Question

# How are model parameters assigned credible distributions?

In model uncertainty characterisation, where aleatory and epistemic uncertainties are present in the input parameters, how are input parameters of the model assigned credible distributions. References would be helpful.

6th Nov, 2015
Bright Uchenna Oparaji
University of Liverpool
Thanks Scott.

Deleted profile
Can you explain what do you mean by credible distributions?
1st May, 2015
Ilya B. Gertsbakh
Ben-Gurion University of the Negev
Have you real-life data for at least a part of the parameters?
5th Nov, 2015
Scott Ferson
University of Liverpool
There is a lot of debate about the right answer to your question.  The maximum entropy people say you don’t need to characterize distributions for them at all.  Just use the (maximally entropic) distributions that match the known constraints on their random values.  The Bayesians tend to want to use standard Bayesian methods, which don’t really address the difference between the aleatory and epistemic uncertainties.  There is no maximum likelihood answer to the question, but you might be able to construct one assuming the parameters have maybe normal shapes whatever the shape of the underlying distribution.  I’m not quite sure what the generalized moment approach would suggest, but this approach is apparently popular among economists right now.
The approach that we suggest is to use confidence boxes if you have random sample data and constraint-based probability boxes when there are no sample data.  These strategies are compatible, and both directly and unambiguously address epistemic and aleatory uncertainties.  Both are described in turn below.
Probability boxes (p-boxes) are described at https://en.wikipedia.org/wiki/Probability_box and at https://en.wikipedia.org/wiki/Probability_bounds_analysis, which include references and some links to many applications in engineering settings.  P-boxes are commonly used when analysts cannot justify the assumption that probability distributions can be precisely specified.  They also arise in modeling whenever intervals need to be combined with probability distributions, or when the dependencies between probability distributions are imperfectly known.  P-boxes are bounds on probability distributions in the same way that intervals can be bounds on imprecisely specified real values.  They can be derived from constraint information about the random values the distribution describes, simply as the envelope of all the distributions that are consistent with that information.  Best-possible p-boxes have been derived for a host of empirical situations such as when you might have information about the range of the variable, or its mean or variance or bounds on them, or maybe something about its shape such as symmetry, unimodality, etc.
These p-boxes can be used immediately in mathematical calculations, and the results will be guaranteed to enclose the true distribution of the output if the p-boxes enclose their respective inputs (and the projection model is correct).  In many cases, the results obtained will also be the best possible bounds that do so, which means that tightening them would require additional empirical information.   There are software tools to create and compute with p-boxes.
When you have sample data, on the other hand, you can do much better by using confidence boxes to characterize the parameters of the distributions.  Confidence boxes (c-boxes) are described at https://sites.google.com/site/confidenceboxes/, which offers on-line access to several recent papers.  By construction, c-boxes tell you confidence intervals for a parameter at any confidence level you like.  Although you can't generally compute with confidence intervals, you can compute with confidence boxes, and you can get arbitrary confidence intervals directly from the results.  C-boxes are imprecise generalizations of confidence distributions (similar, for example, to the t-distribution), and are analogs to posterior distributions with a frequentist interpretation.
The Google site has software in R to compute c-boxes for the parameters of binomial, Bernoulli, Poisson, normal, uniform, exponential, and lognormal distributions, and also for the “non-parametric” case where no distribution shape is to be assumed.  The software also computes the resulting p-boxes for the distributions with those parameters.
In summer 2014, I gave a series of talks on this subject including, coincidentally, at the Institute for Risk and Uncertainty at the University of Liverpool.  Abstracts and slides from those talks are available on request.  We have been working, very s-l-o-w-l-y, on a review paper on this subject.  Our progress can be observed at the collaboration site at https://sites.google.com/site/niharrachallengeproblems/.  (You have to request permission to access the site.)  We are looking for collaborators, if you are interested in contributing.
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6th Nov, 2015
Bright Uchenna Oparaji
University of Liverpool
Thanks Scott.

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