Bayesian Networks and Causal Modelling

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Bayesian Networks and Causal Modelling

• Ann Nicholson

School of Computer Science and Software
Engineering Monash University
2
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

3
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(HE) P(EH) 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(SA)P(TA)

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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
Models causal relationship
Jane has a high temp
Models possible sensor error
Thermometer temp reading
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BN inference

• Evidence observation of specific state
• Task compute the posterior probabilities for
  query node(s) given evidence.

Te
Te
Diagnostic inference
Predictive inference
Intercausal inference
Mixed inference
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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 (DSS2004) (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.

Te
Te
Th
Th
Diagnostic inference
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 isnt general enough

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Defining Intervention II

• We define an intervention on model M as
• M augmented with Ic (M') where
• Ic has the purpose of manipulating C
• Ic is exogenous (has no parents) in M'
• 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 Cn2
• Minus number of possible cyclic graphs
• More exactly (Robinson, 1977)
• f(n) ?(-1)i1 Cni 2i(n-i)f(n-i)
• so for
• n3, f(n)25
• n5, f(n)25,000
• n10, 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 samples
• max P(eH1,?1) max P(eH2,?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(hie) 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
• Learns causal connectedness.
• Learns v-structures. Hence, learns equivalence
  classes.
• 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(eh).
• Applies a trade-off between
• Model simplicity
• Data fit
• Equivalent to that h that maximizes P(h)P(eh)
  --- i.e., P(he).

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MML search algorithms

• MML metrics need to be combined with search.
  This has been done three ways
• Wallace, Korb, Dai (1996) greedy search
  (linear).
• Brute force computation of linear extensions
  (small models only)
• 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
• ODonnell, Korb, Nicholson
• Useful for combining expert and automated methods
• Learning local structure
• Logit models (Neill, Wallace, Korb)
• Hybrid networks - CPT or decision trees
  (ODonnell, Allison, Korb, Hope) (Uses MCMC
  search)

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CaMML

• Information and executables available at
• www.datamining.monash.edu.au/software/camml
• Linear and Discrete versions
• Weka wrapper available