# X h e y pa y b a x pa y x h e x y pa b a x pa 3 3 2 2

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X H E Y pa Y B A X pa Y X H E X Y pa B A X pa } , { ) ( }, , { ) ( } , { ) ( }, , , { ) ( } , , { ) ( }, , { ) ( 3 3 2 2 1 1
My method Mainly two steps Insertion and deletion steps e f g h c d a b e f g h c d a b = 0.5 e f g h c d a b e f g h d = 0.4 c a b e f g h d c a b e f g h d c a b = 0.3 Continue up to t = 0. 5 T 6 T 7 T 1 0 T Insertion step Deletion step
My method Advantage Based on Biological facts (Scale Free Network) No need of thresholds Online approach Scalability Easy to explore sub-networks Fast computation
My method Disadvantage Input order dependency Risky in exploring parents in data with big noise values. (It can be over-fitted to training data) 61 % edges are less order dependent (in part B)
Bayesian network with MCMC Bayesian network Problem 1 Problem 2 A B C D E i i i X pa X P E D C B A P )) ( | ( ) , , , , ( ) | ( ) | ( ) | ( ) | ( ) ( B E P B D P A C P A B P A P ) ( ) ( ) | ( ) | ( D P M P M D P D M P M M P M D P M P M D P ) ( ) | ( ) ( ) | ( Left: in large data set. Right: in small data set.
Bayesian network with MCMC MCMC (Markov Chain Monte Carlo) Inference rule for Bayesian Network Sample from the posterior distribution Proposal Move : Given M_old, propose a new network M_new with proba bility Acceptance and Rejection : ) | ( old new M M Q ) | ( ) | ( ) ( ) | ( ) ( ) | ( , 1 min old new new old old old new new accept M M Q M M Q M P M D P M P M D P P
Bayesian network with MCMC
MCMC in Bayes Net toolbox Hasting factor The proposal probability is calculated from the numb er of neighbours of the model.
Improvement of MCMCs Fan-in The sparse data leads the prior probability to have a non-negligible influence on the posterior P ( M | D ). Limit the maximum number of edges converging on a node, fan-in. If FI ( M ) > a, P(M)=0. Otherwise, P(M)=1. The time complexity reduced largely Acceptable configuration of child and parents in fan-in 3 A A A A B B B C C D A B C D E
Improvement of MCMCs DAG to CPDAG (DAG : Directed Acyclic Graph, CPDAG : Completed Partially Directed Acyclic Graph) X Y X Y P(X, Y) = P(X)P(X|Y) = P(Y|P(Y|X) Set of all equivalent DAGs DAG to CPDAG D E is reversible others are compelled.
Improvement of MCMCs This CPDAG concept bring several advantages: The space of equivalent classes is more reduced. It is easy to trap in local optimum in moving DAG spaces. Incorporating CPDAG to MCMC
MCMCMC Trapping A B A : global optima B : local optima Easy to be trapped in local optima B. Multi chains with different temperatures will be useful to escape from it.
MCMCMC Trapping
MCMCMC A super chain, S Acceptance ratios of a super chain j T i j i j i j M P M D P D S P 1 ) ( ) ( ) ( ) | ( ) | ( ) | ( ) | ( 1 1 i i i i S S Q S S Q l k k l T l l T k k T l l T k k i accept M P M D P M P M D P M P M D P M P M D P P 1 1 1 1 ) ( ) | ( ) ( ) | ( ) ( ) | ( ) ( ) | ( , 1 min
Importance Sampling .

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• Fall '19
• MCMC, Bayesian network, Directed acyclic graph

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