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Bayesian Networks and

Causal Modelling

Ann Nicholson

School of Computer Science and Software Engineering

Monash University

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Overview

Introduction to Bayesian Networks (BNs) Summary of BN research projects Varieties of Causal intervention

» PRICAI2004: K. Korb, L. Hope, A. Nicholson, K. Axnick

Learning Causal Structure» CaMML software

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Probability theory for representing uncertainty

Assigns a numerical degree of belief between 0 and 1 to facts» e.g. “it will rain today” is T/F. » P(“it will rain today”) = 0.2 prior probability

(unconditional) Posterior probability (conditional)

» P(“it wil rain today” | “rain is forecast”) = 0.8 Bayes’ Rule: P(H|E) = P(E|H) x P(H) P(E)

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Bayesian networks

A Bayesian Network (BN) represents a probability distribution graphically (directed acyclic graphs)

Nodes: random variables,» R: “it is raining”, discrete values T/F» T: temperature, cts or discrete variable» C: colour, discrete values {red,blue,green}

Arcs indicate conditional dependencies between variables

P(A,S,T) can be decomposed to P(A)P(S|A)P(T|A)

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Bayesian networks (cont.)

There is a conditional probability distribution (CPD or CPT) associated with each node.

– probability of each state given parent states

“Jane has the flu”

“Jane has a high temp”

“Thermometertemp reading”

XFlu

YTe

QTh

Models causal relationship

Models possible sensor error

P(Flu=T) = 0.05

P(Te=High|Flu=T) = 0.4P(Te=High|Flu=F) = 0.01

P(Th=High|Te=H) = 0.95P(Th=High|Te=L) = 0.1

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BN inference

Evidence: observation of specific state Task: compute the posterior probabilities for query

node(s) given evidence.

Th

Y

Flu

Te

Diagnostic inference

Th

Flu

YTe

Predictive inference

Intercausal inference

Te

Flu TBFlu

Mixed inference

Th

Flu

Te

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Causal Networks

Arcs follow the direction of causal process

Causal Networks are always BNs Bayesian Networks aren't always causal

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Early BN-related projects

DBNS for discrete monitoring (PhD, 1992) Approximate BN inference algorithms based

on a mutual information measure for relevance (with Nathalie Jitnah, 1996-1999)

Plan recognition: DBNs for predicting users actions and goals in an adventure game (with

David Albrecht, Ingrid Zukerman, 1997-2000) DBNs for ambulation monitoring and fall

diagnosis (with biomedical engineering, 1996-2000) Bayesian Poker (with Kevin Korb, 1996-2003)

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Knowledge Engineering with BNs

Seabreeze prediction: joint project with Bureau of Meteorology» Comparison of existing simple rule, expert elicited

BN, and BNs from Tetrad-II and CaMML ITS for decimal misconceptions Methodology and tools to support knowledge

engineering process» Matilda: visualisation of d-separation » Support for sensitivity analysis

Written a textbook: » Bayesian Artificial Intelligence, Kevin B. Korb and

Ann E. Nicholson, Chapman & Hall / CRC, 2004.www.csse.monash.edu.au/bai/book

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Current BN-related projects

BNs for Epidemiology (with Kevin Korb, Charles Twardy)

» ARC Discovery Grant, 2004 » Looking at Coronary Heart Disease data sets» Learning hybrid networks: cts and discrete variables.

BNs for supporting meteorological forecasting process (DSS’2004) (with Ph. D student Tal Boneh, K. Korb, BoM)

» Building domain ontology (in Protege) from expert elicitation» Automatically generating BN fragments» Case studies: Fog, hailstorms, rainfall.

Ecological risk assessment » Goulburn Water, native fish abundance

» Sydney Harbour Water Quality

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Other projects

Autonomous aircraft monitoring and replanning (with Ph.D. student Tim Wilkin, PRICAI2000,

IAV2004) Dynamic non-uniform abstraction for

approximate planning with MDPs (with Ph.D. student Jiri Baum)

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Observation and Intervention

Inference from observations» Predictive reasoning (finding effects)» Diagnostic reasoning (finding causes)

Inference with interventions» Predictive reasoning» Not diagnostic reasoning

Causal reasoning shouldn't go against causality. Th

Flu

Te

Diagnostic inference

Th

Flu

Te

Predictive inference

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Pearlian Determinism

Pearl's reasons for determinism:» Determinism is intuitive» Counterfactuals and causal explanation only make

sense with a deterministic interpretation» Any indeterministic model can be transformed into

a deterministic model

We see no reason for assuming determinism

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Defining Intervention I

Arc cutting» More intuitive» Intervention node

Intervention node» More general interventions» Much easier to implement» To simulate arc cutting: P(C| c, Ic)=1

Arc cutting isn’t general enough

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Defining Intervention II

We define an intervention on model M as: M augmented with Ic (M') where:

1. Ic has the purpose of manipulating C

2. Ic is exogenous (has no parents) in M'

3. Ic directly causes (is a parent of) C

To preserve the original network:

PM'(C| c,¬ Ic) = PM' (C| c)

where c are the original parents of C.We also define P*(C) as the intended distribution.

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Varieties of Intervention: Dependency

The degree of dependency of the effect upon existing parents.

• An independent intervention cuts the child off from its other parents. Thus,

PM'(C| c, Ic) = P*(C)

• A dependent intervention allows any parent interaction.

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Varieties of Intervention : Indeterminism

The degree of indeterminism of the effect.

A deterministic intervention sets the child to one particular state.

A stochastic intervention sets the child to a positive distribution.

Dependency and Determinism characterize any intervention Pearlian interventions are independent and

deterministic

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Varieties of Intervention : Effectiveness

We've found the idea of effectiveness useful.

If P*(C) is what's intended and r is the effectiveness, then

PM'(C | c, Ic) = r × P*(C) + (1-r) × PM'(C | c)

This is a dependent intervention.

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Demo of Causal Intervention Software

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Summary of Causal Intervention

A taxonomy of intervention types More realistic interventions (e.g., partial

effectiveness) A GUI which handles some varieties of

intervention» Pearlian» Partially effective» Extensible to deal with other types of interaction

explicitly

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Learning Causal Structure

This is the real problem; parameterizing models is relatively straightforward estimation problem.

Size of the dag space is superexponential:» Number of possible orderings: n!» Times number of possible arcs: Cn

2

» Minus number of possible cyclic graphs More exactly (Robinson, 1977):

f(n) = (-1)i+1 Cni 2i(n-i)f(n-i)

so for» n=3, f(n)=25» n=5, f(n)=25,000» n=10, f(n) 4.2x1018

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Learning Causal Structure

There are two basic methods:» Learning from conditional independencies (CI

learning)» Learning using a scoring metric (Metric learning)

CI learning (Verma and Pearl, 1991)» Suppose you have an Oracle who can answer yes

or no to any question of the type:

is X conditional independence Y given S?» Then you can learn the correct causal model, up to

statistical equivalence (patterns).

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Statistical Equivalence

Two causal models H1 and H2 are statistically equivalent iff they contain the same variables and joint samples over them provide no statistical grounds for preferring one over the other.

Examples» All fully connected models are equivalent.» A B C and A B C.» A B D C and A B D C.

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Statistical Equivalence (cont.)

(Verma and Pearl, 1991): Any two causal models over the same variables which have the same skeleton (undirected arcs) and the same directed v-structures are statistically equivalent.

Chickering (1995): If H1 and H2 are statistically equivalent, then they have the same maximum likelihoods relative to any joint samplesmax P(e|H1,1) = max P(e|H2,2)

where i is a parameterization of Hi

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Other approaches to structure learning

TETRAD II: Spirtes, Glymour and Scheines (1993). Implemented in their PC algorithm» Doesn't handle well with weak links and small samples

(demonstrated empirically in Dai, Korb, Wallace & Wu (1997)).

Bayesian LBN: Cooper & Herskovits' K2 (1991, 1992)» Compute P(hi|e) by brute force, under the various

assumptions which reduce the computation of PCH(h,e) to a polynomial time counting problem.

» But the hypothesis space is exponential; they go for dramatic simplification by assuming we know the temporal ordering of the variables.

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Learning Variable Order

Reliance upon a given variable order is a major drawback to K2» And many other algorithms (Buntine 1991, Bouckert 1994, Suzuki

1996, Madigan & Raftery 1994)

What's wrong with that?» We want autonomous AI (data mining). If experts can

order the variables they can likely supply models.» Determining variable ordering is half the problem. If

we know A comes before B, the only remaining issue is whether there is a link between the two.

» The number of orderings consistent with dags is exponential (Brightwell & Winkler 1990; number complete). So iterating over all possible orderings will not scale up.

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Statistical Equivalence Learners

Heckerman & Geiger (1995) advocate learning only up to statistical equivalence classes (a la TETRAD II).» Since observational data cannot distinguish btw

equivalent models, there's no point trying to go further.

Madigan, Andersson, Perlman & Volinsky (1996) follow this advice, use uniform prior over equivalence classes.

Geiger and Heckerman (1994) define Bayesian metrics for linear and discrete equivalence classes of models (BGe and BDe)

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Statistical Equivalence Learners

Wallace & Korb (1999): This is not right! These are causal models; they are distinguishable on

experimental data.» Failure to collect some data is no reason to change prior

probabilities. E.g., If your thermometer topped out at 35C, you wouldn't

treat 35C and 34C as equally likely. Not all equivalence classes are created equal:{ A B C, A B C, A B C }{ A B C } Within classes some dags should have greater priors

than others… E.g.,» LightsOn InOffice LoggedOn v.» LightsOn InOffice LoggedOn

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Full Causal Learners

So… a full causal learner is an algorithm that:1. Learns causal connectedness.2. Learns v-structures. Hence, learns equivalence

classes.3. Learns full variable order. Hence, learns full causal

structure (order + connectedness).

TETRAD II: 1, 2. Madigan et al.; Heckerman & Geiger (BGe, BDe): 1,

2. Cooper & Herskovits' K2: 1. Lam and Bacchus MDL: 1, 2 (partial), 3 (partial). Wallace, Neil, Korb MML: 1, 2, 3.

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CaMML

Minimum Message Length (Wallace \& Boulton 1968) uses Shannon's measure of information:

I(m) = - log P(m) Applied in reverse, we can compute P(h,e) from I(h,e). Given an efficient joint encoding method for the

hypothesis & evidence space (i.e., satisfying Shannon's law), MML:

Searches {hi} for that hypothesis h that minimizes I(h) + I(e|h).

Applies a trade-off between» Model simplicity» Data fit

Equivalent to that h that maximizes P(h)P(e|h) --- i.e., P(h|e).

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MML search algorithms

MML metrics need to be combined with search. This has been done three ways:

1. Wallace, Korb, Dai (1996): greedy search (linear).» Brute force computation of linear extensions (small models

only)

2. Neil and Korb (1999): genetic algorithms (linear).» Asymptotic estimator of linear extensions» GA chromosomes = causal models» Genetic operators manipulate them» Selection pressure is based on MML

Wallace and Korb (1999): MML sampling (linear, discrete).» Stochastic sampling through space of totally ordered causal

models» No counting of linear extensions required

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Empirical Results

A weakness in this area --- and AI generally.» Papers based upon very small models, loose comparisons.» ALARM often used --- everything gets it to within 1 or 2 arcs.

Neil and Korb (1999) compared CaMML and BGe (Heckerman & Geiger's Bayesian metric over equivalence classes), using identical GA search over linear models:» On KL distance and topological distance from the true model,

CaMML and BGe performed nearly the same.» On test prediction accuracy on strict effect nodes (those with

no children), CaMML clearly outperformed BGe.

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Extensions to original CaMML

Allow specification of prior on arc » O’Donnell, Korb, Nicholson» Useful for combining expert and automated

methods Learning local structure

» Logit models (Neill, Wallace, Korb)» Hybrid networks - CPT or decision trees

(O’Donnell, Allison, Korb, Hope) (Uses MCMC search)

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CaMML

Information and executables available atwww.datamining.monash.edu.au/software/camml

Linear and Discrete versions Weka wrapper available