1. Representing Uncertainty

2. Aspects of Uncertainty
Suppose you have a flight at 12 noon
When should you leave for SEATAC
What are traffic conditions?
How crowded is security?
Leaving 18 hours early may get you there
But … ?
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3. Decision Theory = Probability + Utility Theory
Min before noon P(arrive-in-time)
20 min
30 min
45 min
60 min
120 min
1080 min
Depends on your preferences
Utility theory: representing & reasoning about preferences
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4. What Is Probability?
Probability: Calculus for dealing with nondeterminism and uncertainty
Cf. Logic
Probabilistic model: Says how often we expect different things to occur
Cf. Function
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5. What Is Statistics? Statistics 1: Describing data
Statistics 2: Inferring probabilistic models from data
Structure
Parameters
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6. Why Should You Care? The world is full of uncertainty
Logic is not enough
Computers need to be able to handle uncertainty
Probability: new foundation for AI (& CS!)
Massive amounts of data around today
Statistics and CS are both about data
Statistics lets us summarize and understand it
Statistics is the basis for most learning
Statistics lets data do our work for us
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7. Outline Basic notions Bayesian networks Statistical learning
Atomic events, probabilities, joint distribution
Inference by enumeration
Independence & conditional independence
Bayes’ rule
Bayesian networks
Statistical learning
Dynamic Bayesian networks (DBNs)
Markov decision processes (MDPs)
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8. Logic vs. Probability Symbol: Q, R … Random variable: Q …
Domain: you specify
e.g. {heads, tails} [1, 6]
Boolean values: T, F
State of the world:
Assignment to Q, R … Z
Atomic event: complete
specification of world: Q… Z
Mutually exclusive
Exhaustive
Prior probability (aka
Unconditional prob: P(Q)
Joint distribution: Prob.
of every atomic event
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9. Syntax for Propositions
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10. Propositions Assume Boolean variables Propositions: A = true B = false
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11. Why Use Probability? E.g. P(AB) = ? P(A) + P(B) - P(AB) A  B A True
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12. Axioms of Probability Theory
All probabilities between 0 and 1
0 ≤ P(A) ≤ 1
P(true) = 1
P(false) = 0.
The probability of disjunction is: A B
A  B
True
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13. Prior Probability Any question can be answered by the
joint distribution
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14. Conditional Probability
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15. Conditional Probability
Def:
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16. Inference by Enumeration
P(toothache)=
= .20 or 20%
This process is called “Marginalization”
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17. Inference by Enumeration
P(toothachecavity =
.20 + ??
.28
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18. Inference by Enumeration
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19. Problems ?? Worst case time: O(dn) Space complexity also O(dn)
Where d = max arity
And n = number of random variables
Space complexity also O(dn)
Size of joint distribution
How get O(dn) entries for table??
Value of cavity &
catch irrelevant -
When computing
P(toothache)
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20. Independence A and B are independent iff:
These two constraints are
logically equivalent
Therefore, if A and B are independent:
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21. Independence A
A  B B
True
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22. Independence Complete independence is powerful but rare
What to do if it doesn’t hold?
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23. Conditional Independence
A&B not independent, since P(A|B) < P(A) A
A  B B
True
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24. Conditional Independence
But: A&B are made independent by C A
A  B
AC
P(A|C) =
P(A|B,C) B
True C
B  C
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25. Conditional Independence
Instead of 7 entries, only need 5
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26. Conditional Independence II
P(catch | toothache, cavity) = P(catch | cavity)
P(catch | toothache,cavity) = P(catch |cavity)
Why only 5 entries in table?
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27. Power of Cond. Independence
Often, using conditional independence reduces the storage complexity of the joint distribution from exponential to linear!!
Conditional independence is the most basic & robust form of knowledge about uncertain environments.
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28. Bayes Rule Simple proof from def of conditional probability: QED:
(Def. cond. prob.)
(Def. cond. prob.)
(Mult by P(H) in line 1)
QED:
(Substitute #3 in #2)
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29. Use to Compute Diagnostic Probability from Causal Probability
E.g. let M be meningitis, S be stiff neck
P(M) = ,
P(S) = 0.1,
P(S|M)= 0.8
P(M|S) =
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30. Bayes’ Rule & Cond. Independence
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31. Bayes Nets
In general, joint distribution P over set of variables (X1 x ... x Xn) requires exponential space for representation & inference
BNs provide a graphical representation of conditional independence relations in P
usually quite compact
requires assessment of fewer parameters, those being quite natural (e.g., causal)
efficient (usually) inference: query answering and belief update
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32. An Example Bayes Net Earthquake Burglary Radio Alarm Nbr1Calls
Pr(B=t) Pr(B=f)
Earthquake
Burglary
Pr(A|E,B)
e,b (0.1)
e,b (0.8)
e,b (0.15)
e,b (0.99)
Radio
Alarm
Nbr1Calls
Nbr2Calls
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33. Earthquake Example (con’t)
Burglary
Alarm
Nbr2Calls
Nbr1Calls
Radio
If I know if Alarm, no other evidence influences my degree of belief in Nbr1Calls
P(N1|N2,A,E,B) = P(N1|A)
also: P(N2|N2,A,E,B) = P(N2|A) and P(E|B) = P(E)
By the chain rule we have
P(N1,N2,A,E,B) = P(N1|N2,A,E,B) ·P(N2|A,E,B)·
P(A|E,B) ·P(E|B) ·P(B)
= P(N1|A) ·P(N2|A) ·P(A|B,E) ·P(E) ·P(B)
Full joint requires only 10 parameters (cf. 32)
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34. BNs: Qualitative Structure
Graphical structure of BN reflects conditional independence among variables
Each variable X is a node in the DAG
Edges denote direct probabilistic influence
usually interpreted causally
parents of X are denoted Par(X)
X is conditionally independent of all nondescendents given its parents
Graphical test exists for more general independence
“Markov Blanket”
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35. Given Parents, X is Independent of Non-Descendants
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36. For Example Earthquake Burglary Radio Alarm Nbr1Calls Nbr2Calls
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37. Given Markov Blanket, X is Independent of All Other Nodes
MB(X) = Par(X)  Childs(X)  Par(Childs(X))
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38. Conditional Probability Tables
Pr(B=t) Pr(B=f)
Earthquake
Burglary
Pr(A|E,B)
e,b (0.1)
e,b (0.8)
e,b (0.15)
e,b (0.99)
Radio
Alarm
Nbr1Calls
Nbr2Calls
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39. Conditional Probability Tables
For complete spec. of joint dist., quantify BN
For each variable X, specify CPT: P(X | Par(X))
number of params locally exponential in |Par(X)|
If X1, X2,... Xn is any topological sort of the network, then we are assured:
P(Xn,Xn-1,...X1) = P(Xn| Xn-1,...X1)·P(Xn-1 | Xn-2,… X1)
… P(X2 | X1) · P(X1)
= P(Xn| Par(Xn)) · P(Xn-1 | Par(Xn-1)) … P(X1)
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40. Inference in BNs The graphical independence representation
yields efficient inference schemes
We generally want to compute
Pr(X), or
Pr(X|E) where E is (conjunctive) evidence
Computations organized by network topology
One simple algorithm:
variable elimination (VE)
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41. P(b|j,m) = P(b) P(e) P(a|b,e)P(j|a)P(m,a)
P(B | J=true, M=true)
Earthquake
Burglary
Radio
Alarm
John
Mary
P(b|j,m) = P(b) P(e) P(a|b,e)P(j|a)P(m,a)
e a
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42. Structure of Computation
Dynamic Programming
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43. Variable Elimination
A factor is a function from some set of variables into a specific value: e.g., f(E,A,N1)
CPTs are factors, e.g., P(A|E,B) function of A,E,B
VE works by eliminating all variables in turn until there is a factor with only query variable
To eliminate a variable:
join all factors containing that variable (like DB)
sum out the influence of the variable on new factor
exploits product form of joint distribution
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44. Example of VE: P(N1) P(N1) = SN2,A,B,E P(N1,N2,A,B,E)
= SN2,A,B,E P(N1|A)P(N2|A) P(B)P(A|B,E)P(E)
= SAP(N1|A) SN2P(N2|A) SBP(B) SEP(A|B,E)P(E)
= SAP(N1|A) SN2P(N2|A) SBP(B) f1(A,B)
= SAP(N1|A) SN2P(N2|A) f2(A)
= SAP(N1|A) f3(A)
= f4(N1)
Earthqk
Burgl
Alarm N1 N2
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45. Notes on VE
Each operation is a simply multiplication of factors and summing out a variable
Complexity determined by size of largest factor
e.g., in example, 3 vars (not 5)
linear in number of vars,
exponential in largest factorelimination ordering greatly impacts factor size
optimal elimination orderings: NP-hard
heuristics, special structure (e.g., polytrees)
Practically, inference is much more
tractable using structure of this sort
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