The following articles and presentations provide important background for understanding and enabling effective use of the metalog distributions. Click here for the Wikipedia article on the metalog distributions. The metalog distributions belong to the class of quantile-parameterized distributions (QPDs). Click here for the Wikipedia article on QPDs. For original papers, presentations and further information, see below.
"The Melalog Distributions", published in Decision Analysis in December 2016,is among the top ten most download papers in the history of the Decision Analysis Journal. It documents the motivation, literature review, definitions, mathematical derivations, and other background research for the equations and other materials implemented on this website.
This paper is open access and free to everyone. Click icon to download directly from the publisher.
The Metalog Distributions (free download)
"The Multivariate Metalog Distributions with Application to Strategic Decision-Making in Golf". This 2023 preprint introduces the multivariate metalog distributions and applies them to a decision analysis of strategic decision-making in golf. Like the univariate metalog distributions (Keelin, 2016), the multivariate metalog distributions are shape- and bounds-flexible, have simple closed-form quantile functions, and are easy to parameterize, fit to data, interpret, and simulate. Multivariate metalogs are illustrated with an easy-to-understand practical application: in golf, choosing the best aimpoint can make the difference between winning and losing.
The Multivariate Metalog Distributions with Application to Strategic Decision-Making in Golf
"Introducing the Metalog Distributions" was published in December 2022 by the Royal Statistical Society in Significance Magazine. This three-page article introduces the metalogs to the statistics community and includes key metalog research results and literature references up until this date.
“The Metalog Distributions: Virtually Unlimited Shape Flexibility, Combining Expert Opinion in Closed Form, and Bayesian Updating in Closed Form”. Metalog properties introduced in this preprint include: 1. Proof that any continuous quantile function can be approximated arbitrarily closely by a metalog, with several new shape flexibility illustrations. 2. A simple, appealing method for combining the metalogs of multiple experts that yields a consensus metalog in closed form. 3. A method for metalog Bayesian updating in closed form in light of new data, presented in a readily understandable way as a fisherman updates his catch probabilities when changing the river on which he fishes.
The Metalog Distributions: Virtually Unlimited Shape Flexibility, Combining Expert Opinion in Closed Form, and Bayesian Updating in Closed Form
"The Metalog Distributions and Extremely Accurate Sums of Lognormals in Closed Form" was presented at the INFORMS Winter Simulation Conference, National Harbor, Maryland, December 9, 2019. The peer-reviewed paper provides the background, foundation, and detail. The PowerPoint, best viewed in slide show mode, provides an overview. The metalog method for summing lognormals in closed form uses nine points from highly accurate simulations of the sum of lognormals to parameterize a nine term log metalog distribution that is guaranteed to run through these points exactly. Thus, the simulations do not need to be redone, and summing lognormals in closed form reduces to using the pre-simulated table of quantile parameters. This table and additional supplementary information can be downloaded from Lumina Decision Systems' Analytica Wiki.
"Quantile Parameterized Distributions", published in Decision Analysis in September, 2011, provides an important theoretical and research-based background for parameterizing flexible continuous probability distributions with CDF data. The metalog distribution is the first published quantile-parameterized distribution (QPD) designed for broad and practical use. To download "Quantile Parameterized Distributions", click the icon.
Key results of the 2011 article are summarized, along with new properties that have since been discovered of metalogs and other QPD's. This article also contains references to recently-published applications of metalogs and other QPD's.
"Quantile Function Methods For Decision Analysis" is a 2013 Stanford Ph.D. dissertation by Brad Powley. This work includes much of the same content as the co-authored "Quantile Parameterized Distributions" paper above but also significant additional contributions. These include a mathematically rigorous definition of QPDs, discussion of transformations of QPDs (which are themselves quantile-parameterized), and a novel theory of tail behavior which applies not only to QPDs but also to a wider range of continuous distributions. To download, click the icon.
