part ii: graphical models. challenges of probabilistic models specifying well-defined probabilistic...

Part II: Graphical models

Challenges of probabilistic models

• Specifying well-defined probabilistic models with many variables is hard (for modelers)

• Representing probability distributions over those variables is hard (for computers/learners)

• Computing quantities using those distributions is hard (for computers/learners)

Representing structured distributions

Four random variables:

X1 coin toss produces heads

X2 pencil levitates

X3 friend has psychic powers

X4 friend has two-headed coin

Domain

{0,1}{0,1}{0,1}{0,1}

Joint distribution0000000100100011010001010110011110001001101010111100110111101111

• Requires 15 numbers to specify probability of all values x1,x2,x3,x4

– N binary variables, 2N-1 numbers

• Similar cost when computing conditional probabilities

€

P(x2, x3, x4 | x1 =1) =P(x1 =1,x2,x3,x4 )

P(x1 =1)

How can we use fewer numbers?






Domain

{0,1}{0,1}{0,1}{0,1}

Statistical independence• Two random variables X1 and X2 are independent if

P(x1|x2) = P(x1)

– e.g. coinflips: P(x1=H|x2=H) = P(x1=H) = 0.5

• Independence makes it easier to represent and work with probability distributions

• We can exploit the product rule:

€

P(x1,x2, x3, x4 ) = P(x1 | x2,x3,x4 )P(x2 | x3, x4 )P(x3 | x4 )P(x4 )

€

P(x1, x2, x3, x4 ) = P(x1)P(x2)P(x3)P(x4 )

If x1, x2, x3, and x4 are all independent…

Expressing independence

• Statistical independence is the key to efficient probabilistic representation and computation

• This has led to the development of languages for indicating dependencies among variables

• Some of the most popular languages are based on “graphical models”


• Introduction to graphical models– representation and inference

• Causal graphical models– causality– learning about causal relationships

• Graphical models and cognitive science– uses of graphical models– an example: causal induction

Graphical models

• Express the probabilistic dependency structure among a set of variables (Pearl, 1988)

• Consist of– a set of nodes, corresponding to variables– a set of edges, indicating dependency– a set of functions defined on the graph that

specify a probability distribution

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Undirected graphical models

• Consist of– a set of nodes– a set of edges– a potential for each clique, multiplied together to yield the distribution over variables

• Examples– statistical physics: Ising model, spinglasses– early neural networks (e.g. Boltzmann machines)

X1

X2

X3 X4

X5

Directed graphical modelsX3 X4

X5

X1

X2

• Consist of– a set of nodes– a set of edges– a conditional probability distribution for each node, conditioned on its parents, multiplied together

to yield the distribution over variables

• Constrained to directed acyclic graphs (DAGs)• Called Bayesian networks or Bayes nets

Bayesian networks and Bayes

• Two different problems– Bayesian statistics is a method of inference– Bayesian networks are a form of representation

• There is no necessary connection– many users of Bayesian networks rely upon

frequentist statistical methods– many Bayesian inferences cannot be easily

represented using Bayesian networks

Properties of Bayesian networks

• Efficient representation and inference– exploiting dependency structure makes it easier

to represent and compute with probabilities

• Explaining away– pattern of probabilistic reasoning characteristic

of Bayesian networks, especially early use in AI

Efficient representation and inferenceFour random variables:


X2 pencil levitates



X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)

The Markov assumption

Every node is conditionally independent of its non-descendants, given its parents

€

P(xi | xi+1,...,xk ) = P(xi | Pa(X i ))

where Pa(Xi) is the set of parents of Xi

€

P(x1,..., xk ) = P(x i | Pa(X i))i=1

k

∏

(via the product rule)

Efficient representation and inferenceFour random variables:


X2 pencil levitates



X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)

P(x1, x2, x3, x4) = P(x1|x3, x4)P(x2|x3)P(x3)P(x4)

1 1

24

total = 7 (vs 15)

Reading a Bayesian network

• The structure of a Bayes net can be read as the generative process behind a distribution

• Gives the joint probability distribution over variables obtained by sampling each variable conditioned on its parents

Reading a Bayesian networkFour random variables:


X2 pencil levitates



X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)


Reading a Bayesian network

• The structure of a Bayes net can be read as the generative process behind a distribution

• Gives the joint probability distribution over variables obtained by sampling each variable conditioned on its parents

• Simple rules for determining whether two variables are dependent or independent

• Independence makes inference more efficient

Computing with Bayes nets

X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)


€

P(x2, x3,x4 | x1 =1) =P(x1 =1, x2,x3, x4 )

P(x1 =1)


X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)


€

P(x1 =1) = P(x1 =1, x2,x3,x4 )x2 ,x3 ,x4

∑sum over 8 values


X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)


€

P(x1 =1) = P(x1 | x3, x4 )P(x2 | x3)P(x3)P(x4 )x2 ,x3 ,x4

∑


X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)


€

P(x1 =1) = P(x1 | x3, x4 )P(x3)P(x4 )x3 ,x4

∑sum over 4 values


• Inference algorithms for Bayesian networks exploit dependency structure

• Message-passing algorithms– “belief propagation” passes simple messages between

nodes, exact for tree-structured networks

• More general inference algorithms– exact: “junction-tree”– approximate: Monte Carlo schemes (see Part IV)

Properties of Bayesian networks

• Efficient representation and inference– exploiting dependency structure makes it easier

to represent and compute with probabilities

• Explaining away– pattern of probabilistic reasoning characteristic

of Bayesian networks, especially early use in AI

Assume grass will be wet if and only if it rained last night, or if the sprinklers were left on:

Explaining away

Rain Sprinkler

Grass Wet

.andif0 sSrR ¬=¬==

),|()()(),,( RSWPSPRPWSRP =

rRsSRSwWP ==== orif1),|(

Explaining away

Rain Sprinkler

Grass Wet

)(

)()|()|(

wP

rPrwPwrP =

Compute probability it rained last night, given that the grass is wet:




Explaining away

Rain Sprinkler

Grass Wet

∑′′

′′′′=

sr

srPsrwP

rPrwPwrP

,

),(),|(

)()|()|(





Explaining away

Rain Sprinkler

Grass Wet

),(),(),(

)()|(

srPsrPsrP

rPwrP

¬+¬+=





Explaining away

Rain Sprinkler

Grass Wet


),()(

)()|(

srPrP

rPwrP

¬+=




Explaining away

Rain Sprinkler

Grass Wet

)()()(

)()|(

sPrPrP

rPwrP

¬+=


Between 1 and P(s)

)(rP>




Explaining away

Rain Sprinkler

Grass Wet

Compute probability it rained last night, given that the grass is wet and sprinklers were left on:

)|(

)|(),|(),|(

swP

srPsrwPswrP =

Both terms = 1




Explaining away

Rain Sprinkler

Grass Wet

Compute probability it rained last night, given that the grass is wet and sprinklers were left on:

)(rP=)|(),|( srPswrP =




Explaining away

Rain Sprinkler

Grass Wet

)(rP=)|(),|( srPswrP =)()()(

)()|(

sPrPrP

rPwrP

¬+= )(rP>

“Discounting” to prior probability.




• Formulate IF-THEN rules:– IF Rain THEN Wet– IF Wet THEN Rain

• Rules do not distinguish directions of inference• Requires combinatorial explosion of rules

Contrast w/ production system

Rain

Grass Wet

Sprinkler

IF Wet AND NOT Sprinkler THEN Rain

• Observing rain, Wet becomes more active. • Observing grass wet, Rain and Sprinkler become more active• Observing grass wet and sprinkler, Rain cannot become less active. No explaining away!

• Excitatory links: Rain Wet, Sprinkler Wet

Contrast w/ spreading activation

Rain Sprinkler

Grass Wet

• Observing grass wet, Rain and Sprinkler become more active• Observing grass wet and sprinkler, Rain becomes less active: explaining away

• Excitatory links: Rain Wet, Sprinkler Wet• Inhibitory link: Rain Sprinkler


Rain Sprinkler

Grass Wet

• Each new variable requires more inhibitory connections• Not modular

– whether a connection exists depends on what others exist– big holism problem – combinatorial explosion

Contrast w/ spreading activationRain

Sprinkler

Grass Wet

Burst pipe


QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

(McClelland & Rumelhart, 1981)

Graphical models

• Capture dependency structure in distributions

• Provide an efficient means of representing and reasoning with probabilities

• Allow kinds of inference that are problematic for other representations: explaining away– hard to capture in a production system– more natural than with spreading activation

Causal graphical models

• Graphical models represent statistical dependencies among variables (ie. correlations)– can answer questions about observations

• Causal graphical models represent causal dependencies among variables (Pearl, 2000)

– express underlying causal structure– can answer questions about both observations and

interventions (actions upon a variable)

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Bayesian networks



X2 pencil levitates



X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)

Nodes: variablesLinks: dependency

Each node has a conditional probability distribution

Data: observations of x1, ..., x4

Causal Bayesian networks



X2 pencil levitates



X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)

Nodes: variablesLinks: causality

Each node has a conditional probability distribution

Data: observations of and interventions on x1, ..., x4

Interventions



X2 pencil levitates



X1 X2

X3X4P(x4) P(x3)

P(x2|x3)P(x1|x3, x4)

hold down pencil

X

Cut all incoming links for the node that we intervene on

Compute probabilities with “mutilated” Bayes net

• Strength: how strong is a relationship?• Structure: does a relationship exist?

E

B C

E

B CB B

Learning causal graphical models

• Strength: how strong is a relationship?

E

B C

E

B CB B

Causal structure vs. causal strength

• Strength: how strong is a relationship?– requires defining nature of relationship

E

B C

w0 w1

E

B C

w0

B B


Parameterization

• Structures: h1 = h0 =

• Parameterization:

E

B C

E

B C

C B

0 01 00 11 1

h1: P(E = 1 | C, B) h0: P(E = 1| C, B)

p00

p10

p01

p11

p0

p0

p1

p1

Generic

Parameterization



E

B C

E

B C

w0 w1w0

w0, w1: strength parameters for B, C

C B

0 01 00 11 1

h1: P(E = 1 | C, B) h0: P(E = 1| C, B)

0w1

w0

w1+ w0

00w0

w0

Linear

Parameterization



E

B C

E

B C

w0 w1w0

w0, w1: strength parameters for B, C

C B

0 01 00 11 1

h1: P(E = 1 | C, B) h0: P(E = 1| C, B)

0w1

w0

w1+ w0 – w1 w0

00w0

w0

“Noisy-OR”

Parameter estimation

• Maximum likelihood estimation:

maximize i P(bi,ci,ei; w0, w1)

• Bayesian methods: as in Part I

• Structure: does a relationship exist?

E

B C

E

B CB B


Approaches to structure learning

• Constraint-based:– dependency from statistical tests (eg. 2)– deduce structure from dependencies E

B CB

(Pearl, 2000; Spirtes et al., 1993)


E

B CB• Constraint-based:– dependency from statistical tests (eg. 2)– deduce structure from dependencies



E

B CB

Attempts to reduce inductive problem to deductive problem

• Constraint-based:– dependency from statistical tests (eg. 2)– deduce structure from dependencies



E

B CB

• Bayesian:– compute posterior

probability of structures,

given observed dataE

B C

E

B C

P(h|data) P(data|h) P(h)

P(h1|data) P(h0|data)

• Constraint-based:– dependency from statistical tests (eg. 2)– deduce structure from dependencies


(Heckerman, 1998; Friedman, 1999)

Bayesian Occam’s Razor

All possible data sets d

P(d

| h

)

h0 (no relationship)

h1 (relationship)

€

P(d

∑ d | h) = 1For any model h,

Causal graphical models

• Extend graphical models to deal with interventions as well as observations

• Respecting the direction of causality results in efficient representation and inference

• Two steps in learning causal models– strength: parameter estimation– structure: structure learning

Uses of graphical models

• Understanding existing cognitive models– e.g., neural network models

• Representation and reasoning– a way to address holism in induction (c.f. Fodor)

• Defining generative models– mixture models, language models (see Part IV)

• Modeling human causal reasoning

Human causal reasoning

• How do people reason about interventions?(Gopnik, Glymour, Sobel, Schulz, Kushnir & Danks, 2004; Lagnado

& Sloman, 2004; Sloman & Lagnado, 2005; Steyvers, Tenenbaum, Wagenmakers & Blum, 2003)

• How do people learn about causal relationships?– parameter estimation (Shanks, 1995; Cheng, 1997)

– constraint-based models (Glymour, 2001)

– Bayesian structure learning (Steyvers et al., 2003; Griffiths & Tenenbaum, 2005)

Causation from contingencies

“Does C cause E?”(rate on a scale from 0 to 100)

E present (e+)

E absent (e-)

C present(c+)

C absent(c-)

a

b

c

d

Two models of causal judgment

• Delta-P (Jenkins & Ward, 1965):

• Power PC (Cheng, 1997):

dccbaacePcePP +−+=−≡Δ −+++ )()|()|(

)()|(1 dcd

P

ceP

Pp

+Δ

=−

Δ≡ −+Power

Buehner and Cheng (1997)

People

P

Power

0.00 0.25 0.50 0.75 1.00 P


People

P

Power

Constant ΔP, changing judgments


People

P

Power

Constant causal power, changing judgments


People

P

Power

ΔP = 0, changing judgments

• Strength: how strong is a relationship?• Structure: does a relationship exist?

E

B C

w0 w1

E

B C

w0

B B


Causal strength

• Assume structure:

ΔP and causal power are maximum likelihood estimates of the strength parameter w1, under different parameterizations for P(E|B,C): – linear ΔP, Noisy-OR causal power

E

B C

w0 w1

B

• Hypotheses: h1 = h0 =

• Bayesian causal inference:

support =

E

B C

E

B CB B

€

P(d | h1) = P(d | w0,w1) p(w0,w1 | h1)0

1

∫0

1

∫ dw0 dw1

€

P(d | h0) = P(d | w0) p(w0 | h0)0

1

∫ dw0

Causal structure

P(d|h1)

P(d|h0)likelihood ratio (Bayes factor) gives evidence in favor of h1

People

ΔP (r = 0.89)

Power (r = 0.88)

Support (r = 0.97)


The importance of parameterization

• Noisy-OR incorporates mechanism assumptions:– generativity: causes increase probability of effects

– each cause is sufficient to produce the effect

– causes act via independent mechanisms(Cheng, 1997)

• Consider other models:– statistical dependence: 2 test

– generic parameterization (cf. Anderson, 1990)

People

Support (Noisy-OR)

2

Support (generic)

Generativity is essential

• Predictions result from “ceiling effect”– ceiling effects only matter if you believe a cause increases the

probability of an effect

P(e+|c+)P(e+|c-)

8/88/8

6/86/8

4/84/8

2/82/8

0/80/8

Support 10050

0

Blicket detector (Dave Sobel, Alison Gopnik, and colleagues)

See this? It’s a blicket machine. Blickets make it go.

Let’s put this oneon the machine.

Oooh, it’s a blicket!

– Two objects: A and B– Trial 1: A B on detector – detector active– Trial 2: A on detector – detector active– 4-year-olds judge whether each object is a blicket

• A: a blicket (100% say yes)

• B: probably not a blicket (34% say yes)

“Backwards blocking” (Sobel, Tenenbaum & Gopnik, 2004)

AB Trial A TrialA B

Possible hypotheses

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

E

BA

Bayesian inference

• Evaluating causal models in light of data:

• Inferring a particular causal relation:

€

P(hi | d) =P(d | hi)P(hi)

P(d | h j )P(h j )j

∑

∑∈

→=→H

jh

jj dhPhEAPdEAP )|()|()|(

Bayesian inference

With a uniform prior on hypotheses, and the generic parameterization

A B

Probability of being a blicket

0.32 0.32

0.34 0.34

Modeling backwards blocking

Assume…

• Links can only exist from blocks to detectors

• Blocks are blickets with prior probability q

• Blickets always activate detectors, but detectors never activate on their own– deterministic Noisy-OR, with wi = 1 and w0 = 0

€

P(B → E | d) =P(h01) + P(h11)

P(h01) + P(h10) + P(h11)=

q

q + q(1− q)

P(E=1 | A=1, B=1): 0 1 1 1

E

BA

E

BA

E

BA

E

BA

P(h00) = (1 – q)2 P(h10) = q(1 – q)P(h01) = (1 – q) q P(h11) = q2


€

P(B → E | d) =P(h11)

P(h10) + P(h11)= q

P(E=1 | A=1, B=0): 0 1 1

P(E=1 | A=1, B=1): 1 1 1

E

BA

E

BA

E

BA

P(h10) = q(1 – q)P(h01) = (1 – q) q P(h11) = q2


Manipulating prior probability(Tenenbaum, Sobel, Griffiths, & Gopnik, submitted)

AB Trial A TrialInitial

Summary

• Graphical models provide solutions to many of the challenges of probabilistic models– defining structured distributions– representing distributions on many variables– efficiently computing probabilities

• Causal graphical models provide tools for defining rational models of human causal reasoning and learning

part ii: graphical models. challenges of probabilistic models specifying well-defined probabilistic...

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