Probabilistic Representation and Reasoning

1
Probabilistic Representation and Reasoning

• Given a set of facts/beliefs/rules/evidence
• Evaluate a given statement
• Determine the probability of a statement
• Find a statement that optimizes a set of
  constraints
• Most probable explanation (MPE) (Setting of
  hidden variables that best explains observations.)

2
Probability Theory

• Random Variables
• Boolean W1,2 (just like propositional logic).
  Two possible values true, false
• Discrete Weather 2 sunny, cloudy, rainy, snow
• Continuous Temperature 2 lt
• Propositions
• W1,2 true, Weather sunny, Temperature 65
• These can be combined as in propositional logic

3
Example

• Consider a car described by 3 random variables
• Gas 2 true, false There is gas in the tank
• Meter 2 empty,full The gas gauge shows the
  tank is empty or full
• Starts 2 yes,no The car starts when you turn
  the key in the ignition

4
Joint Probability Distribution

• Each row is called a primitive event
• Rows are mutually exclusive and exhaustive
• Corresponds to an 8-sided coin with the
  indicated probabilities

5
Any Query Can Be Answered from the Joint
Distribution

• P(Gas false Æ Meter full Æ Starts yes)
  0.0006
• P(Gas false) 0.2, this is the sum of all
  cells where Gas false
• In general To compute P(Q), for any proposition
  Q, add up the probability in all cells where Q is
  true

6
Notations

• P(G,M,S) denotes the entire joint distribution
  (In the book, P is boldface). It is a table or
  function that maps from G, M, and S to a
  probability.
• P(true,empty,no) denotes a single probability
  value
• P(Gastrue Æ Meterempty Æ Startsno)

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Operations on Probability Tables (1)

• Marginalization (summing away)
• ?M,S P(G,M,S) P(G)
• P(G) is called a marginal probability
  distribution. It consists of two probabilities

8
Conditional Probability

• Suppose we observe that Mfull. What is the
  probability that the car will start?
• P(Syes Mfull)
• Definition P(AB) P(A Æ B) / P(B)

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Conditional Probability

• Select cells that match the condition (Mfull)
• Delete remaining cells and M column
• Renormalize the table to obtain P(S,GMfull)
• Sum away Gas ?G P(S,G Mfull)
  P(SMfull)
• Read answer from P(Syes Mfull) cell

10
Operations on Probability Tables
(2)Conditionalizing

• Construct P(G,S M) by normalizing the subtable
  corresponding to Mfull and normalizing the
  subtable corresponding to Mempty

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Chain Rule of Probability

• P(A,B,C) P(AB,C) P(BC) P(C)
• Proof

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Chain Rule (2)

• Holds for distributions too
• P(A,B,C) P(A B,C) P(B C) P(C)
• This means that for each setting of A,B, and C,
  we can substitute into the equation, and it is
  true.

13
Belief Networks (1)Independence

• Defn Two random variables X and Y are
  independent iff
• P(X,Y) P(X) P(Y)
• Example
• X is a coin with P(Xheads) 0.4
• Y is a coin with P(Yheads) 0.8
• Joint distribution

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Belief Networks (2)Conditional Independence

• Defn Two random variables X and Y are
  conditionally independent given Z iff
• P(X,Y Z) P(XZ) P(YZ)
• Example
• P(S,M G) P(S G) P(M G)
• Intuition G independently causes S and M

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Operations on Probability Tables (3)Conformal
Product

• Allocate space for resulting table and then fill
  in each cell with the product of the
  corresponding cells
• P(S,M G) P(S G) P(M G)

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Properties of Conformal Products

• Commutative
• Associative
• Work on normalized or unnormalized tables
• Work on joint or conditional tables

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Conditional Independence Allows Us to Simplify
the Joint Distribution

• P(G,M,S) P(M,S G) P(G) chain rule
• P(M G) P(S G) P(G) CI

18
Bayesian Networks
Gas
Meter
Starts

• One node for each random variable
• Each node stores a probability distribution
  P(node parents(node))
• Only direct dependencies are shown
• Joint distribution is conformal product of node
  distributions
• P(G,M,S) P(G) P(M G) P(S G)

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Inference in Bayesian Networks

• Suppose we observe that Mfull. What is the
  probability that the car will start?
• P(Syes Mfull)
• Before, we handled this by the following steps
• Remove all rows corresponding to Mempty
• Normalize remaining rows to get P(S,GMfull)
• Sum over G ?G P(S,GMfull) P(S Mfull)
• Read answer from the Syes entry in the table
• We want to get the same result, but without
  constructing the joint distribution first.

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Inference in Bayesian Networks (2)

• Remove all rows corresponding to Mempty from all
  nodes
• P(G) unchanged
• P(M G) becomes PG
• P(S G) unchanged
• Sum over G ?G P(G) PG P(S G)
• Normalize to get P(SMfull)
• Read answer from the Syes entry in the table

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Inference with Tables
?G
22
Inference with Tables
?G
Step 1 Delete Mempty rows from all tables
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Inference with Tables
?G
Step 1 Delete Mempty rows from all tables
Step 2 Perform algebra to push summation inwards
(no-op in this case)
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Inference with Tables
?G
Step 1 Delete Mempty rows from all tables
Step 2 Perform algebra to push summation inwards
(no-op in this case)
Step 3 Form conformal product
25
Inference with Tables
Step 1 Delete Mempty rows from all tables
Step 2 Perform algebra to push summation inwards
(no-op in this case)
Step 3 Form conformal product
Step 4 Sum away G
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Inference with Tables
Step 1 Delete Mempty rows from all tables
Step 2 Perform algebra to push summation inwards
(no-op in this case)
Step 3 Form conformal product
Step 4 Sum away G
Step 5 Normalize
27
Inference with Tables
Step 1 Delete Mempty rows from all tables
Step 2 Perform algebra to push summation inwards
(no-op in this case)
Step 3 Form conformal product
Step 4 Sum away G
Step 5 Normalize
Step 6 Read answer from table 0.6469
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Notes

• We never created the joint distribution
• Deleting the Mempty rows from the individual
  table followed by conformal product has the same
  effect as performing the conformal product first
  and then deleting the Mempty rows
• Normalization can be postponed to the end

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Another Example Asia(all variables Boolean)

• Suppose we observe Sneeze
• What is P(Cold Sneeze) P(CoS)?

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Answering the query

• Joint distribution
• ?A,Ca,Sc P(Co) P(A) P(Sn Co,A)
  P(Ca) P(A Ca) P(Sc Ca)
• Apply evidence sn (Sneeze true)
• ?A,Ca,Sc P(Co) P(A) PCo,A P(Ca) P(A
  Ca) P(Sc Ca)
• Push summations in as far as possible
• P(Co) ?A P(A) PCo,A ?Ca P(A Ca) P(Ca)
  ?Sc P(Sc Ca)
• Evaluate
• P(Co) ?A P(A) PCo,A ?Ca P(A Ca) P(Ca)
  PCa
• P(Co) ?A P(A) PCo,A PA
• P(Co) PCo
• PCo
• Normalize and extract answer

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Pruning Leaves

• Leaf nodes not involved in the evidence or the
  query can be pruned.
• Example Scratch

Query
Evidence
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Greedy algorithm for choosing the elimination
order

• nodes set of tables (after evidence)
• V variables to sum over
• while nodes gt 1 do
• Generate all pairs of tables in nodes that share
  at least one variable
• Compute size of table that would result from
  conformal product of each pair (summing over as
  many variables in V as possible)
• Let (T1,T2) be the pair with smallest resulting
  size
• Delete T1 and T2 from nodes
• Add conformal product ?V T1T2 to nodes
• end

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Example of Greedy Algorithm

• Given tables P(Co), PCo,A, P(ACa), P(Ca)
• Variables to sum A, Ca
• Choose PA ?Ca P(ACa) P(Ca)

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Example of Greedy Algorithm (2)

• Given tables P(Co), PCo,A, PA
• Variables to sum A
• Choose PCo ?A PCo,A PA

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Example of Greedy Algorithm (3)

• Given tables P(Co), PCo
• Variables to sum none
• Choose P2Co P(Co) PCo
• Normalize and extract answer

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Bayesian Network For WUMPUS

• P(P1,1,P1,2, , P4,4, B1,1, B1,2, , B4,4)

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Probabilistic Inference in WUMPUS

• Suppose we have observed
• No breeze in 1,1
• Breeze in 1,2 and 2,1
• No pit in 1,1, 1,2, and 1,3
• What is the probability of a pit in 1,3?
• P(P1,3B1,1,B1,2,B2,1, P1,1,P1,2,P2,1)

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What isP(P1,3B1,1,B1,2,B2,1,
P1,1,P1,2,P2,1)?
false
false
query
false
false
true
true
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Prune Leaves Not Involved in Query or Evidence
false
false
query
false
false
true
true
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Prune Independent Nodes
false
false
query
false

P1,1
P2,2
P1,2
P2,1
P1,3
P3,1
P1,4
B1,1
B1,2
B2,1
false
true
true
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Solve Remaining Network
false
false
query
false
P1,1
P2,2
P1,2
P2,1
P1,3
P3,1
B1,1
B1,2
B2,1
?P2,2,P3,1 P(B1,1P1,1,P1,2,P2,1)
P(B1,2P1,1,P1,2,P1,3) P(B2,1P1,1,P2,1,P2,2,P3,
1) P(P1,1) P(P1,2) P(P2,1) P(P2,2)
P(P1,3) P(P3,1)
false
true
true
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Performing the Inference
NORM ?P2,2,P3,1 P(B1,1P1,1,P1,2,P2,1)
P(B1,2P1,1,P1,2,P1,3) P(B2,1P1,1,P2,1,P2,2,P3,
1) P(P1,1) P(P1,2) P(P2,1) P(P2,2)
P(P1,3) P(P3,1)
NORM ?P2,2,P3,1 PP1,3 PP2,2,P3,1 P(P2,2)
P(P1,3) P(P3,1)
NORM PP1,3 P(P1,3) ?P2,2 P(P2,2) ?P3,1
PP2,2,P3,1 P(P3,1)
P(P1,3) h0.69, 0.31i 31 chance of WUMPUS!
We have reduced the inference to a simple
computation over 2x2 tables.
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Summary

• The Joint Distribution is analogous to the truth
  table for propositional logic. It exponentially
  large, but any query can be answered using it
• Conditional independence allows us to factor the
  joint distribution using conformal products
• Conditional independence relationships are
  conveniently visualized and encoded in a belief
  network DAG
• Given evidence, we can reason efficiently by
  algebraic manipulation of the factored
  representation