Goal-Oriented Symbolic Propagation in Bayesian Networks
- Sachit Kachraj
Example 10.3:
Consider the following network (10.1 on P.446) and given the evidence e = {X2 = 1, X5 = 1}
From the table 10.1, X3 and X6 are the only symbolic nodes.
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Goal-Oriented Symbolic Propagation in Bayesian Networks
Therefore the set of feasible monomials is given by
M = Ө
3
X Ө
= {θ
6
300
θ
600
, θ
300
θ
610
, θ
310
θ
600
, θ
310
θ
610
}
= {m1, m2, m3, m4}
From equation (10.12) we get the equation for the un-normalized Conditional probability in a polynomial form.
Our objective here is to calculate the coefficients for each node Xi. The canonical components associated with the symbolic parameters Ө are required for doing so.
As we are dealing with binary variables for any parameter Өi all we can have is {1, 0},
{0, 1}
Intuitively this means that for all the combinations of the extreme values for the parameters
θ
300
θ
310
θ
600
θ
610
0 1 0 1
0
1
1
0
1
0
0
1
1 0 1 0
Are the only possible combinations
All other violating the rule
(OR)
Corresponding to M we can have the following set of canonical components
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Goal-Oriented Symbolic Propagation in Bayesian Networks
Then by instantiating the symbolic parameters to the corresponding values given in its canonical components, all the monomials in 10.18 become either 0 or 1.
This we obtain an expression that only depends on the coefficients
Here the matrix Tij is an identity matrix because all the parameters are associated with the same instantiation of the set of parents
We’ll have
The following figure (10.6) shows the un-normalized conditional probabilities of all nodes given X2 = 1, X5 = 1, associated with four possible canonical components c1, c2, c3 and c4.
This can be obtained by using any of the numeric propagation techniques (from Ch 8 &
Ch 9) that use the JPD to calculate these un-normalized probability values.
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Goal-Oriented Symbolic Propagation in Bayesian Networks
Using these values we can get all the rational functions in Table 10.3
For example from 10.6 we get the following values for X6:
These are the coefficients of the numerator polynomials for X6 = 0, X6 = 1 respectively.
By substituting these values in 10.18 we get
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Goal-Oriented Symbolic Propagation in Bayesian Networks
Adding both the polynomial we get the denominator polynomial
Thus we have
Where
Finally eliminating the dependent parameters θ
310
, θ
610
we get the expressions in table
10.3.
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Goal-Oriented Symbolic Propagation in Bayesian Networks
The problem theory discussed in the last lecture and the related examples are generic and can be used to get a symbolic expression for all the nodes in a given graph. This method can be modified a bit to do the goal oriented symbolic propagation in Bayesian networks.
Let Xi be our goal node for which we want to find the conditional probabilities
P (Xi = j | E = e), where E is the set of evidential nodes with known values E = e.
According to Castillo, Gutierrez and Hadi the un-normalized probabilities are polynomials of the form:
Where mj are monomials in the symbolic parameters Θ
Consider the following Bayesian network with 5 nodes and the Table 1 that shows the corresponding parameters.
Figure 1
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Goal-Oriented Symbolic Propagation in Bayesian Networks
Table 1
Note that some nodes are numeric and some are parametric. X4 is numeric and the other four are parametric.
This algorithm solves the goal oriented problem by calculating the coefficients in equation (4). It is organized into four parts.
Let’s solve this problem for goal node X3 and evidence X2 = 1
PART 1: Identify all relevant Nodes
This involves implementing any of the d-separation algorithms that lets us know about the relevant nodes for a given node. I am yet to make a choice on this. But I already have a couple of them in mind, Geiger’s algorithm and the one proposed by Dr.Vargas himself.
The advantage with the Geiger’s algorithm is that given a node it fetches all the nodes that are d-separated from it at once.
The algorithm proposed by Dr.Vargas claims to be much faster in the sense that it uses the sparseness of the real world Bayesian networks to reduce the number of paths to be traversed. It tells you weather the two given nodes are d-separated or not. We could modify this to actually find the d-separated set.
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Goal-Oriented Symbolic Propagation in Bayesian Networks
Thus, this part involves identifying relevant nodes and eliminating the rest from the graph.
From the above step, eliminating the nodes that are D-separated we get the following graph
X1
X2
And Ө = {Ө1, Ө2, Ө3}
X3
We proceed with the parameters set Ө that also includes the numeric parameters, only to pull them out as a part of coefficients in the third stage.
PART 2: Identify sufficient parameters
With the given evidence, we can identify and remove the parameters that contradict with the evidence.
We do that using the following two rules
Rule 1 : Eliminate the parameters
θ jkπ if x j
≠ k for every Xj € E
θ
200
and
θ
201
are contradictory given that X2 = 1.
Rule 2: Eliminate
θ jkп
if the parents’ instantiations are incompatible with the evidence
(This is just to illustrate the concept: If X1 be the only parent of X2 and given that X1 =
1, then
θ
200
would be a contradiction to the given evidence, so eliminate the parameter
)
For the given problem this case doesn’t exists, since the only evidential node X2 has no children.
So the minimum set of sufficient parameters is given by
Ө = {{ θ
10
, θ
11
}; {
θ
210
,
θ
211
}; {
θ
300
,
θ
310,
θ
301
,
θ
311
}}
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Goal-Oriented Symbolic Propagation in Bayesian Networks
PART 3: Identify the feasible monomials
Once the minimal Θ is identified they are combined to form the monomials by taking the
Cartesian product of the parameters. The infeasible monomials are eliminated by using the following rule
Rule 3: Parameters associated with contradictory conditioning instantiations cannot appear in same monomial
In the monomial
θ
10
θ
210
θ
301
, the first term means that X1 = 0, and the third term means that X1 = 1(since X1 is the parent of X3), hence the contradiction.
Also pull out the numerical parameters out, because they become the part of the coefficients.
Doing that results in the following set of feasible monomials
M = {
θ
10
θ
210
,
θ
11
θ
301
,
θ
11
θ
311
}
PART 4: Calculate coefficients of all the polynomials
The discussion here follows from Example 10.3
For the goal node Xi and for each of its states j build the subset Mj by considering those monomials in M which does not contain any parameter of the form
θ iqπ
with q ≠ j
So M0 = {
θ
10
θ
210
,
θ
11
θ
311
} and M1 = {
θ
10
θ
210
,
θ
11
θ
301
}
Substituting these monomials in Eq 10.18 we get the following polynomials for the unnormalized probabilities P(X1 = j | E = e)
Taking the appropriate combinations of the extreme values for the symbolic parameters we can calculate the values of p
0
(
θ
1
) and p
1
(
θ and P(X1 = 1 | X2= 1 , Ө =
θ
2
) respectively.
2
), which are P(X1 = 1 | X2 = 1, Ө =
θ
1
)
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Goal-Oriented Symbolic Propagation in Bayesian Networks
Using any of the numeric propagation methods we can calculate the values of p
0
(
θ
1
) and p
1
(
θ
2
).
And solving the following equation we get the values of the corresponding coefficients
Similarly taking the canonical components corresponding to X1 = 1 we get
By, substituting in (8) and (9) we get
The sum of the coefficients gives the coefficients for the denominator
Normalizing the values in 13 and 14 with the denominator polynomial we get the probabilities for the goal node as
Eliminating the dependent parameters, we get the final expressions for the conditional probabilities.
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Goal-Oriented Symbolic Propagation in Bayesian Networks
References:
1. Goal Oriented Symbolic Propagation in Bayesian Networks. Castillo, Gutierrez and
Hadi
2. Expert Systems and Probabilistic Network Models, Section 10.6. Castillo, Gutierrez and Hadi
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