1. Computing probabilities using Expect problem-solving Trees: A worked example
Jim Blythe
USC/ISI

2. Overview
Assume that some of the responses from the knowledge server or user might have uncertainty.
Use the structure of Expect’s problem-solving tree (PS tree) to tell us the dependencies between the facts on which the final answer is based.
Use formal rules to produce a bayesian belief net based on the PS tree. (Similar approach to knowledge-based model construction, Wellman et al. 92.)
Use standard procedures for each special form or primitive method to fill conditional probability tables.

3. Example from the anthrax domain
Goal:
(determine-whether (obj production) (of anthrax)
(supports maximum-damage-in-battlefield))
Two subgoals, determine ease of dispensing and strength in environment.
Both depend on whether or not the agent is a dry agent.
Assume it is uncertain whether the agent is dry, and that both subgoals are also uncertain.

4. Sketch of problem-solving tree
Expect builds a PS tree by examining every possible case.
Sketch of tree for this example:
production supports the objective if it is both easy to dispense and strong in the environment
to decide if it’s easy to dispense, decide if it’s a dry or wet agent
If it’s dry, use procedure A
If it’s wet, use procedure B
to decide if it’s strong in the environment, decide if it’s a dry or wet agent
...

5. Start with the Expect PS Tree (only showing enodes)
PS-en1
Determine-whether
Thick lines denote
method expansion
Red borders mean
the value can be
uncertain
PS-en3
AND..
PS-en4
Is-equal ease to easy
PS-en5
Is-equal strength to strong
PS-en6
Estimate ease
PS-en15
Estimate strength
PS-en8
If agent is dry, then .. Else ..
PS-en16
If agent is dry, then .. Else ..
PS-en11
Estimate for dry
PS-en10
Estimate for wet
PS-en9
Is agent dry?
PS-en18
Estimate for dry
PS-en19
Estimate for wet

6. Bayes net creation, step 1: merge enodes that are method expansions
Method expansion nodes have the same value as their parents, so they can be excluded from the final belief net.
PS-en1/3
AND..
PS-en4
Is-equal ease to easy
PS-en5
Is-equal strength to strong
PS-en6/8
If agent is dry, then .. Else ..
PS-en15/16
If agent is dry, then .. Else ..
PS-en11
Estimate for dry
PS-en10
Estimate for wet
PS-en9
Is agent dry?
PS-en18
Estimate for dry
PS-en19
Estimate for wet

7. Step 2: fill possible values from child result types and method definitions.
For this example, each node has a small number of possible
values
true, false
PS-en1/3
AND..
true, false
true, false
PS-en4
Is-equal ease to easy
PS-en5
Is-equal strength to strong
easy, hard
strong, weak
PS-en6/8
If agent is dry, then .. Else ..
PS-en15/16
If agent is dry, then .. Else ..
PS-en11
Estimate for dry
PS-en10
Estimate for wet
PS-en9
Is agent dry?
PS-en18
Estimate for dry
PS-en19
Estimate for wet
easy, hard
easy, hard
true, false
strong, weak
strong, weak

8. Step 3: fill conditional probability tables from templates for special forms and primitivesh t 1 f
(if)
(and)
En4
En5
True
False
true 1
false
true, false
PS-en1/3
AND..
(is-equal)
PS-en4
Is-equal
PS-en5
Is-equal
En15
true
False
Strong 1
Weak
PS-en6/8 If
PS-en15/16 If
PS-en11
Estimate for dry
PS-en10
Estimate for wet
PS-en9
Is agent dry?
PS-en18
Estimate for dry
PS-en19
Estimate for wet
easy, hard
easy, hard
true, false
strong, weak
strong, weak

9. Step 4: further collapse nodes to give final belief net
Collapse each pair where parent has one child and child has one parent. Multiply CPTs: 1 f h t e F T n9
n10
n11
(if * is-equal)
En4
En5
True
False
true 1
false
true, false
PS-en1/3
AND..
(and)
PS-en4
Is-equal/if
PS-en5
Is-equal/if
PS-en11
Estimate for dry
PS-en10
Estimate for wet
PS-en9
Is agent dry?
PS-en18
Estimate for dry
PS-en19
Estimate for wet
easy, hard
easy, hard
true, false
strong, weak
strong, weak

10. Using the belief net
If probability distributions are given for any of the leaf procedures instead of a certain value, we can use the belief net to compute a probability distribution for our final answer.
The result is sound assuming that the PS tree is correct for the case of certainty.
The net can also be used in other directions, eg to compute the probability that the agent was dry based on observations of its effectiveness.

11. Advantage of probabilistic representation
Will handle all cases correctly. Here the leaf nodes are given probabilities and the top node is computed.
1/2
and
easy
strong
dry
e|d
e|w
s|d
s|w 1
1/2
1/4
and
1/2
1/2
easy
strong
1/2 1 1
e|d
e|w
dry
s|d
s|w
and
1/2
1/2
easy
strong
1/2 1 1
e|d
e|w
dry
s|d
s|w

12. Other advantages
Don’t have to mention uncertainty-handling explicitly in methods.
Uses standard belief net evaluation software.
Widely accepted approach, also used in HPKB (e.g. Koller’s work on Spook (Pfeffer et al. 99 )

13. What is required for this approach?
Input probability distributions for nodes from the knowledge server or user, e.g.
P(ease of dispensing | dry), P(dry), ..
Template CPTs for special forms and primitive forms, e.g. if, and, or, is-equal, ..
PS tree must evaluate all possible cases (it usually does)
A way to represent methods with uncertain results (not shown here).

14. Extensions to the approach
Handling numeric values, and other cases with a large number of possible values.
(Should quantise or use abstractions)
Handling concept membership uncertainty.
Make sure the PS Tree includes all cases (e.g. with a covering)
Use a template CPT that builds a distribution over the covering
Representing uncertain methods.
Try to avoid having to extend Expect
Perhaps represent using uncertain methods via a reformulation?