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### But does a BN represent a belief state?In other words, can we compute the full joint distribution of the propositions from it?

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

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)

Grade

SAT

Earthquake

causes

Alarm

effects

JohnCalls

MaryCalls

A More Complex BNIntuitive meaning of arc from x to y: “x has direct influence on y”

Directed acyclic graph

Earthquake

Alarm

JohnCalls

MaryCalls

A More Complex BNSize of the CPT for a node with k parents: 2k

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?

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(BJ) P(B) P(J)

P(BJ|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(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

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

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

Earthquake

Alarm

JohnCalls

MaryCalls

- P(JMABE)= P(JM|A,B,E) P(ABE)= P(J|A,B,E) P(M|A,B,E) P(ABE)(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(ABE) = P(A|B,E) P(B|E) P(E) = P(A|B,E) P(B) P(E)(B and E are independent)
- P(JMABE) = P(J|A)P(M|A)P(A|B,E)P(B)P(E)

Earthquake

Alarm

JohnCalls

MaryCalls

Calculation of Joint ProbabilityP(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

Earthquake

Alarm

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

JohnCalls

MaryCalls

full joint distribution table

Calculation of Joint ProbabilityP(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

Earthquake

Alarm

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

JohnCalls

MaryCalls

full joint distribution table

Calculation of Joint ProbabilitySince a BN defines the full joint distribution of a set of propositions, it represents a belief state

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

Earthquake

Alarm

JohnCalls

MaryCalls

What does the BN encode?Burglary Earthquake

JohnCallsMaryCalls | 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

Reading off independence relationships- How about Burglary Earthquake | Alarm ?
- No! Why?

Earthquake

Alarm

JohnCalls

MaryCalls

Reading off independence relationships- How about Burglary Earthquake | Alarm ?
- No! Why?
- P(BE|A) = P(A|B,E)P(BE)/P(A) = 0.00075
- P(B|A)P(E|A) = 0.086

Earthquake

Alarm

JohnCalls

MaryCalls

Reading off independence relationships- How about Burglary Earthquake | JohnCalls?
- No! Why?
- Knowing JohnCalls affects the probability of Alarm, which makes Burglary and Earthquake dependent

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

- 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

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inferenceSuppose 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 inferenceNow, 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 inferenceNow, 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 EvidenceSuppose we want to compute P(Alarm|Earthquake)

Earthquake

Alarm

JohnCalls

MaryCalls

Top-Down inference with EvidenceSuppose 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

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

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)?

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)?
- 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

- 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)]

How do we read this?

- 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) =

How do we read this?

- 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?

How do we read this?

- 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!

How do we read this?

- 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’)]

Be careful about indices!

Toothache

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…

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

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

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

Homework

- Read R&N 14.1-3

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