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

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.

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.

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 ...

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

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

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

Step 3: fill conditional probability tables from templates for special forms and primitives h 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

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

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.

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

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 )

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).

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?