Chain Event Graphs

Cover
......Page 1
Half Title
......Page 2
Abstract
......Page 3
Copyrights
......Page 5
Dedication
......Page 6
Contents
......Page 8
Preface......Page 12
List of Figures......Page 14
List of Tables......Page 18
Symbols and abbreviations......Page 20
1.1 Some motivation......Page 22
1.2 Why event trees......Page 26
1.3 Using event trees to describe populations......Page 33
1.4 How we have arranged the material in this book......Page 35
1.5 Exercises......Page 37
2.1 Inference on discrete statistical models......Page 38
2.1.2 Two prior-to-posterior analyses......Page 41
2.1.3 Poisson–Gamma and Multinomial–Dirichlet......Page 45
2.1.4 MAP model selection using Bayes Factors......Page 47
2.2 Statistical models and structural hypotheses......Page 49
2.2.1 An example of competing models......Page 50
2.2.2 The parametric statistical model......Page 52
2.3.1 Factorisations of probability mass functions......Page 54
2.3.2 The d-separation theorem......Page 57
2.3.3 DAGs coding the same distributional assumptions......Page 58
2.3.4 Estimating probabilities in a BN......Page 59
2.3.5 Propagating probabilities in a BN......Page 61
2.4 Concluding remarks......Page 64
2.5 Exercises......Page 65
3.1 Models represented by tree graphs......Page 66
3.1.1 Probability trees......Page 67
3.1.2 Staged trees......Page 71
3.2 The semantics of the Chain Event Graph......Page 75
3.3 Comparison of stratified CEGs with BNs......Page 80
3.4.1 The saturated CEG......Page 87
3.4.2 The simple CEG......Page 88
3.4.3 The square-free CEG......Page 89
3.5 Some related structures......Page 90
3.6 Exercises......Page 92
4.1 Encoding qualitative belief structures with CEGs......Page 94
4.1.1 Vertex- and edge-centred events......Page 95
4.1.2 Intrinsic events......Page 98
4.1.3 Conditioning in CEGs......Page 100
4.1.4 Vertex-random variables, cuts and independence......Page 102
4.2 CEG statistical models......Page 109
4.2.1 Parametrised subsets of the probability simplex......Page 111
4.2.2 The swap operator......Page 115
4.2.3 The resize operator......Page 121
4.2.4 The class of all statistically equivalent staged trees......Page 123
4.3 Exercises......Page 127
5.1 Estimating a given CEG......Page 128
5.1.1 A conjugate analysis......Page 129
5.1.2 How to specify a prior for a given CEG......Page 131
5.1.3 Example: Learning liver and kidney disorders......Page 134
5.1.4 When sampling is not random......Page 139
5.2 Propagating information on trees and CEGs......Page 143
5.2.1 Propagation when probabilities are known......Page 144
5.2.2 Example: Propagation for liver and kidney disorders......Page 150
5.2.3 Propagation when probabilities are estimated......Page 152
5.2.4 Some final comments......Page 154
5.3 Exercises......Page 155
Chapter 6. Model selection for CEGs......Page 158
6.1 Calibrated priors over classes of CEGs......Page 160
6.2 Log-posterior Bayes Factor scores......Page 162
6.3 CEG greedy and dynamic programming search......Page 164
6.3.1 Greedy SCEG search using AHC......Page 165
6.3.2 SCEG exhaustive search using DP......Page 169
6.4.1 DP and AHC using a block ordering......Page 175
6.4.2 A pairwise moment non-local prior......Page 178
6.5 Exercises......Page 184
Chapter 7. How to model with a CEG: A real-world application......Page 186
7.1 Previous studies and domain knowledge......Page 188
7.2 Searching the CHDS dataset with a variable order......Page 193
7.3 Searching the CHDS dataset with a block ordering......Page 198
7.4 Searching the CHDS dataset without a variable ordering......Page 204
7.5.1 Exhaustive CEG model search......Page 207
7.5.2 Searching the CHDS dataset using NLPs......Page 208
7.5.3 Setting a prior probability distribution......Page 209
7.6 Exercises......Page 213
Chapter 8. Causal inference using CEGs......Page 214
8.1 Bayesian networks and causation......Page 215
8.1.1 Extending a BN to a causal BN......Page 216
8.1.2 Problems of describing causal hypotheses using a BN......Page 218
8.2 Defining a do-operation for CEGs......Page 222
8.2.1 Composite manipulations......Page 224
8.2.2 Example: student housing situation......Page 226
8.3 Causal CEGs......Page 229
8.3.2 Example: Manipulations of the CHDS......Page 230
8.3.3 Backdoor theorems......Page 235
8.4 Causal discovery algorithms for CEGs......Page 236
8.5 Exercises......Page 239
References......Page 242
Index......Page 252