Knowledge-Based Formulation of Dynamic Decision Models

Knowledge-Based Formulation of Dynamic Decision Models Chenggang Wang, Tze-Yun Leong Medical Computing Laboratory, School of Computing, National University of Singapore, Lower Kent Ridge Road, Singapore 119260 {wangcg, leongty}@comp.nus.edu.sg

Abstract. We present a new methodology to automate decision making over time and uncertainty. We adopt a knowledge-based model construction approach to support automated and interactive formulation of dynamic decision models, i.e., models that explicitly consider the effects of time. Our work integrates and extends different features of the existing frameworks. We incorporate a hybrid knowledge representation scheme that integrates categorical knowledge, probabilistic knowledge, and deterministic knowledge. We provide a set of knowledge-based modification operations for automatic and interactive generation, abstraction, and refinement of the model components. We have built a knowledge base in a real-world domain and shown that it can support automated construction of a reasonable dynamic decision model. The results indicate the practical promise of the proposed design.

1 Introduction Dynamic decision modeling is a very challenging task. The multitude of problems, the domain-specificity, the uncertainty, and the temporal nature of the underlying phenomena all contribute to the intricacy of the dynamic decision modeling process [5]. To automate the model construction process, efforts in developing knowledge-based model construction (KBMC) systems have emerged in recent years. The relevant works include ALTERID [1], Frail3 [4], SUDO-PLANNER [14], and DYNASTY [9]. These systems advocate that the decision models for different problems should be constructed on demand from a knowledge base [13]. This approach facilitates scalability and reusability of the knowledge base. Moreover, the resulting decision models are parsimonious and most relevant to the problems at hand. An important issue in KBMC research is the choice of the target model. All of the existing KBMC systems synthesize only one type of decision models; influence diagram or its variants are the choice in all the systems mentioned. Since each type of decision models can convey only certain information explicitly, some important characteristics of the decision situations are lost when the models are completed. Moreover, most of the existing frameworks do not support knowledge-based model modification. The model construction process terminates when an initial model is completed. If the user is not satisfied with the resulting model, he has to manually modify it separately. We propose a KBMC framework that adopts a new dynamic decision modeling

language, DynaMoL [5], to model target decisions, and provides a set of knowledgebased modification operations for interactive generation, abstraction, and refinement of the model components. The framework incorporates a hybrid knowledge base representation scheme that integrates categorical knowledge, probabilistic knowledge, and deterministic knowledge. We have tested the design by building a knowledge base in a real-world domain and demonstrating how it can support automated formulation of a reasonable dynamic decision model. We illustrate the proposed design and address some future research issues in this paper.

2 System Architecture The proposed system architecture is depicted in Figure 1. Given as input a dynamic decision problem description, the model constructor will access the knowledge base via the KB-manager to formulate a model. Further modification operations can be applied interactively by the user. The final model is then evaluated or solved, with respect to some pre-specified optimality and evaluation criteria, e.g., life expectancy, monetary cost, and/or desirability or utility, by DYNAMO, a prototype implementation of DynaMoL, to determine an optimal course of action.

dynamic decision problem description

user

interface

model constructor model modification

solution

KB-manager

DynaMoL model

DYNAMO

knowledge base

Figure 1. The system architecture

3 An Example Problem For illustration, we consider a simplified dynamic decision problem in colorectal cancer management [2]. In managing the follow up of colorectal cancer patients, a series of diagnostic tests are performed to detect possible recurrence, metastasis, or both recurrence and metastasis of the cancer; treatment is prescribed if cancer is detected. The diagnostic tests available are different in terms of monetary cost, sensitivity, and specificity. The decision is to determine the optimal course of diagnostic tests for

effective disease detection so that timely treatment can be given.

4 The Target Language: DynaMoL DynaMoL has four major components. First, the decision grammar supports problem formulation with multiple interfaces. Second, the presentation convention, in the tradition of graphical decision models, governs parameter visualization in multiple perspectives; two graphical representations are currently included: transition view which corresponds to the Markov state transition diagram, as shown in Figure 2, and influence view which corresponds to the dynamic influence diagram [12] without decision nodes or value nodes, as shown in Figure 3. Third, the mathematical representation, in terms of a Semi-Markov Decision Process (SMDP), provides a concise formulation of the decision problem; it also admits various solution methods. Fourth, the translation convention establishes automatic transformations among the different graphical representations, and from these representations into an SMDP [5].

recurrent rec-met

well metastatic

Figure 2. Transition view for an action. The nodes denote the states; the links denote the possible transitions from one state to another

Figure 3. Influence view for an action. The nodes depict the possible event or chance variables, each with a possible set of outcomes or values, that affect the state transitions, and the links the probabilistic dependences. The states are captured as outcomes of the state variables

DynaMoL makes a good target decision modeling language because the high-level

modeling constructs and the multiple graphical perspectives allow simple and explicit specification of the decision factors and constraints, thus facilitating automated formulation. DynaMoL also supports incremental language extension, which allows the scope of the dynamic decision problems addressed to be gradually expanded.

5 Knowledge Base Representation Our knowledge base representation design is based on the first-order logic-like representations [1, 4, 7, 9] and the fluid multilevel representation [14] in existing KBMC systems. We adopt a hybrid framework that integrates categorical knowledge, probabilistic knowledge and deterministic knowledge. The categorical knowledge captures the definitional and structural relations of the domain concepts. This type of knowledge provides the power of abstraction and inheritance; it supports modeling at multiple levels of details. The probabilistic knowledge captures the interactions (probabilistic dependencies) among the concepts. These relations are needed to support the derivation of missing information in model construction. The deterministic knowledge expresses deterministic rules and declaratory constraints about the domain. 5.1 Categorical Knowledge We use knowledge frames to describe domain concepts, e.g., diseases, tests, treatments, and other entities related to the decision problems. All the concepts may be instantiated as actions and event variables in the probabilistic knowledge base. Every concept definition has at least an attribute indicating the specialization relation, AKO. For all concepts a and b and all individuals α, AKO = {(a, b) | a⊆ b, i.e., ∀α, α∈ a ⇒ α∈b}. The AKO relation imposes a hierarchy or partial ordering of the domain concepts. Figure 4 shows part of such a hierarchy for the example problem. The AKO links will always point upward. A parent of a concept indicates a concept immediately above the given concept in the hierarchy. An ancestor of a concept is situated somewhere above the given concept. A child and a descendent of a concept are analogously defined, but downward in the hierarchy. The lowest common ancestor of a set of concepts {C1, C2,… Cn} is a concept C which is an ancestor of these concepts, and for all children S1, S2,…Sq of C, there exists a Ck, 1 ≤k ≤ n, where Sj, 1≤ j≤ q, is not a parent of Ck. The lowest common ancestors of a set of concepts may not be unique.

event chance event findings

action

Physiological test state

test historical sign symptom result disease fact

complication

Figure 4. Domain knowledge base hierarchy

treatment

5.2 Probabilistic Knowledge To represent the probabilistic knowledge, we must be able to model the action effects, the relations among event variables, and the temporal nature of the decision environment. We use timed concept instances to represent actions and event variables. For example, test-A(t) represents the action to perform test-A at time t. Similarly, test-Aresult(t), which can take either the positive or negative value, represents the event variable test-A-result at time t. It follows that test-A-result(t, positive) is an event. Probabilistic dependency statements, similar to those in [1] except that we separate the logic program clauses as the preconditions in the probabilistic statements for representational clarity and inferential efficiency, represent the action effects and probabilistic relationships among the event variables. They have the form E0|p E1,…, En = Pr(ωE0|ωE1, ωE2… ωEn) ← C1, C2, … C m where n>=0, m>=0. Ei, i = 1,…, n, is an event variable and Cj, j = 1,…, m, is a conditioning literal, which could be an event or an action taken. ω refers to one of the alternative outcomes of the event variables. The left hand side of the equality indicates the possible probabilistic dependence between E1, E2,…, En and E0. The right hand side indicates the conditional probability distribution over the alternative outcomes of the event variable E0 given the outcomes for the event variables E1,…, En. The last part is a logical expression which must be true for this probabilistic relation to be applicable. If P is the above probabilistic sentence, we define precondition(P) to be the conjunction C1, C2, … C m, (probabilistic) antecedent(P) to be the conjunction E1,…,En, and (probabilistic) consequent(P) to be E0. As an example, consider the following dependency, which expresses the probability distribution of metastasis at time t+1 given the outcome of loss-of-appetite at time t when action “observation” is taken and the Dukes’ stage of the patient is C. metastatic(t+1)|ploss-of-appetite(t)=Pr(ωmetastatic(t+1)|ωloss-of-appetite(t)) metastasis (t+1, present) metastasis(t+1, absent) loss-of-appetite(t, yes) 0.6 0.4 loss-of-appetite(t, no) 0.3 0.7 ← observation(t), duke-stage(t,C)

A probabilistic knowledge base is a finite set of probabilistic dependency statements. Event variable e1 directly influences e2, denoted e 1 → e 2 , if there exists a probabilistic rule P such that e 1 ∈ antecedent (P) and e2 = consequent (P). There is a directed path from e1 to e2, if there exists a sequence e 1 → I 1 , I 1 → I 2 ,…, I n → e 2 , where I1, I2,…In are the intermediate nodes on the path between e1 and e2. Since an influence view is a directed acyclic graph, we must ensure that the knowledge base contains no cycles [6]. Since it is not easy to enforce this “acyclicity” requirement for the knowledge base, the KB-manager will include a function which, after new probabilistic rules are added to the knowledge base, will check if they will lead to any cycles in the originally acyclic knowledge base. 5.3 Deterministic Knowledge A deterministic knowledge base is a set of logical statements, called deterministic

dependency statements, of the form L0 ← L1 and L2 … and L n, or L0 ← L1 or L2 … or Ln n>=0, where ← stands for implication, Li, i = 1, …, n, is a condition literal, which can be an event or an action. A deterministic dependency statement expresses the logical relationship or constraint among the events and/or actions; it allows deduction of the implicit values for the preconditions. For example, if a patient is assumed to be dead after having two successive metastasis, the following rule can be included: status(t,dead) ← metastasis(t, yes) and metastasis(t-1, yes). 5.4 Inferences Supported by the KB-Manager With reference to Figure 1, the model constructor can access the knowledge base via some general queries to the KB-manager. This implies that the knowledge base representation is modular with respect to the inferences supported. Therefore, when new constructs are added to the knowledge base representation, or when the structure of the knowledge base is changed, we need not re-implement the model constructor. An example of the queries supported is “what are the event variables that directly influence given at ?” If the knowledge base contains two different rules with identical consequent, the KB-manager will select the more specific rule or just combine the antecedents with some canonical forms, such as noisy-OR [8], generalized noisy-OR [11], noisy-AND, and noisy-ONEOF.

6 Dynamic Decision Model Formulation Based on the example problem, we automate the construction of a dynamic decision model. The resulting well-formed model can then be solved for the optimal course of action by the DYNAMO system. Step 1. Eliciting Decision Context Information The process begins with the model constructor asking the user for some background or context information about the decision problem, e.g., the characteristics of the group of patients concerned, with the help of knowledge base. The problem is to determine a course of optimal actions for the follow-up patients. The decision horizon T is a finitehorizon of 8 time units, with constant decision stage duration of 6 months, and the evaluation criterion is subjective utilities of the final health states for the patients. Step 2.Defining the Action Space The model constructor will then prompt for the alternative actions by showing the user an action hierarchy generated from the categorical knowledge base, as shown in Figure 5. The general concept action for the follow-up problem has two subtypes, test and treatment. A test can be the test for local recurrence, test for metastasis, or test for both. There are different combinations of these tests, leading to different testing strategies. For the example problem, the action space A= {CEA, Ultrasound, Colonoscopy, Sigmoidscopy, Ba-enema, CT-scan, LFT, CXR, None}, which denotes performing a diagnostic test such as CEA, Ultrasound, and Colonscopy to detect metastasis or recurrence, or just do nothing.

action test sigmoid scopy

test for recurrence

test for metastsis

treatment surgery

non-surgical treatment

CT CEA hist- LFT BaCXR chemocolono- enema scan ology ultrasound therapy scopy FOB

radiotherapy

Figure 5. Action hierarchy generated from the knowledge base

Step 3.Defining the State Space The model constructor will first identify the state attribute variables, which denote the different characteristics that constitute and distinguish the states, and define the states in terms of these variables. Some irrelevant states will be reduced according to the domain knowledge and the user’s feedback during the process. For the example problem, the state attribute variables are recurrence, which indicates the presence or absence of recurrence, and metastasis, which indicates the presence or absence of metastasis. A total of 4 states are defined, which are well, recurrent, metastatic, both recurrent and metastatic (rec-met). At this time, the system also generates the state function. A state function is an expression of the form s= Φ Φ Φ Φ(ωA1, ωA2, ….ωAn), where s is a state variable, ωAi is an outcome of the state attribute variable Ai, and Φ Φ Φ Φis a statevalued function which specifies each state variable outcome for every possible combination of outcomes of the state attribute variables. For the above example, state function is such that well := {recurrence = absent, metastasis = absent}, metastatic := {recurrence = absent, metastasis = present}, recurrent := {recurrence = present, metastasis = absent}, rec-met := {recurrence = present, metastasis = present}. The system will then proceed to ask the user about a set of value functions with respect to the utility values defined for each state. For the example problem, we have V(well)=1, V(recurrent)=0.4, V(metastatic)=0.2, V(rec-met)= 0. Step 4.Transition View Construction To specify the transition characteristics, a transition view is constructed for each action defined in the action space for the follow up problem. Figure 1 shows the general structure of the transition view for all the actions. The model constructor will first try to find if there are relevant probabilistic dependency statements, i.e., state-attribute-variablei (t+1) |p state-attribute-variablei (t) defined for all the state attribute variables for a certain action. For any two states, S1, S2 ∈ S, S1 = Φ (A1= a11, A2= a21,…, An= an1), S2 = Φ (A1= a12, A2= a22,…, An= an2), Ai (i=1.n) is a state attribute variable, S2 is accessible from S1 given an action a at time t if for each state attribute variable Ai (i=1..n), P (Ai(t+1, ai2)| Ai(t, ai1)) > 0 ← a(t), i.e., each constituent state attribute of S2 is accessible from the corresponding constituent state attribute of S1. The transition probability from S1 to S2 at time t, P (state (S2,

n

t+1) | state (S1, t)), is defined as ∏ P (Ai(t+1, ai2)| Ai(t, ai1)). This computation i=1 method of transition probabilities is a reasonable approximation as long as the state attribute variables are conditionally independent given the action. However, such probabilistic dependency statements may not exist for all the actions. In this case, if the user can assess such transition functions directly for an action, the transition view is completed. Otherwise, the system will try to identify a set of event variables that constitute the effects of each action and a set of probabilistic influences among the event variables, i.e., the system will specify the transition functions in terms of the influence view components. Step 5.Influence View Construction Influence view construction begins with the incipient influence view, as shown in Figure 6 for the example problem. The basic operation performed by the influence view constructor is backward chaining. The constructor chains backward from each state attribute variable at time t+1, along all the influence relations until reaching the terminating nodes. The terminating nodes can be state attribute variables at time t, event variables at time t-1, or event variables with a marginal probability distribution. By simply backward chaining on the state attribute variables, the constructor generates all the relevant nodes, and avoids generating those nodes which are not the intermediate nodes between the state variables at time t and time t+1. Such nodes are irrelevant in the computation of transition probabilities. The final influence view for the action CEA test in the example problem is shown in Figure 3.

state t

M(t) R(t)

M(t+1) R(t+1)

state t+1

Figure 6. The incipient influence view for an action in the example problem

7 Model Modification Operations We define three sets of model operations to help the user modify the influence view. The model modification process starts with the initial influence view and is complete when two criteria are met. First, the influence view appropriately represents the user’s understanding of the decision problem. Second, the influence view is well-formed with all the appropriate information provided for each node. 7.1 Abstraction Operations A detailed influence view may be too complicated to solve or understand; abstraction on the model is sometimes necessary to omit or suppress detail, for computational tractability and for comprehensibility. Unlike most of the existing work which focus on abstracting the state space [3, 15], we focus on abstracting and refining the network

structure, including abstracting and refining a group of event variables based on concept taxonomies, and summarizing the influence path based on node reduction. In the following discussions, the nodes to be abstracted or the node to be refined are referred to as “original node(s)”, and the node to be abstracted into or the nodes to be refined as “target node(s)”. 7.1.1 Event Variable Abstraction The event variable abstraction operation groups a set of similar variables into a single variable which is a generalization of such variables in the hierarchical knowledge base. Consider the initially constructed influence view, which contains variables such as loss of appetite (LOA), loss of weight (LOW), lymph node involvement (LNI), liver mass (LM), pain and ascites. The user may want to have a more general model. He may choose to group LOA, LOW, LNI, LM, pain and ascites together; and then the system, according to the concept taxonomies in the knowledge base, will suggest that these six variables be abstracted into symptoms of metastasis (SOM). The procedure will first identify the target abstraction variable which is one of the lowest common ancestors of the original nodes. Then, in order to accomplish the probability updating task automatically, the procedure will add the target variable into the influence view, which has no successors and has the original nodes as its only predecessors. In Figure 7, this step is illustrated by dotted lines. The probability distribution P(abstraction node | original nodes), e.g., P(SOM(t, present)|LOA(t, present)), can be inferred from the AKO definition in the hierarchical knowledge base, which is 1. The overall probability distribution table for the abstraction node can be expressed by incorporating a disjunctive definition constraint, stating that a patient will have a symptom of metastasis if he has LOA, LOW, LM, LNI, pain or ascites; otherwise, the probability of having SOM is 0. Given such a fully specified influence view, the original nodes can be removed one by one from the graph probabilistically based on the arc reversal and barren node removal operations [10], which leaves the abstraction node in place of the original nodes in the new graph.

M1 PM

M2

M SM

PS

PS PM

M

SM

Mn Figure 7. A simplified event variable grouping operation example

7.1.2 Influence reduction The influence reduction operation abstracts away the intermediate details (both event variables and relationships among them) between two event variables. Therefore, the final diagram contains only a direct influence between the two original nodes with modified probability distributions. We use an algorithm similar to probabilistic node

reduction in [10] to accomplish this task. 7.2 Event Variable Refinement Event variable refinement is the reverse operation to event variable abstraction. The user can “emphasize” the important event variables by refining them or “de-emphasize” less important, similar variables by abstracting them [3]. For example, in the follow-up model, an influence view which models only the symptoms of metastasis (SOM) may need to be refined to differentiate loss of appetite (LOA), loss of weight (LOW), lymph node involvement (LNI), liver mass (LM), pain, and ascites. Hence, the SOM node needs to be split into nodes of LOA, LOW, LNI, LM, pain, and ascites. All the children of an original node in the concept hierarchy constitute the refinement of the original node. We adopt a strategy similar to that in the event variable abstraction to facilitate the assessment of probability distributions of the variables involved. However, this time the user needs to interactively specify the probability distribution for P (target node | original node), e.g., P (LOW|SOM). This requirement may be relaxed by pre-storing some relevant numbers in the knowledge base. 7.3 Node addition / deletion operations It is possible that new nodes may need to be added to, or existing nodes deleted from, the influence view. In node addition, the knowledge base will be consulted to determine which nodes could be added. In node deletion, the arcs from the node to be removed will be reversed into each successor in order [10]. When the node has no successors and is barren, the node will just be deleted from the model.

8 Related Work Our KBMC design integrates many desirable features of the existing frameworks; it differs from these frameworks mainly by providing a set of knowledge-based model modification operations to support interactive extension and modification of the automatically constructed model. There are some similarities between our framework and Breese’s Alterid [1]. However, Alterid does not consider hierarchical knowledge, so it can not support modeling at different levels of detail; it does not consider time in knowledge representation, so it can not handle dynamic decision model construction. Alterid’s model construction procedure includes both backward chaining and forward chaining. Our influence view constructor employs only backward chaining to avoid the generation of nodes that are irrelevant in the computation of transition probabilities. We have proven that backward chaining is sufficient to generate all relevant event variables in the influence view. Provan’s DYNASTY [9] includes model updating operations to update model components form time interval t to t’. It also incorporates a method for automatic probability updating for such operations; we will explore the feasibility of this method in our future work. However, the system does not consider abstracting an influence path. Since there is no hierarchical knowledge in DYNASTY, the refinement and coarsen-

ing operations mainly depend on the progression or reduction of findings over time. In Wellman’s SUDO-PLANNER [14], domain descriptions can also be expressed at multiple levels of detail. However, in our representation of probabilistic knowledge, the predecessors of a variable are bundled together, rather than represented as a collection of individually asserted predecessors as in SUDO-PLANNER. Illuminated by the elaboration operation in SUDO-PLANNER, which replaces the direct relationship between the two event variables with a more detailed path, we are now exploring if we can introduce similar refinement operation into our framework. Since there is significant difference in the semantics between the probabilistic dependency in our framework and the qualitative influence defined in SUDO-PLANNER, the final definition of the operation may be quite different.

9 Discussion We have presented the design of a knowledge-based dynamic decision model formulation framework. We are now working on the prototype implementation to assess the practical promise of the proposed framework. The overall performance of the framework will then be evaluated based on the scope of decision problems addressed, the ease of encoding domain knowledge in the knowledge base, the speed of model construction, the ease of model revision, and the quality of the resulting model in terms of accuracy, conciseness, and clarity. One potential disadvantage of our influence view construction control strategy is that it cannot avoid exhaustive inclusion in the influence view of all the relevant variable in the knowledge base. Therefore, the influence view may become quite complex sometimes. Although the user can perform abstraction on the model components to reduce the model size, a better solution is to define a set of stopping criteria which takes into account the relative significance of the available information, and the relative utility of further expanding the model. A future research issue of this work involves incorporating a machine learning module into the system, which will learn probabilities from large data sets, so as to reduce the work on the estimation and specification of the numerical parameters involved. Another interesting issue is to extend the framework to handle automated decision model formulation from multiple knowledge sources. Integration of expertise from multiple sources is critical to constructing a comprehensive model. However, the KBMC system should then also address such issues as the wide variety of available information sources, their imprecision, and possible disagreements or inconsistencies among the different sources.

Acknowledgments This work is supported by a strategic research grant no. RP960351 from the National Science and Technology Board and the Ministry of Education in Singapore.

References 1. Breese, J. S.: Construction of belief and decision networks. Computational Intelligence, 8(4): 624-647 (1992) 2. Cao, C. and Leong, T. Y.: Learning Conditional Probabilities for Dynamic Influence Structures in Medical Decision Models, In Proceedings of the 1997 AMIA Annual Fall Symposium (formerly SCAMC), AMIA (1997) 3. Chang, K. C. and Fung, R.: Refinement and Coarsening of Bayesian Networks, Uncertainty in Artificial Intelligence, pages 435-445 (1991) 4. Goldman, R. P. and Charniak, E.: Dynamic construction of belief networks. In Proceedings of the seventh Conference on Uncertainty in Artificial Intelligence, pages 90-97 (1990) 5. Leong, T. Y.: Multiple perspective reasoning. In Aiello, L. C., Doyle, J., and Shapiro, S. (eds) Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR'96), pages 562-573, Morgan Kaufmann (1996) 6. Ngo, L. and Haddawy, P.: A Knowledge-Based Model Construction Approach to Medical Decision Making. In James J. Cimino, editor, Proceedings of 1996 AMIA Annual Fall Symposium, Washington (1996) 7. Ngo, L., Haddawy, P. and Helwig, J.: A Theoretical Framework for Context-Sensitive Temporal Probability Model Construction with Application to Plan Projection. Uncertainty in Artificial Intelligence: Proceedings of the Eleventh Conference, pages 419-426. Morgan Kaufmann (1995) 8. Pearl, J.: Probabilitic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA (1988) 9. Provan, G. M.: Dynamic network updating techniques for diagnostic reasoning. In James Allen, Richard Fikes, and Erik Sandewall, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the Second International Conference (KR91), pages 279-286, San Mateo, CA (1991) 10. Shachter, R. D.: Probabilistic Inference and Influence Diagrams. Operations Research, 36:589-604 (1988) 11. Srinivas, S. (1992). Generalizing the noisy or model to n-ary variables. Technical Memorandum 79, Rockwell International Science Center, Palo Alto Laboratory, Palo Alto, CA 12. Tatman, J. A. and Shachter, R. D.: Dynamic programming and influence diagrams. IEEE Transactions on Systems, Man, and Cybernetics, 20(2):365-279 (1990) 13. Wellman, M. P., Breese, J. S. and Goldman, R. P.: From Knowledge Bases to Decision Models. The Knowledge Engineering Review, 7(1):35-53 (1992) 14. Wellman, M. P.: Formulation of Tradeoffs in Planning Under Uncertainty. Pitman and Morgan Kaufmann (1990) 15. Wellman, M. P. and Liu, C.: State-Space Abstraction for Anytime Evaluation of Probabilistic Networks. Uncertainty in Artificial Intelligence: Proceedings of the Tenth Conference, pages 567-574. Morgan Kaufmann (1994)