b ayesian n etworks. s ignificance of c onditional independence consider grade(cs101), intelligence,...

BAYESIAN NETWORKS

SIGNIFICANCE OF CONDITIONAL INDEPENDENCE

Consider Grade(CS101), Intelligence, and SAT Ostensibly, the grade in a course doesn’t

have a direct relationship with SAT scores but good students are more likely to get good

SAT scores, so they are not independent… It is reasonable to believe that Grade(CS101)

and SAT are conditionally independent given Intelligence

BAYESIAN NETWORK Explicitly represent independence among

propositions Notice that Intelligence is the “cause” of both Grade

and SAT, and the causality is represented explicitly

Intel.

Grade

high low

0.3 0.7

SAT

P(I,G,S) = P(G,S|I) P(I) = P(G|I) P(S|I) P(I)

I \ G ‘A’ ‘B’ ‘C’

low 0.2 0.34 0.46

high 0.74 0.17 0.09

P(I)

P(G|I) P(S|I)

I \ S low high

low 0.95 0.05

high 0.2 0.8

A MORE COMPLEX BN

Burglary Earthquake

Alarm

MaryCallsJohnCalls

causes

effects

Directed acyclic graph

Intuitive meaning of arc from x to y:

“x has direct influence on y”

B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

Size of the CPT for a node with k parents: 2k

A MORE COMPLEX BN

10 probabilities, instead of 31

SIGNIFICANCE OF BAYESIAN NETWORKS

If we know that some variables are conditionally independent, we should be able to decompose joint distribution to take advantage of it

Bayesian networks are a way of efficiently factoring the joint distribution into conditional probabilities

And also building complex joint distributions from smaller models of probabilistic relationships

But… What knowledge does the BN encode about the

distribution? How do we use a BN to compute probabilities of

variables that we are interested in?

WHAT DOES THE BN ENCODE?

Each of the beliefs JohnCalls and MaryCalls is independent of Burglary and Earthquake given Alarm or Alarm

Burglary Earthquake

Alarm

MaryCallsJohnCalls

For example, John doesnot observe any burglariesdirectly

P(BJ) P(B) P(J)P(BJ|A) = P(B|A) P(J|A)


The beliefs JohnCalls and MaryCalls are independent given Alarm or Alarm

For instance, the reasons why John and Mary may not call if there is an alarm are unrelated

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(BJ|A) = P(B|A) P(J|A)P(JM|A) = P(J|A) P(M|A)P(BJ|A) = P(B|A) P(J|A)P(JM|A) = P(J|A) P(M|A)

A node is independent of its non-descendants given its parents


WHAT DOES THE BN ENCODE?Burglary Earthquake

Alarm

MaryCallsJohnCallsA node is

independent of its non-descendants given its parents


Burglary and Earthquake are independent

Burglary and Earthquake are independent

The beliefs JohnCalls and MaryCalls are independent given Alarm or Alarm

For instance, the reasons why John and Mary may not call if there is an alarm are unrelated

LOCALLY STRUCTURED WORLD

A world is locally structured (or sparse) if each of its components interacts directly with relatively few other components

In a sparse world, the CPTs are small and the BN contains much fewer probabilities than the full joint distribution

If the # of entries in each CPT is bounded by a constant, i.e., O(1), then the # of probabilities in a BN is linear in n – the # of propositions – instead of 2n for the joint distribution

BUT DOES A BN REPRESENT A BELIEF STATE?

IN OTHER WORDS, CAN WE COMPUTE THE FULL JOINT DISTRIBUTION OF THE PROPOSITIONS FROM IT?

CALCULATION OF JOINT PROBABILITY

B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

P(JMABE) = ??


B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

P(x1x2…xn) = Pi=1,…,nP(xi|parents(Xi))

full joint distribution table

P(JMABE)= P(J|A)P(M|A)P(A|B,E)P(b)P(e)= 0.9 x 0.7 x 0.001 x 0.999 x 0.998= 0.00062

Since a BN defines the full joint distribution of a set of propositions, it represents a belief state

Since a BN defines the full joint distribution of a set of propositions, it represents a belief state


Burglary EarthquakeJohnCalls MaryCalls | AlarmJohnCalls Burglary | AlarmJohnCalls Earthquake | AlarmMaryCalls Burglary | AlarmMaryCalls Earthquake | Alarm

Burglary Earthquake

Alarm

MaryCallsJohnCalls

A node is independent of its non-descendents, given its parents

READING OFF INDEPENDENCE RELATIONSHIPS

How about Burglary Earthquake | Alarm ? No! Why?

Burglary Earthquake

Alarm

MaryCallsJohnCalls


How about Burglary Earthquake | JohnCalls? No! Why? Knowing JohnCalls affects the probability of

Alarm, which makes Burglary and Earthquake dependent

Burglary Earthquake

Alarm

MaryCallsJohnCalls

INDEPENDENCE RELATIONSHIPS

Rough intuition (this holds for tree-like graphs, polytrees): Evidence on the (directed) road between two

variables makes them independent Evidence on an “A” node makes descendants

independent Evidence on a “V” node, or below the V, makes

the ancestors of the variables dependent (otherwise they are independent)

Formal property in general case : D-separation independence (see R&N)

BENEFITS OF SPARSE MODELS

Modeling Fewer relationships need to be encoded (either

through understanding or statistics) Large networks can be built up from smaller ones

Intuition Dependencies/independencies between variables

can be inferred through network structures Tractable probabilistic inference

PROBABILISTIC INFERENCE IN BAYES NETS

PROBABILISTIC INFERENCE

Is the following problem…. Given:

A belief state P(X1,…,Xn) in some form (e.g., a Bayes net or a joint probability table)

A query variable indexed by q Some subset of evidence variables indexed by

e1,…,ek

Find: P(Xq | Xe1 ,…, Xek)

PROBABILISTIC INFERENCE WITHOUT EVIDENCE

For the moment we’ll assume no evidence variables

Find P(Xq)

PROBABILISTIC INFERENCE WITHOUT EVIDENCE

In joint probability table, we can find P(Xq) through marginalization Computational complexity: 2n-1

(assuming boolean random variables)

In Bayesian networks, we find P(Xq) using top down inference

B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

TOP-DOWN INFERENCESuppose we want to compute P(Alarm)

B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

TOP-DOWN INFERENCESuppose we want to compute P(Alarm)1. P(Alarm) = Σb,e P(A,b,e)2. P(Alarm) = Σb,e P(A|b,e)P(b)P(e)

B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

TOP-DOWN INFERENCESuppose we want to compute P(Alarm)1. P(A) = Σb,e P(A,b,e)2. P(A) = Σb,e P(A|b,e)P(b)P(e)3. P(A) = P(A|B,E)P(B)P(E) +

P(A|B, E)P(B)P(E) +P(A|B,E)P(B)P(E) +P(A|B,E)P(B)P(E)

4. P(A) = 0.95*0.001*0.002 +0.94*0.001*0.998 +0.29*0.999*0.002 +0.001*0.999*0.998= 0.00252

B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

TOP-DOWN INFERENCENow, suppose we want to compute P(MaryCalls)

B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

TOP-DOWN INFERENCENow, suppose we want to compute P(MaryCalls)1. P(M) = P(M|A)P(A) + P(M| A) P(A)

B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

TOP-DOWN INFERENCENow, suppose we want to compute P(MaryCalls)1. P(M) = P(M|A)P(A) + P(M| A) P(A)2. P(M) = 0.70*0.00252 + 0.01*(1-0.0252)

= 0.0117

B E P(A|…)

TTFF

TFTF

0.950.940.290.001

Burglary Earthquake

Alarm

MaryCallsJohnCalls

P(B)

0.001

P(E)

0.002

A P(J|…)

TF

0.900.05

A P(M|…)

TF

0.700.01

TOP-DOWN INFERENCE WITH EVIDENCE

Suppose we want to compute P(Alarm|Earthquake)

TOP-DOWN INFERENCE

Only works if the graph of ancestors of a variable is a polytree

Evidence given on ancestor(s) of the query variable

Efficient: O(d 2k) time, where d is the number of ancestors

of a variable, with k a bound on # of parents Evidence on an ancestor cuts off influence of

portion of graph above evidence node

QUERYING THE BN

The BN gives P(T|C) What about P(C|T)?

Cavity

Toothache

P(C)

0.1

C P(T|C)

TF

0.40.01111

BAYES’ RULE

P(AB) = P(A|B) P(B)= P(B|A) P(A)

So… P(A|B) = P(B|A) P(A) / P(B)

APPLYING BAYES’ RULE Let A be a cause, B be an effect, and let’s say we

know P(B|A) and P(A) (conditional probability tables)

What’s P(B)?



What’s P(B)? P(B) = Sa P(B,A=a) [marginalization]

P(B,A=a) = P(B|A=a)P(A=a) [conditional probability]

So, P(B) = Sa P(B | A=a) P(A=a)



What’s P(A|B)?

HOW DO WE READ THIS?

P(A|B) = P(B|A)P(A) / [Sa P(B | A=a) P(A=a)] [An equation that holds for all values A can take on,

and all values B can take on] P(A=a|B=b) =



and all values B can take on] P(A=a|B=b) = P(B=b|A=a)P(A=a) /

[Sa P(B=b | A=a) P(A=a)]

Are these the same a?




[Sa P(B=b | A=a) P(A=a)]

Are these the same a?

NO!




[Sa’ P(B=b | A=a’) P(A=a’)]

Be careful about indices!

QUERYING THE BN The BN gives P(T|C) What about P(C|T)? P(Cavity|Toothache) =

P(Toothache|Cavity) P(Cavity)

P(Toothache)

[Bayes’ rule]

Querying a BN is just applying Bayes’ rule on a larger scale…

Cavity

Toothache

P(C)

0.1

C P(T|C)

TF

0.40.01111 Denominator computed by

summing out numerator over Cavity and Cavity

NAÏVE BAYES MODELS

P(Cause,Effect1,…,Effectn)= P(Cause) Pi P(Effecti | Cause)

Cause

Effect1 Effect2 Effectn

NAÏVE BAYES CLASSIFIER

P(Class,Feature1,…,Featuren)= P(Class) Pi P(Featurei | Class)

Class

Feature1 Feature2 Featuren

P(C|F1,….,Fk) = P(C,F1,….,Fk)/P(F1,….,Fk)

= 1/Z P(C) Pi P(Fi|C)

Given features, what class?

Spam / Not Spam

English / French/ Latin

…

Word occurrences

MORE COMPLEX NETWORKS

Two limitations of current discussion: “Tree like” networks Evidence at specific locations

Next time: Tractable exact inference without “loops” With loops: NP-hard (compare to CSPs) Approximate inference for general networks

SOME APPLICATIONS OF BN

Medical diagnosis Troubleshooting of hardware/software

systems Fraud/uncollectible debt detection Data mining Analysis of genetic sequences Data interpretation, computer vision, image

understanding

MORE COMPLICATED SINGLY-CONNECTED BELIEF NET

Radio

Battery

SparkPlugs

Starts

Gas

Moves

Region = {Sky, Tree, Grass, Rock}

R2

R4R3

R1

Above

HOMEWORK

Read R&N 14.1-3

b ayesian n etworks. s ignificance of c onditional independence consider grade(cs101), intelligence,...

Documents