The Metalog Distributions

The Metalog Distributions

In economics, business, engineering, science and other fields, continuous uncertainties frequently arise that are not easily- or well-characterized by previously-named continuous probability distributions. Frequently, there is data available from measurements, assessments, derivations, simulations or other sources that characterize the range of an uncertainty. But the underlying process that generated this data is either unknown or fails to lend itself to convenient derivation of equations that appropriately characterize the probability density (PDF), cumulative (CDF) or quantile (inverse CDF) distribution functions.

Introduced in 2016, the metalog distributions are a family of continuous probability distributions that directly address this need. Compared to traditional distributions like the normal, lognormal, beta, extreme value, student-t, the Pearson family of distributions, and dozens of others, metalogs offer the following advantages: much greater shape-flexibility to match data from any source (e.g. empirical data, expert-assessed data, or simulated data); choice of boundedness (unbounded, semi-bounded, or bounded); simplicity of equations; ease of fitting to data with ordinary least squares; closed form 
quantile function that facilitates simulation.


Among the most flexible and easy-to-use continuous probability distributions, the metalogs have an increasingly wide range of practical uses:
  • Converting scientific data from any field into a shape-flexible continuous probability distribution with simple closed-form equations

  • Visualizing data: instantly seeing a shape-flexible (including multi-modal) PDF that corresponds to your data

  • Quantifying expert-assessed uncertainties (e.g. 10/50/90 quantiles with optional lower and/or upper bounds) and providing instant visual feedback

  • Simulating from a shape-flexible closed-form quantile function.
Overall, metalogs are useful for wide range of situations in which CDF data are known, and a simple, flexible, easy-to-use continuous probability distribution is needed to represent that data. See the Metalog Distribution Wikipedia article for additional information and the Metalogs YouTube channel for educational videos.

Featured Application 

"A Decision Analytic Framework for Bayesian Updating of Probability of Success in Clinical Trials"

Tom Keelin demonstrates how to use data from a Phase 1 pilot investigation to create a metalog probability distribution over results for subsequent patients and how to derive a probability of success (POS) for a subsequent clinical trial based on this state of information (SOI). Then, based on new data (Phase 2) that becomes available subsequently, Tom shows how the parameters of the Phase 1 metalog can be updated in closed form based on this new information, how an updated POS for Phase 3 can be calculated, and the conditions under which this updating method is Bayesian. 

We thank the Probability of Success Interest Group of the Society of Decision Professionals for sponsoring this video. We also thank Shaun Comfort (Roche) and Eric Johnson (Glaxo SmithKline) for moderating and for their help in developing the specific (hypothetical) example and data used.

(click image to view video)
Additional videos on the theory and application of metalog distributions are available on the Metalogs YouTube channel.