Solving Influence Diagrams: Exact Algorithms

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 127761, ’Probabilistic Modeling for Decision Support in Integrated Operations,’ by Martin Giese, University of Oslo, and Reidar B. Bratvold, SPE, University of Stavanger, prepared for the 2010 SPE Intelligent Energy Conference and Exhibition, Utrecht, Netherlands, 23-25 March. The paper has not been peer reviewed. A system was designed to assist operational decisions on the basis of real-time sensor readings in a typical scenario: While drilling close to the transition to a high-pressure formation, gas influx is observed. The technology used Bayesian networks and influence diagrams (IDs) to apply the method of decision analysis. The resulting ID was tested with a simulation. Introduction An assumption of integrated operations is that all information provided by the technology will help to make better decisions faster. But for this to happen, information must be presented in an accessible, useful, and decision-relevant format. Decision-analysis tools provide decision makers with logical and consistent recommendations based on the available data. This decision-analysis method was applied to a real case, representative of a typical scenario in drilling operations. The tool enabled good decisions consistently. However, effort is required and challenges exist in the modeling process along with problems regarding the way decision making is treated in the industry. Modeling and Solving A decision tree provides a graphical illustration of all the uncertainties, decisions, and payoffs associated with a decision situation. Although application of decision trees is intuitive for simple problems, the method loses its appeal for complex decisions. A symmetric decision tree grows exponentially with the number of variables in the decision domain, which often makes its use impractical. For complex decisions, the ID can provide a level of insight and transparency that cannot be achieved with the equivalent decision tree. The ID is a compact graphical-model representation for reasoning under uncertainty and was developed to assist structuring and analysis of complex decisions. The ID has two advantages—structuring and probabilistic evaluation. The structuring provides a graphical representation of the inter-relationships among the decision-basis elements (i.e., alternatives, information, and preferences), while the evaluation appraises the resulting decision structure and provides an assessment of its expected utility given the quantified decision basis. The representational framework of an ID captures the probabilistic relationships among decisions (alternatives), key uncertainties (information), and utilities (values or preferences). The ID model comprises four types of nodes. Decision nodes specify a set of alternatives. Chance nodes represent the key uncertainties in the decision domain. Deterministic nodes represent either constants or intermediate calculations. Value nodes represent the criteria used to make the decisions.

Influence diagrams are a powerful graphic representation for decision models, complementary to decision trees. Influence diagrams and decision trees are different graphic representations for the same underlying mathematical model and operations. This article describes the elements of an influence diagram, and shows several familiar decision problems represented as decision trees and as influence diagrams. The authors also contrast the information highlighted in each graphic representation, demonstrate how to calculate the expected utilities of decision alternatives modeled with an influence diagram, provide an overview of the conceptual basis of the solution algorithms that have been developed for influence diagrams, discuss the strengths and limitations of influence diagrams relative to decision trees, and describe the mathematical operations that are used to evaluate both decision trees and influence diagrams. They use clinical examples to illustrate the mathematical operations of the influence-diagram-evaluation algorithm; these operations are arc reversal, chance node removal by averaging, and decision node removal by policy determination. Influence diagrams may be helpful when problems have a high degree of conditional independence, when large models are needed, when communication of the probabilistic relationships is important, or when the analysis requires extensive Bayesian updating. The choice of graphic representation should be governed by convenience, and will depend on the problem being analyzed, on the experience of the analyst, and on the background of the consumers of the analysis.

Chapter 5 - Decision Analysis Fundamentals

We review the development of research in Decision Analysis (DA) over the past fifty years. After presenting the axiomatic foundations and discussing the DA process, we start with value-focused thinking as a problem structuring method. We then analyze the model building phase, with a focus on graphical models for decision-making under uncertainty: belief networks, decision trees, and influence diagrams. Next, we analyze how DA research has dealt with uncertainty focusing on the areas of elicitation, aggregation, and evaluation. We then discuss sensitivity analysis, describing local and global techniques, from one-way sensitivity analysis to the value of information. Finally, we review the literature on information acquisition and discuss the role of information value in this context. • We investigate the intersections between Decision Analysis and Operations Research in the last 50 years. • We review advances in value-focused thinking. • We investigate progress in the development of graphical models. • We detail the literature development in addressing uncertainty and sensitivity analysis. • We present the developments in information acquisition.

TWO ALGORITHMS THAT LEARN DNF BY DISCOVERING RELEVANT FEATURES

Dynamic decision making without expected utility: An operational approach

Conversion of a colorectal cancer guideline into clinical decision trees with assessment of validity

Motivation and Objectives Neuroblastoma (NB) is the most common pediatric solid tumor characterized by clinical and molecular risk factors. The mortality is about fifty percent and this makes exploration of new and more effective risk factors for improving stratification mandatory. Hypoxia is a condition of low oxygen tension occurring in poorly vascularized areas of the tumor associated with poor prognosis. We had previously defined a robust gene expression signature measuring the hypoxic component of NB tumors (NB-hypo) that is a novel, independent risk factor (Fardin et al., 2010). Integrating classical risk factors with NB-hypo could improve the stratification of NB patients. We wanted to develop a prognostic classifier of NB patients’ outcome blending existing knowledge on clinical and molecular risk factors with the prognostic NB-hypo signature. Furthermore, we were interested in the decision tree classifier that outputs explicit rules easily translated into the clinical setting. Methods A total of 182 NB patients were enrolled on the bases of availability of gene expression profile by Affymetrix GeneChip HG-U133plus2.0, clinical and molecular information. NB tumor stage was defined according to the International NB Staging System (INSS). Age at diagnosis was dichotomized as greater or equal than 1 year and less than one year. MYCN status was amplified or normal. Good and poor outcome were defined as the patient’s status alive or dead 5 years after diagnosis respectively. The risk group was assigned according to the International Neuroblastoma Risk Group (INRG) Consensus Pretreatment Classification Schema. The 182 NB patients cohort was clustered in High and low hypoxia by k-means analysis of the 62 probsets constituting the NB-hypo signature previously described to measure tumor hypoxia (Fardin et al., 2009). We utilized the k-means algorithm implemented in the WEKA software (Hall et al., 2009) setting up number of clusters to 2, 500 iterations, preserving instances order and using Manhattan distance. The classification was performed by induction of decision trees. We utilized the Weka J48 implementation of the popular C4.5 algorithm (Kotsiantis, 2007; Murthy, 1998) and we set up the following options: pruning parameter was 0.25, pruning method was sub-tree raising and minimum number of instances per leaf was 2. Each leaf of the decision tree classifier identifies a non overlapping group of patients and each decision node identifies a branch which splits the dataset. We utilized Fisher’s exact test to measure the statistical significance of groups and branches. Fisher’s test was utilized in the context of decision trees to design a top-down approach to prune out non statistically significant branches (Liu et al., 2010). For each leaf we counted the number of correctly classified (named n) and the number of incorrectly classified instances (named m). We considered the marginal totals y and ¬y which represent poor and good outcome patients respectively. We designed a 2x2 contingency table of the two possible outcomes (Good or Poor) against the number of instances included in a give leaf and the remaining instances. Application of the Fisher exact test to this table generates a p value giving the probability of observing 2x2 table, or more extreme tables, knowing the marginal totals (y, ¬y) and assuming independency among the patients in a specific leaf and those in other leaves. Results and Discussion Patients were stratified in good and poor outcome on the bases of the following risk factors: Age at diagnosis, INSS stage, MYCN status and NB-hypo. The algorithm generated a decision tree classifier composed by 3 decision nodes and 7 leaves covering 87% of non overlapping good outcome patients and 100% of non overlapping poor outcome patients. Each path from the root to a leaf utilizes some, but not all, considered risk factors. Interestingly NB-hypo was included in the decision node that stratified stage 3 tumors demonstrating its usefulness in NB patients’ stratification. The leaves classifying good outcome patients had the very low error of 2% indicating a good performance of the algorithm in predicting this class. In contrast, the classification of poor outcome patients produced the leaf with the highest error of 13%. This leaf includes stage 4 tumors that are traditionally difficult to stratify by any known risk factor. To test statistical significance of the splits performed by the algorithm and the groups of patients, we utilized the Fisher’s exact test with a confidence set at 95%. The results showed that the groups obtained at each split were statistically significant. Furthermore, analysis of single groups of patients identified by leaves demonstrated that some, but no all reached the significance threshold. The significant leaves included the NB-hypo among the utilized risk factors. These results further strengthen the value of NB-hypo in predicting patients’ outcome. Lack of significance was often associated with a rather low number of patients in the leaf. This is a limitation of the divide-and-conquer approach of the algorithm, applied to relatively small patients’ cohorts, that recurrently splits the dataset. We then assessed the concordance of the predictions with INRG risk assessment. High Risk patients were correctly included in the leaves classifying poor outcome patients and Low Risk patients mapped correctly in the good outcome leaves. Interestingly, NB-hypo generated a leaf identifying a new group of poor outcome patients, sharing the high NB-hypo, whose characteristic fell into both High and Intermediate Risk. We collected and analyzed the results of 1000 10-fold cross validations and we observed that most classifiers had 7 leaves and only 5 out of 10^4 deviated from this pattern. The recurrence of 7 leaves demonstrated the high stability of the decision tree classifier that we generated. Analysis of the pruning parameters revealed optimal performance in the range of 0.1-0.3 in line with what used in this study. The path to reach each leaf can be easily transposed into a “if…than…” rule that, in turn provides an easy readout of the classifier, precious for translating the classification into the clinical setting. In conclusion, we demonstrated that the decision tree algorithm C4.5 can derive explicit rules for NB patients stratification if classical risk factors are blended with the NB hypo signature. These rules are statistically significant and quite stable and suitable to be conveyed to the clinic to design new therapies perhaps taking hypoxia into consideration as a potential target. Acknowledgements The work was supported by the Fondazione Italiana per la Lotta al Neuroblastoma, the Associazione Italiana per la Ricerca sul Cancro, the Società Italiana Glicogenosi, the Fondazione Umberto Veronesi and the Ministero della Salute Italiano. Davide Cangelosi and Fabiola Blengio are recipients of a fellowship from the Fondazione Italiana per la Lotta al Neuroblastoma. References Fardin P, Barla A, Mosci S, Rosasco L, Verri A, Varesio L. (2009) The l1-l2 regularization framework unmasks the hypoxia signature hidden in the transcriptome of a set of heterogeneous neuroblastoma cell lines. BMC Genomics, 10:474. doi:10.1186/1471-2164-10-474 Fardin P, Barla A, Mosci S, Rosasco L, Verri A, et al. (2010) A biology-driven approach identifies the hypoxia gene signature as a predictor of the outcome of neuroblastoma patients, Journal of Molecular Cancer 9:1. 185. doi:10.1186/1476-4598-9-185 Hall M, Eibe F, Holmes G, Pfahringer B, Reutemann P, Witten IH. (2009) The WEKA Data Mining Software: An Update: SIGKDD Explorations. doi: 10.1145/1656274.1656278 Kotsiantis S.B. (2007) Supervised Machine Learning: A Review of Classification Techniques. Informatica 31:249-268. Liu W, Chawla S, Cieslak D, Chawla N. (2010) A Robust Decision Tree Algorithm for Imbalanced Data Sets. In: SDM. 766-777. Murthy S.K. (1998) Automatic Construction of Decision Trees from Data: A Multi-Disciplinary Survey, Data Mining Knowledge Discovery 2:4. 345-389. doi: 10.1023/A:1009744630224

Influence diagrams (IDs) are a popular framework representing a decision maker’s belief and preferences about a sequence of decisions to be made under uncertainty. The quantification of IDs which consists of defining conditional probabilities for chance nodes and utility functions for value nodes is not always obvious. In fact, decision makers cannot always provide exact utilities and in some cases, it is more convincible to describe utilities with interval values than precise ones. This paper is about extending general IDs (with precise single-valued parameters) into IDs with interval-valued utilities. Such extension is interesting because it enables one to model decision making processes in the situation that utilities are represented by interval values. This paper extend Gibbs sampling algorithm to the Bayesian network (BN) containing interval-valued probabilities for approximate inference and propose a method to evaluate IDs with interval-valued utilities based on the BN inference.

Graphical representations have played a central role in decision analysis. Although decision trees remain popular, more general graphical languages can be used to encode relationships among variables of a decision basis. Influence diagrams, introduced in the late 1970s, provide a general graphical representation for decision analysis. We publish, in this first of two volumes of a special issue of Decision Analysis, the original manuscript on influence diagrams, along with a retrospective by the authors. We also include two new articles focusing respectively on the use of graphical models for group coordination challenges and on handling the unintended side effects of interventions.

Decision Model for Acute Appendicitis Treatment With Decision Tree Technology—A Modification of the Alvarado Scoring System

Manually tuning the parameters or "knobs" of a complex software system is an extremely difficult task. Ideally, the process of software tuning should be automated, allowing software systems to reconfigure themselves as needed in response to changing conditions. We present a methodology that uses a probabilistic, graphical model known as an influence diagram as the foundation of an effective, automated approach to software tuning. We have used our methodology to simultaneously tune four knobs from the Berkeley DB embedded database system, and our results show that an influence diagram can effectively generalize from training data for this domain.

This article aims to present a case study of a real decision problem, with the main objective of assisting an animal production company, on the installation of a new factory. When evaluating the various alternatives, organizations seek clear, objective and, if possible, mathematically represented criteria. However, decision-making involves a cognitive mental process resulting from the selection of the most appropriate course of action, based on various tangible and intangible criteria (Saaty, 2000) arbitrarily chosen by decision makers. In addition to the various criteria, we also have the participation of several decision makers with the common objective of choosing of the best place to guarantee the highest growth of the company. Several methods have been developed to aid decision-making in this scenario. In the described case, we used one of those methods, namely AHP (Analytic Hierarchy Process), to hierarchize the possible location for the installation of the new factory, being the alternatives the following: Carregado, Benavente and Covilha. In 1980, Professor L. Saaty of the University of Pittsburgh, created AHP, which allows using qualitative and/or quantitative criteria in the evaluation process. This method starts with the construction of a decision tree where the criteria, sub-criteria and the various alternatives are detailed. In a second step, those criteria are compared among the various elements at the same level of hierarchy, using a matrix created by the author - Fundamental Ratio Scale in pair-wise comparison of Saaty -, with values from 1 to 9, depending on their relevance, being 1 of equal importance and the 9 of absolute importance. The third step in the model is to calculate the relative priority of each criteria and to analyze the consistency of the judgments of the decision makers (i.e. the consistency of their responses/ comparisons). At final step, the composite priority matrix for the various alternatives is obtained, reflecting the alternatives hierarchy. According to Forman and Peniwati (1998), the behavior of a decision makers’ group is a factor that will determine how the information will be analyzed and aggregated. Should the group acts as a unit (when individuals in a group essentially desire the best for the organization they represent, regardless of their own preferences, values and goals, acting in tune and making their judgments so that the group behaves as a new individual), a variant of the AHP known as Individual Judgment Aggregation (IJA) is used. For groups that prefer to maintain individual analysis, there is Individual Priority Aggregation (IPA) alternative. In both cases it is possible to assign different weights to the decision makers in the process or to consider them with same degree of importance in terms of the decision. In both cases, the group decision is treated mathematically. In the application of the AHP to our case, two decision makers considered 3 criteria subdivided into several sub-criteria: implementation (sub-criteria: suppliers, cost of installation and distance to current clients), labor (sub-criteria: availability and qualification) and incentives (sub-criteria: social security, and taxes). In this case study, decision-makers' consensus was chosen, using IJA. The application of this method to this real case allowed us to create a model of analysis and support decision, contributing with another tool for this complex decision process.

Formalizing aspects of human judgment under uncertainty in support of decision making is a topic central to the fields of artificial intelligence, decision analysis, and psychology. Automated aids based on probability judgments were first proposed in the 1960s [Edwards, 1962; Edwards et al., 1968], however computers were not sufficiently accessible at that time to make the vision practical. By the 1980s techniques in the artificial intelligence community for expert decision support emphasized logical and deterministic models [de Kleer and Williams, 1987; Genesereth, 1984], as well as non-probabilistic methods [Buchanan and Shortliffe, 1984] for handling uncertainty. Over the last ten years advances in graphical models, such as Bayesian networks [Pearl, 1988] and influence diagrams [Howard and Matheson, 1981], have led to a resurgence in the use of decision-theoretic approaches for automated decision making [Heckerman et al., 1992; Breese et al., 1992; Abramson et al., 1996]. Interest in these methods has been motivated by the recognition that heuristic decision making in complex uncertain environments can lead to suboptimal choices [Tversky and Kahneman, 1974; Kleinmuntz, 1985]. Normative systems constructed on the principles of Bayesian probability, multiattribute utility theory, and decision analysis have great potential for automated reasoning and improving human decision making. This paper is a merging of ideas from decision analysis and techniques for probabilistic reasoning developed in the artificial intelligence community. The methods are being applied on a large scale in automated diagnostic procedures for Microsoft software [Heckerman et al., 1995], thus bringing to fruition the notion of normative decision systems envisioned by Ward Edwards in 1962.

A political model using influence diagrams and the Analytic Hierarchy Process: the Camp David negotiations