Prior distributions for Bayesian inference about extremes

Extreme events although very rare can have very large, and very unwelcome effects.
To obtain some reassurance that people are highly protected from such harm some
understanding is needed as to the size and likelihood of extremes. But because extreme
events are very rare, so is direct evidence of their likelihood. A particular extreme may
never have been observed. And yet, we still wish to understand its likelihood so that
we can protect ourselves from its effects.
A very beautiful theorem of probability theory appears to offer a way of assessing the
probability of the occurrence of events well beyond the data that have been observed.
Although its preconditions set an ideal that rarely if ever has been met, a wider theory
has been developed that is directly applicable to practical problems. But there is still a
catch: the forecasts of extremes are subject to huge uncertainty, sometimes rendering
them practically useless.
One way of tightening the estimates of extremes that has been increasingly recognised
over the past 30 years or so is to use the Bayesian paradigm for inference about extremes
to bring into account informative priors, especially those obtained by elicitation from
experts. That way specific observations of the quantity of interest are supplemented by
other data.
Elicitation is a complex process that needs to be conducted according to strict protocols
to yield from experts both valid estimates of quantities and of the uncertainty of those
estimates. Typically only a small number of quantiles can be produced for each quantity.
But for Bayesian inference the priors need to be full probability distributions.
This thesis examines options for stretching, so to speak, a small number of elicited
quantiles into a full distribution. It points to the disadvantages of fitting quantiles to a
parametric textbook distribution. It presents a new way of obtaining a wide variety of
distributions defined as Gaussian processes over series of knots. It also considers the
use of minimal distributions in the sense that a bare minimum of assumptions is made.
Finally it reports the results of an elicitation exercise based on these principles.