PC-stable : an order-independent constraint-based approach

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Table of contents

Abstract
Acknowledgements
1 Introduction
1.1 Expansion of dangerous’ gene families in vertebrates
1.2 Learning causal graphical models
1.2.1 The state-of-the-art approaches
1.2.1.1 Constraint-based methods
1.2.1.2 Search-and-score methods
1.2.2 3o2: a novel network reconstruction method
2 Graphical Notation and Terminology
2.1 Directed Acyclic Graphs
2.1.1 Graphical model terminology
2.1.2 Interpretation of a DAG
2.1.2.1 The d-separation criterion
2.1.2.2 Markov equivalence
2.2 Ancestral Graphs
2.2.1 Complications from latent and selection variables
2.2.1.1 Spurious correlations
2.2.1.2 Spurious causal eects
2.2.2 Graphical model terminology
2.2.3 Interpretation of a MAG
2.2.3.1 The m-separation criterion
2.2.3.2 Markov equivalence
2.2.3.3 Finding adjacencies in a PAG
3 Constraint-based Methods
3.1 The PC algorithm
3.1.1 Learning the skeleton and sepsets
3.1.2 Orientating the skeleton
3.1.3 Propagating the orientations
3.1.4 Statistical tests for conditional independence
3.1.4.1 Chi-square conditional independence test
3.1.4.2 G-square conditional independence test
3.2 Variations on the PC algorithm
3.2.1 PC-stable: an order-independent constraint-based approach
3.2.1.1 Order-independent skeleton learning
3.2.1.2 Order-independent orientation steps
3.2.2 Causal Inference with latent and selection variables
3.2.2.1 The FCI algorithm
3.2.2.2 Improvements of the FCI algorithm
4 Bayesian methods
4.1 Learning a Bayesian Network
4.2 Scoring funtions
4.2.1 Bayesian scores
4.2.2 Information-theoretic scores
4.2.2.1 The LL scoring function
4.2.2.2 The MDL/BIC criterion
4.2.2.3 The NML criterion
4.3 The search-and-score paradigm
4.3.1 The exact discovery
4.3.2 The heuristic search approaches
4.4 Hybrid approaches
5 Information-theoretic methods
5.1 Information-theoretic measures
5.1.1 The Shannon entropy
5.1.2 The data processing inequality
5.1.3 The multivariate information
5.1.4 The cross-entropy of a Bayesian network
5.2 State-of-the-art of information-theoretic methods
5.2.1 The Chow & Liu approach
5.2.2 Relevance Network
5.2.3 ARACNe
5.2.4 Minimum Redundancy Networks
5.2.5 The MI-CMI algorithm
5.3 Hybrid approaches using Mutual Information Tests
6 3o2: a hybrid method based on information statistics
6.1 Information theoretic framework
6.1.1 Inferring isolated v-structures vs non-v-structures
6.1.2 Inferring embedded v-structures vs non-v-structures
6.1.3 Uncovering causality from a stable/faithful distribution
6.1.3.1 Negative 3-point information as evidence of causality
6.1.3.2 Propagating the orientations
6.2 Robust reconstruction of causal graphs from nite datasets
6.2.1 Finite size corrections of maximum likelihood
6.2.2 Complexity of graphical models
6.2.3 Probability estimate of indirect contributions
6.3 The 3o2 scheme
6.3.1 Robust inference of conditional independencies
6.3.2 The 3o2 algorithm
6.3.2.1 Reconstruction of network skeleton
6.3.2.2 Orientation of network skeleton
6.4 Extension of the 3o2 algorithm
6.4.1 Probabilistic estimation of the edge endpoints
6.4.2 Allowing for latent variables
6.4.3 Allowing for missing data
6.5 Implemented pipelines
6.5.1 The 3o2 pipeline
6.5.2 The discoNet pipeline
7 Evaluation on Benchmark Networks
7.1 Using simulated datasets from real-life networks
7.2 Using simulated datasets from simulated networks
7.3 Using simulated datasets from undirected networks
7.3.1 Generating datasets from undirected networks
7.3.2 Reconstructing simulated undirected networks
8 Interplay between genomic properties on the fate of gene dupli- cates with the 3o2 method
8.1 Enhanced retention of ohnologs
8.1.1 Enhanced retention of dangerous’ ohnologs
8.1.2 Retained ohnologs are more dangerous’ than dosage balanced
8.2 Going beyond simple correlations in genomic data
8.2.1 The Mediation framework in the context of causal inference
8.2.2 The Mediation analysis applied to genomic data
8.3 Retention of dangerous’ ohnologs through a counterintuitive mechanism
8.4 Reconstructing the interplay of genomic properties on the retention of ohnologs
8.4.1 Genomic properties of interest
8.4.2 Non-adaptive retention mechanism supported by the 3o2 causal models
9 Reconstruction of zebrash larvae neural networks from brain- wide in-vivo functional imaging
9.1 The zebrash calcium functional imaging
9.1.1 The calcium functional imaging
9.1.2 The zebrash as vertebrate model
9.1.3 The SPIM technique
9.1.4 The experimental setup
9.1.5 From uorescent signal to neuron activity
9.2 Neural network reconstructions with the 3o2 method
9.2.1 Preprocessing of SPIM images by neuron clustering
9.2.2 Reconstruction of temporal activity patterns
9.2.2.1 Alternating neural activity in optokinetic reponse
9.2.2.2 Neural activity related to hunting behaviour
10 Reconstruction of the hematopoiesis regulation network with the 3o2 method
10.1 The hematopoiesis regulation network
10.2 Regulation network reconstruction with the 3o2 method
10.2.1 Dataset and interactions of interest
10.2.2 The interactions recovered by 3o2 and alternative methods
11 Reconstruction of mutational pathways involved in tumours pro- gression with the 3o2 method
11.1 The breast cancer dataset
11.1.1 A brief overview of breast cancer
11.1.2 The issue of missing values
11.2 3o2 mutational pathways in breast cancer
11.2.1 Information content of complete vs. incomplete datasets
11.2.2 3o2 cascades of mutations in breast cancer
11.2.3 Temporal cascade patterns of advantageous mutations
12 Conclusions
A Complementary evaluations on real-life networks
A.1 Evaluation of the PC method by signicance level
A.2 Evaluation of the Aracne reconstruction method
A.3 Evaluation of the Bayesian methods by score
A.4 Evaluation of 3o2 by score
A.5 Evaluation of 3o2 against Bayesian and MMHC methods
A.6 Execution time comparisons
B Complementary evaluations on simulated networks
B.1 Description of the benchmark networks
B.2 Evaluation of 3o2 by score
B.3 Evaluation of the PC method by signicance level
B.4 Evaluation of the MMHC method by signicance level
B.5 Comparison between 3o2, PC and Bayesian hill-climbing
B.6 Evaluation of the Bayesian methods by score
Bibliography