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, 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 now open access and free to everyone. To download it, click the icon on the right. To download it directly from the publisher, click here.
“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.
"The Metalog Distributions" were explained in an invited lecture at Stanford University, Department of Management Science and Engineering, on February 28, 2017. To download a pdf of the PowerPoint slides, click the icon.
"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.