Exact Inference Algorithms for Probabilistic Reasoning

1
Exact Inference Algorithms for Probabilistic
Reasoning

• COMPSCI 276
• Fall 2009

2
Exact Inference Algorithms Bucket-elimination

• COMPSCI 276, Fall 2009
• Set 4 Rina Dechter

(Reading class notes chapter 4 , Darwiche
chapter 6)
3
Belief Updating
Smoking
lung Cancer
Bronchitis
X-ray
Dyspnoea
P (lung canceryes smokingno, dyspnoeayes )
?
4
(No Transcript)
5
Probabilistic Inference Tasks

• Belief updating E is a subset X1,,Xn, Y
  subset X-E, P(YyEe)
• P(e)?
• Finding most probable explanation (MPE)
• Finding maximum a-posteriory hypothesis
• Finding maximum-expected-utility (MEU) decision
  

6
Belief updating is NP-hard

• Each sat formula can be mapped to a Bayesian
  network query.
• Example (u,v,w) and (u,w,y) sat?

7
Motivation

• How can we compute P(D)?, P(DA0)? P(AD0)?
• Brute force O(k4)
• Maybe O(4k2)

Given
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
Belief updating P(Xevidence)?
P(ae0)
P(a)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
(No Transcript)
20
Bucket elimination Algorithm BE-bel (Dechter
1996)
21
Bucket elimination Algorithm BE-bel (Dechter
1996)
22
BE-BEL
23
Student Network example

• P(J)?

Intelligence
Difficulty
Grade
SAT
Apply
Letter
Job
24
(No Transcript)
25
The induced-width

• width is the max number of parents in the
  ordered graph
• Induced-width width of induced graph
  recursively connecting parents going from last
  node to first.
• Induced-width w(d) the max induced-width over
  all nodes
• Induced-width of a graph max w(d) over all d

26
Two orderings of the moral graph
D1 ACB FDG is depicted in Figure 4.7a. In
this case, the ordered graph and its in-duced
ordered graph are identical, since all the
earlier neighbors of each node are already
connected. The maximum induced width is 2.
Indeed, in this case, the maximum arity of
functions recorded by the elimination algorithms
is 2. For d2 A FDCBG the induced graph is
depicted in Figure 4.7c. The width of C is
initially 2 (see Figure 4.7b) while its induced
width is 3. The maximum induced width over all
variables for d2 is 4, and so is the recorded
function's dimensionality.
27
Complexity of elimination
The effect of the ordering
28
BE-BEL
More accurately O(r exp(w(d)) where r is the
number of cpts. For Bayesian networks rn. For
Markov networks?
29
(No Transcript)
30
The impact of observations
31
BE-BEL
Use the ancestral graph only
32
(No Transcript)
33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
Probabilistic Inference Tasks

• Belief updating
• Finding most probable explanation (MPE)
• Finding maximum a-posteriory hypothesis
• Finding maximum-expected-utility (MEU) decision
  

38
(No Transcript)
39
Finding Algorithm elim-mpe (Dechter 1996)
Elimination operator
40
Generating the MPE-tuple
41
Algorithm BE-MPE
42
BE-MPE
BE-MPE
43
(No Transcript)
44
BE-MAP
45
Algorithm BE-MAP
Variable ordering Restricted Max buckets
should Be processed after sum buckets
46
More accurately O(r exp(w(d)) where r is the
number of cpts. For Bayesian networks rn. For
Markov networks?
47
Finding small induced-width

• NP-complete
• A tree has induced-width of ?
• Greedy algorithms
• Min width
• Min induced-width
• Max-cardinality
• Fill-in (thought as the best)
• See anytime min-width (Gogate and Dechter)

48
Min-width ordering
Proposition algorithm min-width finds a
min-width ordering of a graph
49
Greedy orderings heuristics
min-induced-width (miw) input a graph G (VE),
V 1 vn output An ordering of the nodes
d (v1 vn). 1. for j n to 1 by -1 do 2.
r ? a node in V with smallest degree. 3. put r in
position j. 4. connect r's neighbors E ? E union
(vi vj) (vi r) in E (vj r) 2 in E, 5.
remove r from the resulting graph V ?V -
r. min-fill (min-fill) input a graph G
(VE), V v1 vn output An ordering of
the nodes d (v1 vn). 1. for j n to 1 by
-1 do 2. r ?a node in V with smallest fill edges
for his parents. 3. put r in position j. 4.
connect r's neighbors E ?E union (vi vj) (vi
r) 2 E (vj r) in E, 5. remove r from the
resulting graph V ?V r.
Theorem A graph is a tree iff it has both width
and induced-width of 1.
50
Different Induced-graphs
51
Min-induced-width
52
Min-fill algorithm

• Prefers a node who add the least number of
  fill-in arcs.
• Empirically, fill-in is the best among the greedy
  algorithms (MW,MIW,MF,MC)

53
Chordal graphs and Max-cardinality ordering

• A graph is chordal if every cycle of length at
  least 4 has a chord
• Finding w over chordal graph is easy using the
  max-cardinality ordering
• If G is an induced graph it is chordal chord
• K-trees are special chordal graphs (A graph is a
  k-tree if all its max-clique are of size k1,
  created recursively by connection a new node to k
  earlier nodes in a cliques
• Finding the max-clique in chordal graphs is easy
  (just enumerate all cliques in a max-cardinality
  ordering

54
Max-cardinality ordering
Figure 4.5 The max-cardinality (MC) ordering
procedure.