CSCE 582
Computation of the Most Probable
Explanation in Bayesian Networks
using Bucket Elimination
-Hareesh Lingareddy
University of South Carolina
Bucket Elimination
- Algorithmic framework that generalizes
dynamic programming to accommodate
algorithms for many complex problem solving
and reasoning activities.
- Uses “buckets” to mimic the algebraic
manipulations involved in each of these
problems resulting in an easily expressible
algorithmic formulation
Bucket Elimination Algorithm
- Partition functions on the graph into “buckets” in
backwards relative to the given node order
- In the bucket of variable X we put all functions
that mention X but do not mention any variable
having a higher index
- Process buckets backwards relative to the node
order
- The computed function after elimination is
placed in the bucket of the ‘highest’ variable in its
scope
Algorithms using Bucket Elimination
-
Belief Assessment
Most Probable Estimation(MPE)
Maximum A Posteriori Hypothesis(MAP)
Maximum Expected Utility(MEU)
Belief Assessment
• Definition
- Given a set of evidence compute the posterior probability of
all the variables
– The belief assessment task of Xk = xk is to find
bel ( xk )  P( X k  xk | e)  k
n
  P( x | pa( x ), e)
X { xk } i 1
i
i
where k – normalizing constant
• In the Visit to Asia example, the belief assessment problem
answers questions like
– What is the probability that a person has tuberculosis, given
that he/she has dyspnoea and has visited Asia recently ?
Belief Assessment Overview
• In reverse Node Ordering:
– Create bucket function by multiplying all functions
(given as tables) containing the current node
– Perform variable elimination using Summation
over the current node
– Place the new created function table into the
appropriate bucket
Most Probable Explanation (MPE)
• Definition
– Given evidence find the maximum probability
assignment to the remaining variables
– The MPE task is to find an assignment xo = (xo1, …, xon)
such that
n
P( x o )  max  P( xi | pa( xi ), e)
X
i 1
Differences from Belief Assessment
– Replace Sums With Max
– Keep track of maximizing value at each
stage
– “Forward Step” to determine what is the
maximizing assignment tuple
Elimination Algorithm for Most Probable
Explanation
Finding MPE = max ,,,,,,, P(,,,,,,,)
MPE= MAX{,,,,,,,} (P(|)* P(|)* P(|,)* P(|,)* P()*P(|)*P(|)*P())
Bucket :
P(|)*P()
Bucket :
P(|)
Bucket :
P(|,), =“no”
Bucket :
P(|,)
H(,)
Bucket :
P(|)
H(,,)
Bucket :
P(|)*P()
Hn(u)=maxxn ( ПxnFnC(xn|xpa))
H(,,)
Bucket :
H(,)
Bucket :
H()
H()
MPE probability
H()
Elimination Algorithm for Most Probable
Explanation
Forward part
Bucket :
P(|)*P()
’ = arg max P(’|)*P()
Bucket :
P(|)
’ = arg max P(|’)
Bucket :
P(|,), =“no”
’ = “no”
Bucket :
P(|,) H(,) H()
’ = arg max P(|’,’)*H(,’)*H()
Bucket :
P(|) H(,,)
’ = arg max P(|’)*H(’,,’)
Bucket :
P(|)*P() H(,,)
’ = arg max P(’|)*P()* H(’,’,)
Bucket :
H(,)
’ = arg max H(’,)
Bucket :
H() H()
’ = arg max H()* H()
Return: (’, ’, ’, ’, ’, ’, ’, ’)
MPE Overview
• In reverse node Ordering
– Create bucket function by multiplying all functions (given
as tables) containing the current node
– Perform variable elimination using the Maximization
operation over the current node (recording the
maximizing state function)
– Place the new created function table into the appropriate
bucket
• In forward node ordering
– Calculate the maximum probability using maximizing state
functions
Maximum Aposteriori Hypothesis
(MAP)
• Definition
– Given evidence find an assignment to a subset of
“hypothesis” variables that maximizes their
probability
– Given a set of hypothesis variables A = {A1, …, Ak},
A  X ,the MAP task is to find an assignment
ao = (ao1, …, aok) such that
P(a o )  max
A
n
 P( x | pa( x ), e)
X  A i 1
i
i