The Northern Ireland group held a meeting on Friday, 12 April, 2019 at 3pm in the Maths and Physics Teaching Centre at Queen's University Belfast.
The speaker was Professor Simon Wilson of Trinity College Dublin, who talked about quantifying system risk in the context of space vehicles exploding prematurely on re-entry to the earth's atmosphere. This was a problem where there were few observed events and hence measured data were lacking.
The problem was one in which (a) there is substantial expert opinion, but access to it is limited; and (b) the top event and its causes are not completely observed. Professor Wilson's solution was to propose a (Bayesian) belief network approach comprising four steps: 1) Elicit a fault tree and convert to a belief network; 2) Elicit priors on primary event probabilities; 3) Use these to get an initial assessment of the top event probability; and 4) Update this with (in)complete data when available.
Prior elicitation was a key component and experts were used to specify the prior distribution p(θ) on the (discrete) quantities of interest, θ. The next step involved mapping the (perhaps) messy, qualitative and quantitative information to a precise probability distribution.
Professor Wilson outlined the detailed procedure based on a fault tree. Pairwise comparisons were used to compare the probabilities of events with a 'cornerstone' event. The expert gives an interval (plo; phi) for the pi: (all i) and is then asked if pi compared to pj (all i; j) is more or less likely, and to categorise the results as: (a) equally, (b) moderately (c) strongly, (d) very strongly, and (e) absolutely. This five-point scale is then mapped to a numeric score qij and subsequently, via a weighting process, to the 95% probability region of a Beta distribution.
With these prior arrangements, Professor Wilson illustrated how to analyse full and partial data via a likelihood-based approach. With partial data it was necessary to marginalise the likelihood the complete likelihood over the unobserved events. Inference was then carried by a random-walk Metropolis algorithm on the logit(pi) with a Gaussian proposal.
The application of this method to the re-entry vehicle problem with a speci c fault tree involved three separate sets of elicitations, three experts and the resulting prior mean was 0.17 ( central 95% interval 0.11 to 0.24). Incorporating the very sparse data (five vehicles - all successfully de-orbited - hence no explosions) led to a posterior estimate and central 95% interval of 0.10, (0.02,0.25), respectively.
The results were clear. The priors seemed to over-estimate the risk. The downward posterior estimate reflected the sparse data. However, Simon concluded by saying that it was an interesting technique that raised many questions.
The talk was received with acclaim. The various sources of subjectivity entering the elicitation procedure were discussed as were some of the assumptions underpinning the structure of the fault tree. The independence assumptions required were also discussed.