Transcript Bayesian Networks and Causal Modelling

Bayesian Networks
and
Causal Modelling
Ann Nicholson
School of Computer Science
and Software Engineering
Monash University
1
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”
FXlu
P(Flu=T) = 0.05
TY
e
P(Te=High|Flu=T) = 0.4
P(Te=High|Flu=F) = 0.01
Models causal relationship
“Jane has a
high temp”
Models possible sensor error
“Thermometer
temp reading”
TQh
P(Th=High|Te=H) = 0.95
P(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.
Flu
Flu
Te
Y
Te
Y
Th
Th
Diagnostic
inference
Predictive
inference
Flu
Flu
TB
Te
Flu
Te
Th
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
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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
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Autonomous aircraft monitoring and
replanning (with Ph.D. student Tim Wilkin, PRICAI2000,
IAV2004)
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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)
Flu
Flu

Inference with interventions
Te
Te

» Predictive reasoning
» Not diagnostic reasoning
Causal reasoning shouldn't go against causality.
Th
Th
Diagnostic
inference
Predictive
inference
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Pearlian Determinism
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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
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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.
2.
3.

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.
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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
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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)
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.)
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(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(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).
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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

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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).
»
2.
Neil and Korb (1999): genetic algorithms (linear).
»
»
»
»

Brute force computation of linear extensions (small models
only)
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 at
www.datamining.monash.edu.au/software/camml
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Linear and Discrete versions
Weka wrapper available
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