Bounded probability may be useful to express epistemic uncertainty when assessors find it difficult to specify it with precise probabilities as point values of e.g. P(event A) = 40% or a specific distributions e.g. P(number < 5) = 40% (see section on subjective probability). To express with a bounded probability is instead to say that P(event A) is between 30% and 50%. Probabilities can be bounded in two sides or just one side. Remember that a probability is by definition always bounded between 0 and 1, so a one sided bound always implies a two sided bound either 0 or 1.
The idea of bounded probability have been used to separate between aleatory and epistemic uncertainty, where the probability represents aleatory uncertainty and the imprecision of the probability represents epistemic uncertainty. So called probability boxes can be used to represent aleatory and epistemic uncertainty about a variable. For example, variable X is normally distributed with parameters mu and sigma. Sigma is know to be 1. Uncertainty about the parameter mu is represented by an interval. In that case, variable X can be represented by a probability box. There is a nice theory to combine several variables characterized by probability boxes, which works well as long as the model is not repeating the variables to much since the epistemic uncertainty becomes wider and wider for every convolution. I call this a probability box for a variable or for aleatory uncertainty. The intervals for the parameters are given by experts and there is (to my know knowledge,)no way to learn from data under the probability box specification and keep it as a probability box.
In this discussion I am interested in the use of bounded probability to quantify epistemic uncertainty. Again we can think of a probability box to represent our uncertainty about parameter mu. Both the probability distribution and the bound on it represents epistemic uncertainty. The bound can be an indication of second order uncertainty, i.e. that we have poor knowledge to support the specification of uncertainty on the parameter. Consider the case where the estimation of parameter mu is backed up by two large experimental studies compared to a wild guess by an expert. One could argue that the uncertainty coming from the differences in knowledge backing up the estimate could be represented by a flat versus more peaked probability distribution for mu. Yes, it is possible to do that and use precise probabilities to represent epistemic uncertainty. However, if the expert is making a wild guess, maybe she would be more comfortable in specifying a bounded probability instead of a precise one.
During the last years, I have been thinking on how to make inference when epistemic uncertainty is quantified by bounded probability. I found out that one way is to use sets of probability distributions for parameters in Bayesian inference. Robust Bayesian inference is originally about studying the influence of the choice of prior in a Bayesian analysis. However, it can also be used in line with theory of imprecise probability to quantify epistemic uncertainty by bounded probability. There are plenty of examples for the implementation of Bayesian models with sets of priors for conjugate models. More recently, we see applications of Robust Bayesian inference with models relying on MCMC sampling.
An important thing that I found out is that there is no need to specify something like probability boxes on every variable and then combine them according to the assessment model. Instead, quantifying epistemic uncertainty in the quantity of interest is about finding the bounds under the sets of priors. It is an optimization problem. Robust Bayesian inference allows for continuous integrating of data combined with expert judgement where any weakness coming from the experts or the lack or non-informativeness in data, to be represented by wider bounds on the probabilities representing epistemic uncertainty.
Uncertainty about the quantity of interest will be a bounded probability. Visualization on the quantity of interest can be a probability box (note that the quantity of interest is epistemic). It is a bit tricky to visualize uncertainty about a variable when epistemic uncertainty is expressed by bounded probability. One way is to collapse the probability box for epistemic uncertainty into an interval and create a probability box for aleatory uncertainty for the variable. But that will remove the information in the first and second order uncertainty. This is used for viz and not for propagation.
An advantage with Robust Bayesian inference is that is it not that different from Bayesian inference, and it allows a risk analyst to shift or even combine both types within the same assessment. One must carefully think why bounds on probabilities are needed in the first place. A main argument against Bayesian inference is the problem to specify precise probability distributions. Robust Bayesian inference addresses this -what we see is that it pushes the questions to be that is can be difficult to specify the set of probability distributions. This stresses the need for good and structured expert knowledge elicitation in risk assessment. This cannot be solved by the choice of expression of epistemic uncertainty.
There are other ways to reach bounded probability as a quantification of epistemic uncertainty. Imprecise probability can in addition to Bayesian inference, come from likelihood based inference, frequentistic inference or Dempster-Shafer theory.
Links added by Scott:
interval probabilities, confidence boxes, probability boxes, robust Bayes distribution bands, or other bounding structures from the theory of imprecise probabilities.