Efficient Inference for General Efficient Inference for General Hybrid Bayesian NetworksHybrid Bayesian Networks

Wei SunWei Sun

PhD in Information TechnologyPhD in Information Technology

wsun@gmu.eduwsun@gmu.edu

George Mason University, 2007George Mason University, 2007

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Acknowledgements

Sincere gratitude goes to: Dr. KC Chang Dr. Kathryn Laskey Dr. Kristine Bell Dr. James Gentle

Financial support from MDA sponsored projects.

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Overview

Inference algorithms development Probabilistic inference for Bayesian network (BN) is NP-hard in

general.

It is well known that no exact solution is possible in nonlinear, non-Gaussian case.

This dissertation develops efficient approximate inference algorithms for hybrid Bayesian networks in which arbitrary continuous variables with nonlinear relationships are mixed with discrete variables.

Model performance evaluation for hybrid BN It is typically evaluated using extensive simulation with possibly

complicated inference algorithms. This dissertation develops approximate analytical performance

prediction using Gaussian mixture models.

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Outline

Background

Research objective and contributions

Literature review

Efficient inference algorithms for general hybrid BN.

BN model performance evaluation.

Summary

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Simple Example of BN

Vehicle Identification

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Advantages using BN Model

Conditional independence simplifies specification and inference.

Joint distribution using general probability model: 2*2*2*3*3=72 71 probabilities need to be specified.

BN model: P(T,W,F,R,S)=P(T)P(W)P(R|T,W)P(F|T)P(S|T,F)only 22 probabilities need to be specified.Note: for real-life problems, the savings could be significant.

Modularity, decomposability, transparent modeling, efficient reasoning.

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Real Applications

- big networks; -more interactions between variables; - nonlinear relationships; - hybrid variables; - arbitrary distributions.

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Research Overview

Many real-world problems are naturally modeled by hybrid BN with both categorical and continuous variables: nonlinear relationships, arbitrary distributions, big size.

Objective: develop efficient approximate inference algorithms that can perform acceptably in problems with nonlinearity, heterogeneity, and non-Gaussian variable.

Approach: message passing, unscented transformation, function estimation, Gaussian mixture model are integrated in a unified manner.

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Contributions

Novel Inference Algorithms development Unscented message passing (UMP-BN) for arbitrary continuous

Bayesian networks Hybrid message passing (HMP-BN) for general mixed Bayesian

networks

Performance evaluation methods Approximate analytical method to predict BN model performance

without extensive simulations It can help the decision maker to understand the model

prediction performance and help the modeler to build and validate model effectively.

Software development Inference algorithms coding in MATLAB are added in BN

Toolbox as extra inference engines.

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Literature Review

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Definition of BN

A Bayesian network is a directed, acyclic graph consisting of nodes and arcs: Nodes: variables Arcs: probabilistic dependence relationships Parameters: for each node, there is a conditional probability distribution

(CPD).

CPD of Xi: P(Xi|Pa(Xi)) where Pa(Xi) represents all parents of Xi

Discrete: CPD is typically represented as a table, also called CPT. Continuous: CPD involves functions, such as P(Xi|Pa(Xi)) = f(Pa(Xi), w),

where w is a random noise.

Joint distribution of variables in BN is

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Probabilistic Inference in BN

Task: find the posterior distributions of query nodes given evidence Bayes’ Rule:

Both exact and approximate inference using BNs are NP-hard. Tractable inference algorithms exist only for special classes of BNs.

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Classify BNs by Network Structure

Multiply - connected networksSingly-connected networks (a.k.a. polytree)

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Classify BNs by Node Types

Node types Discrete: conditional probability

distribution is typically represented as a table.

Continuous: Gaussian or non-Gaussian distribution; conditional probability distribution is specified using functions:

P(Xi|Pa(Xi)) = f(Pa(Xi), w) where w is a random noise; the function could be linear/nonlinear.

Hybrid model: discrete and continuous variables are mixed in the networks.

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Conditional Linear Gaussian (CLG)

CLG – Conditional Linear Gaussian model is the simplest hybrid Bayesian networks: All continuous variable are Gaussian The functional relationships between continuous variables and

their parents are linear. There is no continuous parent for any discrete node.

Given any assignment of all discrete variables in CLG, it represents a multivariate Gaussian distribution.

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Conditional Hybrid Model (CHM)

The conditional hybrid model (CHM) is a special hybrid BN: There is no continuous parent for any discrete node. Continuous variable can be arbitrary. The functional relationships between variables can be nonlinear.

Only difference between the CHM and general hybrid BN is the restriction that there is no continuous parent for any discrete node.

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Examples of CHM and CLG

Conditional Hybrid Model (CHM) CLG model

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Taxonomy of BNs

Research Focus

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Inference Algorithms Review - 1 Exact Inference

Pearl’s message passing algorithm (MP) [Pearl88] In MP, messages (probabilities/likelihood) propagate between variables. After

finite number of iterations, each node has its correct beliefs. It only works for pure discrete or pure Gaussian and singly-connected network

(inference is done in linear time).

Clique tree (a.k.a. Junction tree) [LS88,SS90,HD96] and related algorithms Includes variable elimination, arc reversal, symbolic probabilistic inference (SP

I). It only works on pure discrete or pure Gaussian networks or simple CLGs For CLGs, clique tree algorithm is also called Lauritzen’s algorithm (1992). It r

eturns the correct mean and variance of the posterior distributions for continuous variables even though the true distribution might be Gaussian mixture.

It does not work for general hybrid model and is intractable for complicated CLGs.

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Inference Algorithms Review - 2

Approximate Inference Model simplification

Discretization, linearization, arc removal etc. Performance degradation could be significant.

Sampling method Logic sampling [Hen88] Likelihood weighting [FC89] Adaptive Importance Sampling (AIS-BN) [CD00], EPIS-BN [YD03],

Performs well in case of unlikely evidence, but only work for pure discrete networks Markov chain Monte Carlo.

Gibbs sampling Cutset sampling [BD06], can be used in importance sampling too. It outperforms AIS-BN, but

only works for discrete BNs.

Loopy propagation [MWJ99]: use Pearl’s message passing algorithm for networks with loops. This become a popular topic recently.

For pure discrete or pure Gaussian networks with loops, it usually converges to approximate answers in several iterations.

For hybrid model, message representation and integration are open issues. Numerical hybrid loopy propagation [YD06], computational intensive.

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Methodologies for efficient inferencein hybrid BN models

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Pearl’s Message Passing Algorithm

In polytree, any node d-separate the sub-network above it from the sub-network below it. For a typical node X in a polytree, evidence can be divided into two exclusive sets, and processed separately:

Define messages and messages as:

Multiply-connected network may not be partitioned into two separate sub-networks by a node.

Then the belief of node X is:

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Pearl’s Message Passing in BNs

In message passing algorithm, each node maintains Lambda message and Pi message for itself. Also it sends Lambda message to every parent it has and Pi message to its children.

After finite-number iterations of message passing, every node obtains its correct belief.

For polytree, MP returns exact For polytree, MP returns exact belief; belief; For networks with loop, MP is For networks with loop, MP is called loopy propagation that often called loopy propagation that often gives good approximation to gives good approximation to posterior distributions.posterior distributions.

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Message is represented by MEAN and VARIANCE regardless of the distribution.

Message propagations between continuous variables are equivalent to fusing multiple estimates and estimating functional transformation of distributions.

Unscented transformation uses deterministic sampling scheme to obtain good estimates of the first two moments of continuous random variable subject to an nonlinear function transformation.

Continuous Message Representation and Propagation

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Unscented Transformation (UT)

Unscented transformation is a Unscented transformation is a deterministic sampling deterministic sampling methodmethod Approximate the first two Approximate the first two

moments of a continuous moments of a continuous random variable transformed random variable transformed via an arbitrary nonlinear via an arbitrary nonlinear function. function.

UT bases on the principle UT bases on the principle that it is easier to that it is easier to approximate a probability approximate a probability distribution than a nonlinear distribution than a nonlinear function.function.

deterministic sample deterministic sample points are chosen and points are chosen and propagated via the nonlinear propagated via the nonlinear function.function.

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Unscented Transformation Example

A cloud of 5000 samples drawn from a Gaussian prior is propagated through an arbitrary highly nonlinear function and the true posterior sample mean and covariance are calculated, which can be regarded as a ground truth of the two approaches, EKF and UT.

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Unscented Message Passing (UMP-BN)(For arbitrary continuous BN)

Conventional Pearl’s EquationsConventional Pearl’s Equations Derived generalized Equations to Derived generalized Equations to handle continuous variables.handle continuous variables.

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UMP-BN Algorithm

Wei Sun and KC Chang. “Wei Sun and KC Chang. “Unscented Message Passing for Arbitrary Continuous Variables in Bayesian NetworksUnscented Message Passing for Arbitrary Continuous Variables in Bayesian Networks””In Proceedings of the 22nd AAAI Conference on Artificial Intelligence, Vancouver, Canada, July 2007. In Proceedings of the 22nd AAAI Conference on Artificial Intelligence, Vancouver, Canada, July 2007.

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UMP-BN: Numerical Experiments

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Numerical Results - 1

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Numerical Results - 2

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Hybrid Message Passing

Message passing in pure discrete networks is well established.

Message passing in pure continuous networks could be approximated using UMP-BN.

So why not partitioning the network, passing messages separately, then fusing intermediate messages.

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Network Partition for CHM BNs

Definition: In a conditional hybrid BN model (CHM), a discrete variable is called a discrete parent if and only if it has at least one continuous child.

Definition: The set of all discrete parents in a CHM is called the interface nodes of the network.

The interface nodes in the hybrid Bayesian network model can partition the network into separated network segments each one has either pure discrete or pure continuous variables.

The interface nodes “d-separate” continuous variables from other non-interface discrete variables. Furthermore, if we have multiple continuous network segments, the interface node d-separate continuous variables in different continuous sub-networks.

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Network Partition Example

Discrete Discrete variablevariable

Continuous Continuous variablevariable

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Step 1: Inference in Continuous Segment

T

H

F K

B

G R S

M YObserved as Observed as “G=g”“G=g”

Observed as “M=m, Observed as “M=m, Y=y”Y=y”

For any assignment of For any assignment of interface nodes, one can apply interface nodes, one can apply UMP-BN to compute ‘mean’ UMP-BN to compute ‘mean’ and ‘covariance’ of posterior and ‘covariance’ of posterior distribution for hidden distribution for hidden continuous variable given continuous variable given local evidence.local evidence.

P(T|K=1, M=m, Y=y)P(T|K=2, M=m, Y=y)

P(R|K=1, M=m, Y=y)P(R|K=2, M=m, Y=y)

P(S|K=1, M=m, Y=y)P(S|K=2, M=m, Y=y)

GaussianGaussianComponentComponent

p(M=m, Y=y|K=1) = ap(M=m, Y=y|K=2) = b likelihoodlikelihood

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Step 2: Network Transformation

T

H

F K

B

G R S

M YObserved as “G=g”Observed as “G=g”

Observed as “M=m, Observed as “M=m, Y=y”Y=y”

Dummy

H

F K

B

G

Where is a normalizing Where is a normalizing constant.constant.

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Step 3: Messages Integration

TK

R S

M Y

Observed as “M=m, Observed as “M=m, Y=y”Y=y”

Now “K” has the posterior distribution Now “K” has the posterior distribution given all evidence saved as the mixing given all evidence saved as the mixing prior for the intermediate messages of prior for the intermediate messages of hidden continuous variables. hidden continuous variables.

P(K = i | E)P(K = i | E)

Mean and variance of Mean and variance of the Gaussian mixture.the Gaussian mixture.

Mean and variance of the Mean and variance of the iithth Gaussian component. Gaussian component.

Mixing Mixing prior prior

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Hybrid Message Passing Algorithm (HMP-BN)

Wei Sun and KC Chang. “Wei Sun and KC Chang. “Hybrid Message Passing for Mixed Bayesian NetworksHybrid Message Passing for Mixed Bayesian Networks”. In Proceedings of ”. In Proceedings of the 10th International Conference on Information Fusion, Quebec, Canada, July 2007.the 10th International Conference on Information Fusion, Quebec, Canada, July 2007.

Wei Sun and KC Chang. “Wei Sun and KC Chang. “Message Passing for General Bayesian Networks: Representation, Propagation Message Passing for General Bayesian Networks: Representation, Propagation and Integrationand Integration”. Submitted to IEEE Transactions on Aerospace Electronic Systems. September, 2007.”. Submitted to IEEE Transactions on Aerospace Electronic Systems. September, 2007.

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Numerical Experiment - I

T

H

F K

B

G R S

M YObserved as “G=g”Observed as “G=g”

Observed as “M=m, Observed as “M=m, Y=y”Y=y”

Synthetic Hybrid Model Synthetic Hybrid Model - 1- 1

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Results of Model I

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Results of Model I – Cont.

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Results of Model I – Unlikely Evidence

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Numerical Experiment - II

Synthetic Hybrid Model - Synthetic Hybrid Model - 22

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Results of Model II

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Performance Summary

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Complexity of HMP-BN

The complexity of HMP-BN grows exponentially with the size of interface nodes.

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Unscented Hybrid Loopy Propagation

UD

X

Weighted sum of continuous message.Weighted sum of continuous message.where is the function specified in CPD of X.where is the function specified in CPD of X.

Non-negative constant. Non-negative constant.

Weighted sum of continuous message.Weighted sum of continuous message.where is the inverse function. where is the inverse function.

Complexity is reduced significantly! Only depends on the size of discrete parents in local CPD.Complexity is reduced significantly! Only depends on the size of discrete parents in local CPD.

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Performance Evaluation for hybrid BN model

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Performance Metric

- Probability of Correct Classification

(Pcc):

For discrete variable For discrete variable TT in a hybrid BN, in a hybrid BN,

Pcc is the probability mass function of T Pcc is the probability mass function of T

given each state of given each state of TT. .

Diagonal of Pcc = [0.69, 0.75, 0.77, 0.82, 0.78]Diagonal of Pcc = [0.69, 0.75, 0.77, 0.82, 0.78]

An example of Pcc:An example of Pcc:

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Theoretical Pcc Computing

T: discrete target variable. E: evidence set {Ed, Ec}.

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Continuous Evidence Only

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Approximation with Gaussian Mixture

Approximate analytical performance prediction. Approximate analytical performance prediction.

KC Chang and Wei Sun. “KC Chang and Wei Sun. “Performance Modeling for Dynamic Bayesian NetworksPerformance Modeling for Dynamic Bayesian Networks”. ”. In Proceedings of SPIE Conference, Volume 5429, Orlando, 2004.In Proceedings of SPIE Conference, Volume 5429, Orlando, 2004.

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Numerical Experiment

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Experiment Results

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Summary

This dissertation presents new approaches for hybrid BN inference. Develop novel inference algorithms

UMP-BN [SC07a]: efficient message passing for arbitrary continuous BNs.

HMP-BN [SC07b]: partitioning the network into separated network segments each part is either pure discrete or pure continuous; conducting message passing in each segment; message integration via the set of interface nodes.

Evaluate hybrid BN model performance Predict probability of correct classification (Pcc) Approximate analytical performance prediction using

Gaussian mixture. Avoid extensive simulations with possibly complicated

inference algorithm.

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Future Research Directions

Theoretical convergence study of loopy belief propagation will be fruitful.

Current results by UMP-BN and HMP-BN can be applied for further importance sampling.

To reduce the complexity of HMP-BN, loopy version of hybrid message passing algorithm instead of network partition is in process.

For network where discrete variable has continuous parents, our network partition method can not be applied. Further development of efficient message passing between different types of variable is needed.

Dynamic BN inference typically use sequential estimation methods. The algorithms proposed in this dissertation could be integrated with other approaches for DBN inference.

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Publications

Wei Sun and KC Chang. “Message Passing for General Hybrid Bayesian networks: Representations, Propagations and Integrations”. Submitted to IEEE Transactions on Aerospace and Electronic Systems, September, 2007.

Wei Sun and KC Chang. “Convergence Study of Message Passing in Arbitrary Continuous Bayesian Networks.” To appear in SPIE Conference, Orlando, March, 2008.

Wei Sun and KC Chang. “Hybrid Message Passing for General Mixed Bayesian Networks”. In Proceedings of the 10th International Conference on Information Fusion, Quebec, Canada, 2007.

Wei Sun and KC Chang. “Unscented Message Passing for Arbitrary Continuous Bayesian Networks”. In Proceedings of the 22nd AAAI Conference on Artificial Intelligence, Vancouver, Canada, 2007.

Wei Sun and KC Chang. “Probabilistic Inference Using Importance Sampling for Hybrid Bayesian Networks”. In Proceedings of SPIE Conference, Volume 5809, Orlando, 2005.

KC Chang and Wei Sun. “Performance Modeling for Dynamic Bayesian Networks”. In Proceedings of SPIE Conference, Volume 5429, Orlando, 2004.

KC Chang and Wei Sun. “Comparing Probabilistic Inference for Mixed Bayesian Networks”. In Proceedings of SPIE Conference, Volume 5096, Orlando, 2003. .

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Thank Thank you!you!

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Backup

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Taxonomy of BNs

Research Focus

Not exclusiveDifferent perspectives

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Gaussian Mixture Example

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Historical data shows the weight of the

apple is Normally distributed ~ N(350, 10).

Bayesian Inference Motivation

Measurement noise is white Gaussian with variance 20 (gram)^2.

330 gramsNoisy measurement

Shall we believe it?

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Bayesian Inference Motivation – Cont.

What can we do with nonlinearity, non-Gaussian and big problems?

Bayes’ RuleBayes’ Rule

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Complexity of HMP-BN

The biggest set of discrete parents for one continuous sub-networks.