)
3.5.1 Re-expressing Bayes Theorem
63(1)
3.5.2 Bayesian Paradigm
63(4)
3.5.3 Conjugacy
67(5)
3.5.4 Sources for Priors
72(2)
3.5.5 Noninformative Priors
74(2)
3.5.6 Evidence-Centered Design and the Bayesian Paradigm
76(5)
4 Basic Graph Theory and Graphical Models
81(24)
4.1 Basic Graph Theory
82(4)
4.1.1 Simple Undirected Graphs
83(1)
4.1.2 Directed Graphs
83(1)
4.1.3 Paths and Cycles
84(2)
4.2 Factorization of the Joint Distribution
86(5)
4.2.1 Directed Graph Representation
86(2)
4.2.2 Factorization Hypergraphs
88(2)
4.2.3 Undirected Graphical Representation
90(1)
4.3 Separation and Conditional Independence
91(4)
4.3.1 Separation and D-Separation
91(2)
4.3.2 Reading Dependence and Independence from Graphs
93(1)
4.3.3 Gibbs-Markov Equivalence Theorem
94(1)
4.4 Edge Directions and Causality
95(2)
4.5 Other Representations
97(8)
4.5.1 Influence Diagrams
97(2)
4.5.2 Structural Equation Models
99(1)
4.5.3 Other Graphical Models
100(5)
5 Efficient Calculations
105(52)
5.1 Belief Updating with Two Variables
106(5)
5.2 More Efficient Procedures for Chains and Trees
111(11)
5.2.1 Propagation in Chains
112(4)
5.2.2 Propagation in Trees
116(3)
5.2.3 Virtual Evidence
119(3)
5.3 Belief Updating in Multiply Connected Graphs
122(13)
5.3.1 Updating in the Presence of Loops
122(1)
5.3.2 Constructing a Junction Tree
123(11)
5.3.3 Propagating Evidence Through a Junction Tree
134(1)
5.4 Application to Assessment
135(10)
5.4.1 Proficiency and Evidence Model Bayes Net_Fragments
137(2)
5.4.2 Junction Trees for Fragments
139(4)
5.4.3 Calculation with Fragments
143(2)
5.5 The Structure of a Test
145(4)
5.5.1 The Q-Matrix for Assessments Using Only Discrete Items
146(1)
5.5.2 The Q-Matrix for a Test Using Multi-observable Tasks
147(2)
5.6 Alternative Computing Algorithms
149(8)
5.6.1 Variants of the Propagation Algorithm
150(1)
5.6.2 Dealing with Unfavorable Topologies
150(7)
6 Some Example Networks
157(40)
6.1 A Discrete IRT Model
158(8)
6.1.1 General Features of the IRT Bayes Net
161(1)
6.1.2 Inferences in the IRT Bayes Net
162(4)
6.2 The "Context" Effect
166(6)
6.3 Compensatory, Conjunctive, and Disjunctive Models
172(6)
6.4 A Binary-Skills Measurement Model
178(12)
6.4.1 The Domain of Mixed Number Subtraction
178(2)
6.4.2 A Bayes Net Model for Mixed-Number Subtraction
180(4)
6.4.3 Inferences from the Mixed-Number Subtraction Bayes Net
184(6)
6.5 Discussion
190(7)
7 Explanation and Test Construction
197(44)
7.1 Simple Explanation Techniques
198(3)
7.1.1 Node Coloring
198(2)
7.1.2 Most Likely Scenario
200(1)
7.2 Weight of Evidence
201(8)
7.2.1 Evidence Balance Sheet
202(3)
7.2.2 Evidence Flow Through the Graph
205(4)
7.3 Activity Selection
209(6)
7.3.1 Value of Information
209(4)
7.3.2 Expected Weight of Evidence
213(2)
7.3.3 Mutual Information
215(1)
7.4 Test Construction
215(9)
7.4.1 Computer Adaptive Testing
216(1)
7.4.2 Critiquing
217(3)
7.4.3 Fixed-Form Tests
220(4)
7.5 Reliability and Assessment Information
224(17)
7.5.1 Accuracy Matrix
225(5)
7.5.2 Consistency Matrix
230(1)
7.5.3 Expected Value Matrix
230(2)
7.5.4 Weight of Evidence as Information
232(9)
Part II Learning and Revising Models from Data
8 Parameters for Bayesian Network Models
241(38)
8.1 Parameterizing a Graphical Model
241(3)
8.2 Hyper-Markov Laws
244(2)
8.3 The Conditional Multinomial-Hyper-Dirichlet Family
246(4)
8.3.1 Beta-Binomial Family
247(1)
8.3.2 Dirichlet-Multinomial Family
248(1)
8.3.3 The Hyper-Dirichlet Law
248(2)
8.4 Noisy-OR and Noisy-AND Models
250(4)
8.4.1 Separable Influence
254(1)
8.5 DiBello's Effective Theta Distributions
254(13)
8.5.1 Mapping Parent Skills to θ Space
256(1)
8.5.2 Combining Input Skills
257(3)
8.5.3 Samejima's Graded Response Model
260(3)
8.5.4 Normal Link Function
263(4)
8.6 Eliciting Parameters and Laws
267(12)
8.6.1 Eliciting Conditional Multinomial and Noisy-AND
269(3)
8.6.2 Priors for DiBello's Effective Theta Distributions
272(1)
8.6.3 Linguistic Priors
273(6)
9 Learning in Models with Fixed Structure
279(52)
9.1 Data, Models, and Plate Notation
279(8)
9.1.1 Plate Notation
280(2)
9.1.2 A Bayesian Framework for a Generic Measurement Model
282(2)
9.1.3 Extension to Covariates
284(3)
9.2 Techniques for Learning with Fixed Structure
287(10)
9.2.1 Bayesian Inference for the General Measurement Model
288(1)
9.2.2 Complete Data Tables
289(8)
9.3 Latent Variables as Missing Data
297(1)
9.4 The EM Algorithm
298(7)
9.5 Markov Chain Monte Carlo Estimation
305(10)
9.5.1 Gibbs Sampling
308(1)
9.5.2 Properties of MCMC Estimation
309(3)
9.5.3 The Metropolis-Hastings Algorithm
312(3)
9.6 MCMC Estimation in Bayes Nets in Assessment
315(9)
9.6.1 Initial Calibration
316(5)
9.6.2 Online Calibration
321(3)
9.7 Caution: MCMC and EM are Dangerous!
324(7)
10 Critiquing and Learning Model Structure
331(40)
10.1 Fit Indices Based on Prediction Accuracy
332(3)
10.2 Posterior Predictive Checks
335(7)
10.3 Graphical Methods
342(5)
10.4 Differential Task Functioning
347(3)
10.5 Model Comparison
350(4)
10.5.1 The DIC Criterion
350(3)
10.5.2 Prediction Criteria
353(1)
10.6 Model Selection
354(4)
10.6.1 Simple Search Strategies
355(1)
10.6.2 Stochastic Search
356(1)
10.6.3 Multiple Models
357(1)
10.6.4 Priors Over Models
357(1)
10.7 Equivalent Models and Causality
358(4)
10.7.1 Edge Orientation
358(1)
10.7.2 Unobserved Variables
358(2)
10.7.3 Why Unsupervised Learning cannot Prove Causality
360(2)
10.8 The "True" Model
362(9)
11 An Illustrative Example
371(40)
11.1 Representing the Cognitive Model
372(10)
11.1.1 Representing the Cognitive Model as a Bayesian Network
372(5)
11.1.2 Representing the Cognitive Model as a Bayesian Network
377(2)
11.1.3 Higher-Level Structure of the Proficiency Model; i.e., p(θ/λ) and p(λ)
379(2)
11.1.4 High Level Structure of the Evidence Models; i.e., p(r)
381(1)
11.1.5 Putting the Pieces Together
382(1)
11.2 Calibrating the Model with Field Data
382(15)
11.2.1 MCMC Estimation
383(6)
11.2.2 Scoring
389(3)
11.2.3 Online Calibration
392(5)
11.3 Model Checking
397(8)
11.3.1 Observable Characteristic Plots
398(3)
11.3.2 Posterior Predictive Checks
401(4)
11.4 Closing Comments
405(6)
Part III Evidence-Centered Assessment Design
12 The Conceptual Assessment Framework
411(56)
12.1 Phases of the Design Process and Evidentiary Arguments
414(10)
12.1.1 Domain Analysis and Domain Modeling
414(4)
12.1.2 Arguments and Claims
418(6)
12.2 The Student Proficiency Model
424(14)
12.2.1 Proficiency Variables
424(4)
12.2.2 Relationships Among Proficiency Variables
428(5)
12.2.3 Reporting Rules
433(5)
12.3 Task Models
438(5)
12.4 Evidence Models
443(10)
12.4.1 Rules of Evidence (for Evidence Identification)
444(4)
12.4.2 Statistical Models of Evidence (for Evidence Accumulation)
448(5)
12.5 The Assembly Model
453(5)
12.6 The Presentation Model
458(2)
12.7 The Delivery Model
460(1)
12.8 Putting It All Together
461(6)
13 The Evidence Accumulation Process
467(40)
13.1 The Four-Process Architecture
468(6)
13.1.1 A Simple Example of the Four-Process Framework
471(3)
13.2 Producing an Assessment
474(14)
13.2.1 Tasks and Task Model Variables
474(4)
13.2.2 Evidence Rules
478(8)
13.2.3 Evidence Models, Links, and Calibration
486(2)
13.3 Scoring
488(19)
13.3.1 Basic Scoring Protocols
489(4)
13.3.2 Adaptive Testing
493(4)
13.3.3 Technical Considerations
497(3)
13.3.4 Score Reports
500(7)
14 Biomass: An Assessment of Science Standards
507(42)
14.1 Design Goals
507(3)
14.2 Designing Biomass
510(5)
14.2.1 Reconceiving Standards
510(3)
14.2.2 Defining Claims
513(1)
14.2.3 Defining Evidence
514(1)
14.3 The Biomass Conceptual Assessment Framework
515(20)
14.3.1 The Proficiency Model
515(4)
14.3.2 The Assembly Model
519(4)
14.3.3 Task Models
523(6)
14.3.4 Evidence Models
529(6)
14.4 The Assessment Delivery Processes
535(10)
14.4.1 Biomass Architecture
536(2)
14.4.2 The Presentation Process
538(2)
14.4.3 Evidence Identification
540(1)
14.4.4 Evidence Accumulation
541(2)
14.4.5 Activity Selection
543(1)
14.4.6 The Task/Evidence Composite Library
543(1)
14.4.7 Controlling the Flow of Information Among the Processes
544(1)
14.5 Conclusion
545(4)
15 The Biomass Measurement Model
549(34)
15.1 Specifying Prior Distributions
550(11)
15.1.1 Specification of Proficiency Variable Priors
552(2)
15.1.2 Specification of Evidence Model Priors
554(6)
15.1.3 Summary Statistics
560(1)
15.2 Pilot Testing
561(5)
15.2.1 A Convenience Sample
561(3)
15.2.2 Item and other Exploratory Analyses
564(2)
15.3 Updating Based on Pilot Test Data
566(13)
15.3.1 Posterior Distributions
566(9)
15.3.2 Some Observations on Model Fit
575(2)
15.3.3 A Quick Validity Check
577(2)
15.4 Conclusion
579(4)
16 The Future of Bayesian Networks in Educational Assessment
583(18)
16.1 Applications of Bayesian Networks
583(3)
16.2 Extensions to the Basic Bayesian Network Model
586(7)
16.2.1 Object-Oriented Bayes Nets
586(2)
16.2.2 Dynamic Bayesian Networks
588(4)
16.2.3 Assessment-Design Support
592(1)
16.3 Connections with Instruction
593(3)
16.3.1 Ubiquitous Assessment
594(2)
16.4 Evidence-Centered Assessment Design and Validity
596(1)
16.5 What We Still Do Not Know
597(4)
A Bayesian Network Resources 601(6)
A.1 Software
601(3)
A.1.1 Bayesian Network Manipulation
602(1)
A.1.2 Manual Construction of Bayesian Networks
603(1)
A.1.3 Markov Chain Monte Carlo
603(1)
A.2 Sample Bayesian Networks
604(3)
References 607(32)
Author Index 639
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