An influence diagram is a graphical structure for modeling uncertain variables and decisions and explicitly revealing probabilistic dependence and the flow of information. It is an intuitive framework in which to formulate problems as perceived by decision makers and to incorporate the knowledge of experts. At the same time, it is a precise description of information that can be stored and manipulated by a computer. We develop an algorithm that can evaluate any well-formed influence diagram and determine the optimal policy for its decisions. Since the diagram can be analyzed directly, there is no need to construct other representations such as a decision tree. As a result, the analysis can be performed using the decision maker's perspective on the problem. Questions of sensitivity and the value of information are natural and easily posed. Modifications to the model suggested by such analyses can be made directly to the problem formulation, and then evaluated directly.
An influence diagram is a network representation of probabilistic inference and decision analysis models. The nodes correspond to variables that can be either constants, uncertain quantities, decisions, or objectives. The arcs reveal probabilistic dependence of the uncertain quantities and information available at the time of the decisions. The influence diagram focuses attention on relationships among the variables. As a result, it is increasingly popular for eliciting and communicating the structure of a decision or probabilistic model. This paper develops the framework for assessment and analysis of linear-quadratic-Gaussian models within the influence diagram representation. The "Gaussian influence diagram" exploits conditional independence in a model to simplify elicitation of parameters for the multivariate normal distribution. It is straightforward to assess and maintain a positive (semi-)definite covariance matrix. Problems of inference and decision making can be analyzed using simple transformations to the assessed model, and these procedures have attractive numerical properties. Algorithms are also provided to translate between the Gaussian influence diagram and covariance matrix representations for the normal distribution.influence diagram, Gaussian decision model, multivariate normal assessment
An influence diagram is a network representation for probabilistic and decision analysis models. The nodes correspond to variables which can be constants, uncertain quantities, decisions, or objectives. The arcs reveal the probabilistic dependence of the uncertain quantities and the information available at the time of the decisions. The detailed data about the variables are stored within the nodes, so the diagram graph is compact and focuses attention on the relationships among the variables. Influence diagrams are effective communication tools and recent developments also allow them to be used for analysis. We develop algorithms to address questions of inference within a probabilistic model represented as an influence diagram. We use the conditional independence implied by the diagram's structure to determine the information needed to solve a given problem. When there is enough information we can solve it, exploiting that conditional independence. These same results are applied to problems of decision analysis. This methodology allows the construction of computer tools to maintain and evaluate complex models.
Although a number of algorithms have been developed to solve probabilistic inference problems on belief networks, they can be divided into two main groups: exact techniques which exploit the conditional independence revealed when the graph structure is relatively sparse, and probabilistic sampling techniques which exploit the "conductance" of an embedded Markov chain when the conditional probabilities have non extreme values. In this paper, we investigate a family of Monte Carlo sampling techniques similar to Logic Sampling [Henrion, 1988] which appear to perform well even in some multiply-connected networks with extreme conditional probabilities, and thus would be generally applicable. We consider several enhancements which reduce the posterior variance using this approach and propose a framework and criteria for choosing when to use those enhancements.
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