CSE 591 - FALL 03.
Chitta Baral
Department of Computer Science and Engineering
Arizona State University
Tempe, AZ 85287-5406 USA
http://www.public.asu.edu/∼cbaral/cse571-f99/
October 12, 2003
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Probability, Bayes Nets and Causality - Image Processing and Analysis | CSE 591, Study notes of Computer Science
Material Type: Notes; Class: Introduction to Image Processing and Analysis; Subject: Computer Science and Engineering; University: Arizona State University - Tempe; Term: Fall 2003;
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CSE 591 - FALL 03.
Chitta Baral
Department of Computer Science and Engineering
Arizona State University
Tempe, AZ 85287-5406 USA
http://www.public.asu.edu/
∼ cbaral/cse571-f99/
October 12, 2003
PROBABILITY, BAYES NETS AND CAUSALITY
Probability, Bayes nets and Causality
means: belief in
A
under the assumption that
B
is known with
absolute certainty.
P
A
B
P
A
A
and
B
are independent.
P
A
B, C
P
A
C
A
and
B
are conditionally independent
given
C
Dawid’s notation: (
A B | C )
P than that of joint events.Bayesian philosophers see the conditional relationship as more basic
(^) (
A
B
P
A
B
P
B
Probability, Bayes nets and Causality
Bayesian Networks
Goal:
to provide convenient means of expressing substantive assumptions
to facilitate economical representations of joint probability functions
to facilitate efficient inferences from observations
relationshipIdea: Directed acyclic graphs is used to represent causal or temporal
Basic decomposition scheme
P
A
B
P
A
B
P
B
P
x
1 , x
2 , x
3 ) =
P
x 1 ∧ x 2 ∧ x 3
P
x
1 | x
2 , x
3 ) P
(^) ( x
2
∧
x
3 ) =
P
x
1 | x
2 , x
3 ) P
(^) ( x
2 | x
3 ) P
(^) ( x
3 )
Probability, Bayes nets and Causality
Inference with Bayesian Networks
Prediction and abduction
x
- a set of observations
y
- a set of variables deemed important for prediction or diagnosis
Need to compute
P
y
| x
).
P
y
| x
) =
p ( y, x
p ( x ) = ∑ s P
y, x, s
∑ y,s
P
y, x, s
An example:
The Network ∗
P
tampering
P
f ire
Directed Edges: (
tampering, alarm
f ire, alarm
f ire, smoke
alarm, leaving
leaving, report
Probability, Bayes nets and Causality
P local probability distributions:
(^) ( alarm
f ire, tampering
P
alarm
f ire,
tampering
P
alarm
f ire, tampering
P
alarm
f ire,
tampering
P
smoke
f ire
P
smoke,
f ire
P
leaving
alarm
P
leaving
alarm
P
report
leaving
P
report
leaving
Different kinds of inferences ∗
Diagnostic inferences:
P
f ire
report
Causal inferences (prediction):
P
leaving
tampering
Intercausal inferences:
P
f ire
alarm, tampering
Mixed inferences:
P
alarm
report, f ire
P An illustration:
(^) ( tampering
report, smoke
P (^) ( tampering,report,smoke
)
P
(^) ( report,smoke
)
Probability, Bayes nets and Causality
× 0. 9 × 0.
×
×
Similarly, we can also compute
f 1 ( alarm
T, tampering
F
f 1 ( alarm
F, tampering
T
(^) ) and
f 1 ( alarm
F, tampering
F
∑We can now write the denominator as: tampering,leaving,alarm
P
tampering
P
leaving
alarm
P
report
T
leaving
f 1 ( alarm, tampering
∑
tampering,leaving
P
tampering
P
report
T
leaving
∑
alarm
P
leaving
alarm
f 1 ( alarm, tampering
Let us denote
∑ alarm
P
leaving
alarm
f 1 ( alarm, tampering
by
f 2 ( leaving, tampering
). We can compute it as we compute
f 1
= The denominator can now be written as:
∑
tampering,leaving
P
tampering
P
report
T
leaving
f 2 ( leaving, tampering
∑
tampering
P
tampering
∑
leaving
P
report
T
leaving
f 2 ( leaving, tampering
Let us denote
∑
leaving
P
report
T
leaving
Probability, Bayes nets and Causality
f 2 ( leaving, tampering
) by
f 3 ( tampering
) and compute it like
the other
f i s.
∑The denominator can now be written as: tampering
P
tampering
f 3 ( tampering
Main Issues and challenges
Computing the conditional probabilities efficiently
Inference in general networks in NP-hard
(say for trees).Many efficient algorithms are defined for particular kind of networks ∗
Algorithm based on message passing architecture for trees.
Join-tree propagation
Cutset conditioning
Hybrid combinations of the above two
Approximation methods: stochastic simulation.
Probability, Bayes nets and Causality
distributions can not.)Causal networks can predict the effect of actions. (Simple joint
Stability and autonomy
the network without changing the others.Autonomy: It is possible to change one parent child relationship in
minimum of extra information.Stability: One can predict the effect of external interventions with
of autonomy, the change is local.merely the immediate changes implied by the intervention. Becausefunction for each of the many possible interventions, we specifyAutonomy and intervention: Instead of specifying a new probability
LetDefinition: Causal Bayesian network
P
v
) be a probability distribution on a set
V
of variables, and let
P
x ( v
) denote the distribution resulting from the intervention
do
X
x
) which sets any subset
X
of variables to constants
x
.
Denote by
P
the set of all interventional distributions
P
x ( v ),
X
V
12
Probability, Bayes nets and Causality
including
P
v
) which represents no intervention. A DAG
G
is said to
be a
causal Bayesian network
compatible with
P
iff the following
three conditions hold for every
P
x
P
P
x ( v
) is Markov relative to
G
P
x ( v i ) = 1, for all
V
i
∈
X
, whenever
v i
is consistent with
X
x
.
P
x ( v i | pa
i ) =
P
v i | pa
i ) for all
V
i
∈
X
, whenever
pa
i
is consistent
with
X
x
.
Properties:
for all
v
consistent with
x
:
P
x ( v
) =
∏
{ i | V i ∈ X
}
P
(^) ( v i | pa
i )
For all
i ,
P
(^) ( v i | pa
i ) =
P
pa
i ( v i )
parents, corresponds to causal effects.)(The above ensures, conditional probabilities with respect to
Probability, Bayes nets and Causality
S
1
remains true regardless of what we learn or know about the
season or the pavement.
Falling barometer predicts rain, does not explain it.
Probability, Bayes nets and Causality
Functional Causal Models
Two views of non-determinism
due to our ignorance of the underlying boundary condition.Nature’s laws are deterministic, and randomness surfaces merelyLaplace’s (1814) conception of natural phenomena:
All relationships are inherently stochastic.Modern (quantum mechanical) conception of physics:
Why Pearl’s book uses Laplace’s conception of causality
sciencesbesides the fact that it is used in genetics, econometrics and social
It is more general. ∗
round;relationships (with stochastic inputs), but not the other wayEvery stochastic model can be emulated by many functional
Probability, Bayes nets and Causality
F
is a set of functions
f 1 ,... , f
n }
giving rise to a set of structural
equations of the form:
x
i
=
f i ( pa
i , u
i ),
i
= 1
,... , n
Types of queries that can be answered using functional causal models
- Prediction
: Would the pavement be slippery if we
find
the
sprinkler off?
- Interventions
: Would the pavement be slippery if we
make sure
that the sprinkler is off?
- Counterfactuals
: Would the pavement be slippery
had
the
and the sprinkler is on?sprinkler been off, given that the pavement is in fact not slippery
Prediction using Markovian causal models:
member ofCausal diagram: A graph obtained by having edges from each
P A
i
to
X
i .
called semi-Markovian.If the causal diagram is acyclic then the corresponding model is
Probability, Bayes nets and Causality
the values of
X
variables will be uniquely determined by the
U
variables.
The joint distribution
P
x
1 ,... , x
n ) is determined uniquely by
the distribution
P
u
) of the error variables.
is calledIf in addition the error terms are mutually independent, the model
Markovian
Theorem (Pearl and Verma): Every Markovian causal model
M
induces a distribution
P
x
1 ,... , x
n ) that satisfies the Markov
condition relative to the causal diagram
G
associated with
M
(^) , that
is each variable
X
i
is independent on all its non-descendants, given
its parents
P A
i
in
G
Theorem (Drudgel and Simon): For every Bayesian network
G
characterized by a distribution
P
(^) , there exists a function model that
generates a distribution identical to
P
over the probabilistic specificationAdvantages of doing prediction using causal-functional specification ∗
When organizing knowledge using Markov causal models reliable