Robust Discretization of Continuouos Probability Distributions for Value-of-Information Analysis
Authors: Bratvold Reidar Brumer , Thomas Philip 
Published: May 2016

Abstract
One of the most useful features of decision analysis is its ability to distinguish between constructive and wasteful information gathering. Value-of-Information (VOI) analysis evaluates the benefits of collecting additional information before making a decision.
VOI models describe the relationship between the uncertain quantities of interest, the reliability of the information and the decision criteria. Many uncertainty quantities are continuous in nature and the probability distributions that describe their uncertainty are discretized, and presented in decision trees, to simplify analysis.
A three-point discretization of a continuous distribution is standard, and often preserves the main characteristics (central tendency, spread) of distributions that are close to symmetric. However, VOI studies rarely, if ever, include an analysis on the sensitivity of the VOI to the quality of the discretization. In this work we investigate a variety of discretization techniques, for a range of typical information gathering situations. The investigation utilizes a robust model that accurately calculates the VOI for any combination of continuous distributions. The key criterion for assessing a discretization techniques is whether or not they it has a significant impact on the decision to collect the information. The goal of the work is to provide practical guidance on the level and technique of discretization required.

Introduction.
Most of what petroleum engineers or geoscientists do involves "acquiring" information, with the aim of improving decision-making. "Information" is used here in a broad sense to cover such activities as acquiring of data, performing technical studies, hiring consultants, or performing diagnostic tests. In fact, other than to meet applicable regulatory requirements, the main reason for collecting any information, or doing any technical analysis, should be to make better decisions. The fundamental question for any information-gathering process is then whether the likely improvement in decision-making is worth the cost of obtaining the information. This is the question that the VOI technique is designed to answer.
The oil and gas literature includes a number of papers (e.g. Begg et al, 2002; Bratvold et al, 2008) and books (e.g. Bratvold and Begg, 2010; Newendorp and Schuyler, 2003) where VOI analysis is introduced and described in detail. It is common to use decision trees to structure and evaluate the VOI decision. If the underlying uncertainty is continuous in nature, one of the common discretization methodologies such as Extended Swanson-Megill, Extended Pearson-Tukey, or the McNamee-Celona is often used to discretize the underlying uncertainty into a few, usually 2 or 3, degrees(Bickel et al. 2011). In general, the discrete probabilities and values are selected with the aim of matching the moments (mainly the mean and variance) of the continuous representation. In this paper we refer to this approach as the Low Resolution Decision Tree (LRDT) approach.
A few papers discuss the calculation of VOI when the uncertain event of interest is continuous (Chavez & Henrion, 2004; Arild, Lohne, and Bratvold, 2008; Bickel, 2012). However, the VOI calculation approach in these papers is presented using different terminology than what is used in the LRDT approach. Furthermore, the papers are either limited to relatively simple problems or have very different representations of the reliability or quality of the information gathered. Despite the fact the LRDT approach can have significant errors, VOI calculation are rarely conducted using the full continuous representation of the uncertain event of interest. We suspect that the reason lies in the issue discussed above.