1. Probabilistic Reasoning

2. 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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3. Bayes Rule Simple proof from def of conditional probability: QED:
(Def. cond. prob.)
(Def. cond. prob.)
قانون بیز
(Mult by P(H) in line 1)
QED:
(Substitute #3 in #2)
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4. 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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5. Bayes’ Rule & Cond. Independence
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6. 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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7. BN What do the arrows really mean?
Topology may happen to encode causal structure
Topology only guaranteed to encode conditional independence
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8. P(X1, X2,... Xn ) = P(X1)P(X2)... P(Xn)
Independence
If X1, X2,... Xn are mutually independent, then
P(X1, X2,... Xn ) = P(X1)P(X2)... P(Xn)
Joint can be specified with n parameters
cf. the usual 2n-1 parameters required
While extreme independence is unusual,
Conditional independence is common
BNs exploit this conditional independence
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9. An Example Bayes Net P(E) 0.002 Earthquake Burglary Alarm Nbr1Calls
Pr(B=t))
0.001
Earthquake
Burglary
Pr(A|E,B)
e,b
e,b
e,b
e,b
Alarm
Nbr1Calls
Nbr2Calls
A P(N1)
T 0.9
F 0.05
A P(N2)
T 0.7
F 0.01
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10. Earthquake Example (con’t)
Burglary
Alarm
Nbr2Calls
Nbr1Calls
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)
10 =
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11. 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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12. Given Parents, X is Independent of Non-Descendants
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13. For Example Earthquake Burglary Radio Alarm Nbr1Calls Nbr2Calls
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14. Given Markov Blanket, X is Independent of All Other Nodes
MB(X) = Par(X)  Childs(X)  Par(Childs(X))
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15. 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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16. 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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17. Bayes Nets Representation Summary
Bayes nets compactly encode joint distributions
Guaranteed independencies of distributions can be deduced from BN graph structure
D-separation gives precise conditional independence guarantees from graph alone
A Bayes’ net’s joint distribution may have further (conditional) independence that is not detectable until you inspect its specific distribution
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18. Inference in BNs
Inference: calculating some useful quantity from a joint probability distribution
The graphical independence representation
yields efficient inference schemes
Posterior probability:
Most likely explanation:
Computations organized by network topology
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19. 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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20. Inference by Enumeration
Given unlimited time, inference in BNs is easy
Recipe:
State the marginal probabilities you need
Figure out ALL the atomic probabilities you need
Calculate and combine them
E.g.
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21. In this simple method, we only need the BN to synthesize the joint entries
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22. Inference by Enumeration?
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23. Variable Elimination Why is inference by enumeration so slow?
You join up the whole joint distribution before you sum out the hidden variables
You end up repeating a lot of work!
Idea: interleave joining and marginalizing!
Called “Variable Elimination”
Still NP-hard, but usually much faster than inference by enumeration
We’ll need some new notation to define VE 2
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24. Factor 1 Joint distribution: P(X,Y) Selected joint: P(x,Y)
Entries P(x,y) for all x, y
Sums to 1
Selected joint: P(x,Y)
A slice of the joint distribution
Entries P(x,y) for fixed x, all y
Sums to P(x)
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25. Factor 2 Family of conditionals: P(X |Y) Single conditional: P(Y | x)
Multiple conditionals
Entries P(x | y) for all x, y
Sums to |Y|
Single conditional: P(Y | x)
Entries P(y | x) for fixed
x, all y
Sums to 1
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26. Specified family: P(y | X) Entries P(y | x) for fixed y, but for all x
Sums to … who knows!
In general, when we write P(Y1 … YN | X1 … XM)
It is a “factor,” a multi-dimensional array
Its values are all P(y1 … yN | x1 … xM)
Any assigned X or Y is a dimension missing (selected) from the array
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27. Example: Traffic Domain
Random Variables
R: Raining
T: Traffic
L: Late for class!
First query: P(L)
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28. Operation 1: Join Factors
First basic operation: joining factors
Combining factors:
Just like a database join
Get all factors over the joining variable
Build a new factor over the union of the variables involved
Example: Join on R
Computation for each entry: pointwise products
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29. Example: Multiple Joins
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30. © Daniel S. Weld

31. Operation 2: Eliminate Second basic operation: marginalization
Take a factor and sum out a variable
Shrinks a factor to a smaller one
A projection operation
Example:
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32. Multiple Elimination
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33. P(L) : Marginalizing Early!
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34. Marginalizing Early-2
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35. Evidence If evidence, start with factors that select that evidence
No evidence uses these initial factors:
Computing P(L|+r) , the initial factors become:
We eliminate all vars other than query + evidence
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36. Evidence II esult will be a selected joint of query and evidence
E.g. for P(L | +r), we’d end up with:
To get our answer, just normalize this
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37. General Variable Elimination
Query:
Start with initial factors:
Local CPTs (but instantiated by evidence)
While there are still hidden variables (not Q or evidence):
Pick a hidden variable H
Join all factors mentioning H
Eliminate (sum out) H
Join all remaining factors and normalize
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38. Variable Elimination Bayes Rule
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39. Example 2
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40. © Daniel S. Weld

41. Notes on VE
Each operation is a simply multiplication of factors and summing out a variable
Complexity determined by size of largest factor
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)
On tree-structured graphs, variable elimination runs in polynomial time, like tree-structured CSPs
Practically, inference is much more tractable using structure of this sort
© Daniel S. Weld