The quest for conditional independence in prospectivity modeling: weights-of-evidence, boost weights-of-evidence, and logistic regression

Abstract

The objective of prospectivity modeling is prediction of the conditional probability of the presence \mathsf{T}=\mathrm{1} or absence \mathsf{T}=\mathrm{0} of a target \mathsf{T} given favorable or prohibitive predictors \mathsf{B}, or construction of a two classes \left\{\mathrm{0},\mathrm{1}\right\} classification of \mathsf{T}. A special case of logistic regression called weights-of-evidence (WofE) is geologists’ favorite method of prospectivity modeling due to its apparent simplicity. However, the numerical simplicity is deceiving as it is implied by the severe mathematical modeling assumption of joint conditional independence of all predictors given the target. General weights of evidence are explicitly introduced which are as simple to estimate as conventional weights, i.e., by counting, but do not require conditional independence. Complementary to the regression view is the classification view on prospectivity modeling. Boosting is the construction of a strong classifier from a set of weak classifiers. From the regression point of view it is closely related to logistic regression. Boost weights-of-evidence (BoostWofE) was introduced into prospectivity modeling to counterbalance violations of the assumption of conditional independence even though relaxation of modeling assumptions with respect to weak classifiers was not the (initial) purpose of boosting. In the original publication of BoostWofE a fabricated dataset was used to “validate” this approach. Using the same fabricated dataset it is shown that BoostWofE cannot generally compensate lacking conditional independence whatever the consecutively processing order of predictors. Thus the alleged features of BoostWofE are disproved by way of counterexamples, while theoretical findings are confirmed that logistic regression including interaction terms can exactly compensate violations of joint conditional independence if the predictors are indicators.

Keywords

general weights of evidence / joint conditional independence / naïve Bayes model / Hammersley–Clifford theorem / interaction terms / statistical significance

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Acknowledgments

The authors would like to thank two anonymous reviewers for their thorough and constructive efforts to help us improve our manuscript. The authors gratefully acknowledge financial funding by the German Federal Ministry for Economic Affairs and Energy (BMWi) within the frame of “Zentrales Innovationsprogramm Mittelstand” (ZIM) on Entwicklung eines Verfahrens zur dreidimensionalen Prognose von verdeckten Rohstofflagerstätten am Beispiel des Erzgebirges. Last but not least the authors greatly appreciate H. Konstanze Zschoke’s, MSc Geophysics, painstaking effort to convert our manuscript from the high-quality typesetting system LaTeX into the word processor MS Word.

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2016 Higher Education Press and Springer-Verlag Berlin Heidelberg