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Graphical Models - Inference -

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Mainly based on F. V. Jensen, 'Bayesian Networks and Decision Graphs', Springer ... Dyspnoea - Inference (Variable Elimination) Bayesian Networks. Advanced. I WS 06/07 ... – PowerPoint PPT presentation

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Title: Graphical Models - Inference -


1
Graphical Models- Inference -
Mainly based on F. V. Jensen, Bayesian Networks
and Decision Graphs, Springer-Verlag New York,
2001.
Advanced I WS 06/07
Variable Elimination
  • Wolfram Burgard, Luc De Raedt, Kristian
    Kersting, Bernhard Nebel

Albert-Ludwigs University Freiburg, Germany
2
Outline
  • Introduction
  • Reminder Probability theory
  • Basics of Bayesian Networks
  • Modeling Bayesian networks
  • Inference
  • Excourse Markov Networks
  • Learning Bayesian networks
  • Relational Models

3
Elimination in Chains
  • Forward pass
  • Using definition of probability, we have

- Inference (Variable Elimination)
4
Elimination in Chains
  • By chain decomposition, we get

- Inference (Variable Elimination)
5
Elimination in Chains
  • Rearranging terms ...

- Inference (Variable Elimination)
6
Elimination in Chains
  • Now we can perform innermost summation

- Inference (Variable Elimination)
7
Elimination in Chains
X
  • Rearranging and then summing again, we get

- Inference (Variable Elimination)

8
Elimination in Chains with Evidence
  • Similar due to Bayes Rule
  • We write the query in explicit form

- Inference (Variable Elimination)
9
Variable Elimination
  • Write query in the form
  • Iteratively
  • Move all irrelevant terms outside of innermost
    sum
  • Perform innermost sum, getting a new term
  • Insert the new term into the product

Semantic Bayesian Network
Eliminate irrelevant variables
- Inference (Variable Elimination)
10
Asia - A More Complex Example
- Inference (Variable Elimination)
11
We want to compute P(d)Need to eliminate
v,s,x,t,l,a,b
  • Initial factors

- Inference (Variable Elimination)
12
We want to compute P(d)Need to eliminate
v,s,x,t,l,a,b
  • Initial factors

Eliminate v
- Inference (Variable Elimination)
13
We want to compute P(d)Need to eliminate
s,x,t,l,a,b
  • Initial factors

Eliminate s
- Inference (Variable Elimination)
Summing on s results in a factor with two
arguments fs(b,l). In general, result of
elimination may be a function of several variables
14
We want to compute P(d)Need to eliminate
x,t,l,a,b
  • Initial factors

- Inference (Variable Elimination)
Eliminate x
Note that fx(a)1 for all values of a
15
We want to compute P(d)Need to eliminate t,l,a,b
  • Initial factors

- Inference (Variable Elimination)
Eliminate t
16
We want to compute P(d)Need to eliminate l,a,b
  • Initial factors

- Inference (Variable Elimination)
Eliminate l
17
We want to compute P(d)Need to eliminate a,b
  • Initial factors

- Inference (Variable Elimination)
Eliminate a,b
18
Variable Elimination (VE)
  • We now understand VE as a sequence of rewriting
    operations
  • Actual computation is done in elimination step
  • Exactly the same procedure applies to Markov
    networks
  • Computation depends on order of elimination

- Inference (Variable Elimination)
19
Dealing with Evidence
  • How do we deal with evidence?
  • Suppose get evidence V t, S f, D t
  • We want to compute P(L, V t, S f, D t)

- Inference (Variable Elimination)
20
Dealing with Evidence
  • We start by writing the factors

- Inference (Variable Elimination)
21
Dealing with Evidence
  • We start by writing the factors
  • Since we know that V t, we dont need to
    eliminate V
  • Instead, we can replace the factors P(V) and
    P(TV) with

- Inference (Variable Elimination)
22
Dealing with Evidence
  • We start by writing the factors
  • Since we know that V t, we dont need to
    eliminate V
  • Instead, we can replace the factors P(V) and
    P(TV) with
  • This selects the appropriate parts of the
    original factors given the evidence
  • Note that fP(v) is a constant, and thus does not
    appear in elimination of other variables

- Inference (Variable Elimination)
23
Dealing with Evidence
Given evidence V t, S f, D t, compute P(L,
V t, S f, D t )
  • Initial factors, after setting evidence

- Inference (Variable Elimination)
24
Dealing with Evidence
Given evidence V t, S f, D t, compute P(L,
V t, S f, D t )
  • Initial factors, after setting evidence
  • Eliminating x, we get

- Inference (Variable Elimination)
25
Dealing with Evidence
Given evidence V t, S f, D t, compute P(L,
V t, S f, D t )
  • Initial factors, after setting evidence
  • Eliminating x, we get
  • Eliminating t, we get

- Inference (Variable Elimination)
26
Dealing with Evidence
Given evidence V t, S f, D t, compute P(L,
V t, S f, D t )
  • Initial factors, after setting evidence
  • Eliminating x, we get
  • Eliminating t, we get
  • Eliminating a, we get

- Inference (Variable Elimination)
27
Dealing with Evidence
Given evidence V t, S f, D t, compute P(L,
V t, S f, D t )
  • Initial factors, after setting evidence
  • Eliminating x, t, a, we get
  • Finally, when eliminating b, we get

- Inference (Variable Elimination)
28
Dealing with Evidence
Given evidence V t, S f, D t, compute P(L,
V t, S f, D t )
  • Initial factors, after setting evidence
  • Eliminating x, t, a, we get
  • Finally, when eliminating b, we get

- Inference (Variable Elimination)
29
Outline
  • Introduction
  • Reminder Probability theory
  • Basics of Bayesian Networks
  • Modeling Bayesian networks
  • Inference (VE, Junction Tree)
  • Excourse Markov Networks
  • Learning Bayesian networks
  • Relational Models
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