Parameter Learning
• In order to fully specify the Bayesian network and thus fully represent the joint probability distribution, it is necessary to specify for each node X the probability distribution for X conditional upon X's parents
prior possibility P(Y) ( rankings, recent history of their performance)
Introduction
• First half is over • The outcome of the first period may be
treated as a random variable X, the óbserved evidence' that influence your prediction of the final value of Y.
• Prior confidence --------belief • Process--------belief propagation dynamics
causal relationships
statistical dependence between
Bayesian Networks
• DAG: Directed Acyclic Graph • CPT: Conditioanl Probability Tables
• P(Y|X)= PX |YPY Hale Waihona Puke XIntroduction
• A train of consequences • Before the match has started, prior
possibilityP(Y); after observing the evidence X1, posterior probability P(Y|X1); new evidence X2 is acquired, then
d-seperation
d-seperation is a sufficient condition for conditional independence
Markov Blankets
Markov Blanket of a node is the set consisting of its parents, children and spouses
Data
conditional probability tables
Directed Markov Assumption
• Each variable is independent of its nondescendents in the DAG given the values of its parents.
Directed Markov Assumption
Markov Blankets
u1 un
z1
zn
Y1 Yn
feature A is independent of all other features given MB(A)={u1,...un,z1,...zn, Y1,...,Yn}
Markov Blankets
• For any value xi of Xi,
Advantage of BN: X 1,...aX l.ln oa .w.u,ro osne tolryeed sstta iime maan rttiene c gjoad ie nyrtedsl,w aistttc irvriebe a lua yttion lrnlbay n:sfo
small number of parameters
PY|X1,X2PX P 1,X X1 2,|X Y2 P YPX2P |Y X P 2P X1 X |Y 1PY PX2P |Y X P 2 Y |X1
Introduction
• Bayes theorem allows us to transform our prior probabilistic knowledge of an event into a more robust, posterior knowledge of its probability.
d-seperation
• some terminology • head-to-tail at node B • tail-to tail at A
A B C A
B
C
A
C
• head-to-head at B
B
d-seperation
A
B
C D
E
Z is observed: A,B are in Z; C, D or E are not in Z
Hybrid Random Fields Bayes Networks
Jill 5/2/2013
Introduction
• For some mysterious reasons, • Bayes networks are often introduced in the
literature with examples concerning weather forecase: • Is it likely to be sunny on Sunday given the fact that it is raining on Saturday, and that my granny's back hurts?
Introduction
• make prediction on whether your favorite football team will win, lose or draw tonight
• outcome of the match ---random variableY • Y=w, l, d • Rely on your original estimate of a certain