The Metalog Distributions

Notation & Conventions

This page documents notation and conventions used throughout the metalog system.

As illustrated on the right, we use "y" to denote a probability as shown on the y-axis of a CDF (cumulative distribution function) and "x" to denote a quantity on the x-axis.

Metalog input parameters are user-specified CDF data points.  Given m input parameters, we denote their x- and y-components respectively as x = (x1, ... ,xm) and y = (y1, ... ,ym). For the unbounded-metalog example on the right, x = (40, 50, 70) and y = (0.1, 0.5, 0.9). We use the terms "input data" and "input parameters" interchangeably.

For semi-bounded metalogs, in addition to x and y, the user specifies a lower bound bl.  For bounded metalogs, the user specifies an upper bound bu as well.

In the metalog system, we specify the CDF and PDF (probability density function) as functions of y rather than x, and denote them as M(y) and m(y) respectively. Specified in this way, M(y) is called a "quantile function", which is the inverse of a CDF specified as a function of x. Thus, M(y) is the quantity on the x-axis of both the CDF and PDF plots, and m(y) is the quantity on the vertical axis of the PDF plot. When specified in this way, the PDF is the locus of points (x, y') = (M(y), m(y)) that results from varying y from 0 to 1.