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

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are needed to see this picture.

R Suzic, thesis 2006

A part of the human fysiology: Bayes net describing

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:

GRF:s are MRF:s!!

(see proof by Cheung in course pack)

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.

Graph learning:M3 vs M4,M3’ vs M4’

Causal Inference:M4’ vs M4’’(Unreliable!)

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?

Simpson’s Paradox

A more useful model was:

S -> T -> R <- S, where S is sex