Directed - Bayes Nets Undirected - Markov Random Fields Gibbs Random Fields Causal graphs and causality GRAPHICAL MODELS
Dec 30, 2015
Directed - Bayes NetsUndirected - Markov Random
FieldsGibbs Random Fields
Causal graphs and causality
GRAPHICAL MODELS
Graphical Model Technology
• B-Course: Server at Helsinki University
• Bayes Net: Kevin Murphy’s package (EM oriented)
• Course code directory (MCMC-oriented)
• Genie (Carnegie- Mellon University)
• Numerous commercial and academic packages of varying quality
Graphical Model Technology
• Inference model, directed: The pdf of a variable C is chosen from CPT
(Conditional Probability Table) set depending on values of parent variables.
• Inference model, undirected: The pdf of a variable is chosen from CPT set
depending on values of all neighbours... Inference model, GRF
The pdf of a variable is the product of a number of ‘energy functions’ involving cliques containing node.
Bayes’ Net (BN)
Directed Graph, No cycles.Sample can be generated along edges, each variablecan be sampled when its parents are sampled
Useful for engineered systems,diagnostic systems…
Conditional Probability Table--CPT
Complex BN application: Situation Awareness
QuickTime™ and a decompressor
are needed to see this picture.
R Suzic, thesis 2006
Markov Random Fields
Common in Imaging:Distribution of nodeconditional on the rest of the nodes is the same asdistribution of nodeconditional on neighbors--MARKOV property
Gibbs sampling: sample unknown nodes conditional on neighbors: MCMC with acceptance probability 1!
Segmentation in MR- MRF sampling
Clustering (in spectrum)to get prel segmentation. Smoothing with MRFremoves ‘pepper and salt’.
From BN to MRFA BN can be changed toan equivalent MRF byMORALIZATION: Find unmarriedparents and marry them.The MRF graph can howeverdescribe a larger set of pdfs.
The opposite way is not possible:Many MRFs have no equivalent BN
From BN to MRFExact inference in BNis possible by transformingmoralized graph to junction tree: Every edge in left graph mustlive in some node of right tree. Feasible only if node sets are small
Moralized graph from BNJunction tree or tree-decomposition
Gibbs Random Field (GRF)Maximal cliques C of G:{i,h,g,e}, {e,f,d}, {e,c},{d,b},{c,b,a}.
A GRF is a probability distribution overnode values that falls apart into CLIQUE ENERGY FUNCTIONS
G:
Graphical Model Technology
• Train model using historic/simulation data Where are the edges? Which are the dependencies?
• Use model: From partial set of variables,infer values of missing variables
• Flexible, Intuitive -- but Error Prone!
Learning Graphical model from Data, MCMC style
• Decide on model type (BN, MRF, Chain Graph)• If directed, decide ordering of Nodes to prevent
comparing equivalent models• Find appropriate formula for computing Bayes’
factor in favor of edge present.• Run MCMC: in each step propose to delete/add
edge, decide acceptance or not of proposal.• Trace is sample of posterior graph structure.
Learning Graphical model from Data, EM style
• Inference of most likely tree model is easy• (Chow Liu, 1968): Use Dirichlet Prior
dependency test, select largest Bayes’ factor edge which does not create cycle, and include it.
• This is essentially Kruskal’s algorithmfor shortest spanning tree.
Causality Reasoning
M4’: A dependent on B, but given C, A and B are independent AB|C, not AB M4’’: A independent of B, but given C, A dependent on B not AB|C, AB
Does this suggest that C causes A and B in M4’,and C is caused by A and B in M4’’??Can be decided from observational data!
Testing Treatment
T --> R Simple model. Is there an edge between T (treatment) and R (recovery)?
But what about Gender perspective?