The metalog distribution is a flexible continuous probability distribution designed for ease of use in practice. Together with its transforms, the metalog family of continuous distributions is unique because it embodies all of following properties: virtually unlimited shape flexibility; a choice among unbounded, semi-bounded, and bounded distributions; ease of fitting to data with linear least squares; simple, closed-form quantile function (inverse CDF) equations that facilitate simulation; a simple, closed-form PDF; and Bayesian updating in closed form in light of new data. Moreover, like a Taylor series, metalog distributions may have any number of terms, depending on the degree of shape flexibility desired and other application needs. Applications where metalog distributions can be useful typically involve fitting empirical data, simulated data, or expert-elicited quantiles to smooth, continuous probability distributions. Fields of application are wide-ranging, and include economics, science, engineering, and numerous other fields. The metalog distributions, also known as the Keelin distributions, were first published in 2016 by Tom Keelin.
History The history of probability distributions can be viewed, in part, as a progression of developments towards greater flexibility in shape and bounds when fitting to data. The normal distribution was first published in 1756, and Bayes' theorem in 1763. The normal distribution laid the foundation for much of the development of classical statistics. In contrast, Bayes' theorem laid the foundation for the state-of-information, belief-based probability representations. Because belief-based probabilities can take on any shape and may have natural bounds, probability distributions flexible enough to accommodate both were needed. Moreover, many empirical and experimental data sets exhibited shapes that could not be well matched by the normal or other continuous distributions. So began the search for continuous probability distributions with flexible shapes and bounds. Early in the 20th century, the Pearson family of distributions, which includes the normal, beta, uniform, gamma, student-t, chi-square, F, and five others, emerged as a major advance in shape flexibility. These were followed by the Johnson distributions. Both families can represent the first four moments of data (mean, variance, skewness, and kurtosis) with smooth continuous curves. However, they have no ability to match fifth or higher-order moments. Moreover, for a given skewness and kurtosis, there is no choice of bounds. For example, matching the first four moments of a data set may yield a distribution with a negative lower bound, even though it might be known that the quantity in question cannot be negative. Finally, their equations include intractable integrals and complex statistical functions, so that fitting to data typically requires iterative methods. Early in the 21st century, decision analysts began working to develop continuous probability distributions that would exactly fit any specified three points on the cumulative distribution function for an uncertain quantity (e.g., expert-elicited 0.10 , 0.50 {\displaystyle 0.10,0.50} , and 0.90 {\displaystyle 0.90} quantiles). The Pearson and the Johnson family distributions were generally inadequate for this purpose. In addition, decision analysts also sought probability distributions that would be easy to parameterize with data (e.g., by using linear least squares, or equivalently, multiple linear regression). Introduced in 2011, the class of quantile-parameterized distributions (QPDs) accomplished both goals. While being a significant advance for this reason, the QPD originally used to illustrate this class of distributions, the Simple Q-Normal distribution, had less shape flexibility than the Pearson and Johnson families, and lacked the ability to represent semi-bounded and bounded distributions. Shortly thereafter, Keelin developed the family of metalog distributions, another instance of the QPD class, which is more shape-flexible than the Pearson and Johnson families, offers a choice of boundedness, has closed-form equations that can be fit to data with linear least squares, and has closed-form quantile functions, which facilitate Monte Carlo simulation.
Definition and quantile function The metalog distribution is a generalization of the logistic distribution, where the term "metalog" is short for "metalogistic". Starting with the logistic quantile function, x = Q ( y ) = μ + s ln ( y 1 − y ) {\displaystyle x=Q(y)=\mu +s{\mbox{ }}\ln {\Bigl (}{y \over {1-y}}{\Bigr )}} , Keelin substituted power series expansions in cumulative probability y = F ( x ) {\displaystyle y=F(x)} for the μ {\displaystyle \mu } and the s {\displaystyle s} parameters, which control location and scale, respectively.
μ = a 1 + a 4 ( y − 0.5 ) + a 5 ( y − 0.5 ) 2 + a 7 ( y − 0.5 ) 3 + a 9 ( y − 0.5 ) 4 + … {\displaystyle \mu =a_{1}+a_{4}(y-0.5)+a_{5}(y-0.5)^{2}+a_{7}(y-0.5)^{3}+a_{9}(y-0.5)^{4}+\dots }
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