1 / 56

# Bayesian Networks - PowerPoint PPT Presentation

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…

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Bayesian Networks' - whitby

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Bayesian Networks

• 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

P(I)

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

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

Intel.

P(S|I)

P(G|I)

SAT

Earthquake

causes

Alarm

effects

JohnCalls

MaryCalls

A More Complex BN

Intuitive meaning of arc from x to y: “x has direct influence on y”

Directed acyclic graph

Earthquake

Alarm

JohnCalls

MaryCalls

A More Complex BN

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

• 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?

Earthquake

Alarm

JohnCalls

MaryCalls

What does the BN encode?

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

P(BJ) P(B) P(J)

P(BJ|A) =P(B|A) P(J|A)

For example, John doesnot observe any burglariesdirectly

Earthquake

Alarm

JohnCalls

MaryCalls

What does the BN encode?

• The beliefs JohnCalls and MaryCalls are independent given Alarm or Alarm

P(BJ|A) =P(B|A) P(J|A)

P(JM|A) =P(J|A) P(M|A)

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

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

Earthquake

Alarm

JohnCalls

MaryCalls

What does the BN encode?

Burglary and

Earthquake are

independent

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

• 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

• 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?

Earthquake

Alarm

JohnCalls

MaryCalls

Calculation of Joint Probability

P(JMABE) = ??

Earthquake

Alarm

JohnCalls

MaryCalls

• P(JMABE)= P(JM|A,B,E)  P(ABE)= P(J|A,B,E)  P(M|A,B,E)  P(ABE)(J and M are independent given A)

• P(J|A,B,E) = P(J|A)(J and B and J and E are independent given A)

• P(M|A,B,E) = P(M|A)

• P(ABE) = P(A|B,E)  P(B|E)  P(E) = P(A|B,E)  P(B)  P(E)(B and E are independent)

• P(JMABE) = P(J|A)P(M|A)P(A|B,E)P(B)P(E)

Earthquake

Alarm

JohnCalls

MaryCalls

Calculation of Joint Probability

P(JMABE)= 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

Earthquake

Alarm

P(x1x2…xn) = Pi=1,…,nP(xi|parents(Xi))

JohnCalls

MaryCalls

 full joint distribution table

Calculation of Joint Probability

P(JMABE)= 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

Earthquake

Alarm

P(x1x2…xn) = Pi=1,…,nP(xi|parents(Xi))

JohnCalls

MaryCalls

 full joint distribution table

Calculation of Joint Probability

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

P(JMABE)= 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

Earthquake

Alarm

JohnCalls

MaryCalls

What does the BN encode?

Burglary  Earthquake

JohnCallsMaryCalls | Alarm

JohnCalls Burglary | Alarm

JohnCalls Earthquake | Alarm

MaryCalls Burglary | Alarm

MaryCalls Earthquake | Alarm

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

Earthquake

Alarm

JohnCalls

MaryCalls

• How about Burglary Earthquake | Alarm ?

• No! Why?

Earthquake

Alarm

JohnCalls

MaryCalls

• How about Burglary  Earthquake | Alarm ?

• No! Why?

• P(BE|A) = P(A|B,E)P(BE)/P(A) = 0.00075

• P(B|A)P(E|A) = 0.086

Earthquake

Alarm

JohnCalls

MaryCalls

• How about Burglary  Earthquake | JohnCalls?

• No! Why?

• Knowing JohnCalls affects the probability of Alarm, which makes Burglary and Earthquake dependent

• 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)

• 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

• 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)

• For the moment we’ll assume no evidence variables

• Find P(Xq)

• 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

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference

Suppose we want to compute P(Alarm)

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference

Suppose we want to compute P(Alarm)

P(Alarm) = Σb,eP(A,b,e)

P(Alarm) = Σb,e P(A|b,e)P(b)P(e)

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference

• Suppose we want to compute P(Alarm)

• P(Alarm) = Σb,eP(A,b,e)

• P(Alarm) = Σb,e P(A|b,e)P(b)P(e)

• P(Alarm) = 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)

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference

• Suppose we want to compute P(Alarm)

• P(A) = Σb,eP(A,b,e)

• P(A) = Σb,e P(A|b,e)P(b)P(e)

• 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)

• 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

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference

Now, suppose we want to compute P(MaryCalls)

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference

Now, suppose we want to compute P(MaryCalls)

P(M) = P(M|A)P(A) + P(M|A) P(A)

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference

Now, suppose we want to compute P(MaryCalls)

P(M) = P(M|A)P(A) + P(M|A) P(A)

P(M) = 0.70*0.00252 + 0.01*(1-0.0252) = 0.0117

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference with Evidence

Suppose we want to compute P(Alarm|Earthquake)

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference with Evidence

Suppose we want to compute P(A|e)

P(A|e) = Σb P(A,b|e)

P(A|e) = Σb P(A|b,e)P(b)

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference with Evidence

• Suppose we want to compute P(A|e)

• P(A|e) = Σb P(A,b|e)

• P(A|e) = Σb P(A|b,e)P(b)

• P(A|e) = 0.95*0.001 +0.29*0.999 + = 0.29066

• 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

Toothache

Querying the BN

• The BN gives P(T|C)

• P(AB) = P(A|B) P(B) = P(B|A) P(A)

• So… P(A|B) = P(B|A) P(A) / P(B)

• 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)?

• 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)?

• P(B) = Sa P(B,A=a) [marginalization]

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

• So, P(B) = SaP(B | A=a) P(A=a)

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(A|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(A|B)?

• P(A|B) = P(B|A)P(A)/P(B) [Bayes rule]

• P(B) = SaP(B | A=a) P(A=a) [Last slide]

• So, P(A|B) = P(B|A)P(A) / [SaP(B | A=a) P(A=a)]

• P(A|B) = P(B|A)P(A) / [SaP(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) =

• P(A|B) = P(B|A)P(A) / [SaP(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) = P(B=b|A=a)P(A=a) / [SaP(B=b | A=a) P(A=a)]

Are these the same a?

• P(A|B) = P(B|A)P(A) / [SaP(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) = P(B=b|A=a)P(A=a) / [SaP(B=b | A=a) P(A=a)]

Are these the same a?

NO!

• P(A|B) = P(B|A)P(A) / [SaP(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) = P(B=b|A=a)P(A=a) / [Sa’P(B=b | A=a’) P(A=a’)]

Toothache

Querying the BN

• The BN gives P(T|C)

• P(Cavity|Toothache) = P(Toothache|Cavity) P(Cavity) P(Toothache)[Bayes’ rule]

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

Denominator computed by summing out numerator over Cavity and Cavity

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

Cause

Effect1

Effect2

Effectn

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

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

Given features, what class?

Naïve Bayes Classifier

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

Spam / Not Spam

English / French/ Latin

Class

Feature1

Feature2

Featuren

Word occurrences

• 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

• Medical diagnosis

• Troubleshooting of hardware/software systems

• Fraud/uncollectible debt detection

• Data mining

• Analysis of genetic sequences

• Data interpretation, computer vision, image understanding

Gas

SparkPlugs

Starts

Moves

More Complicated Singly-Connected Belief Net

R1

Above

R2

R4

R3