This type of approach is very much needed in the area of engineering and science I have been working in. Yet I have never been acquainted to this technique before in a formal way: How different models can be compared over experimental data. Here is a view of Bayesian approach in the nuclear world (dosimetry or scram rate at plants.) In many cases, while you think you are building a theory with some modeling, what you are in fact doing is applying some type of belief in a model and hope the data confirm it. A pretty specific example is that of criticality computations in nuclear engineering. Criticality computation is a way to assess whether a certain amount of nuclear material can become critical (i.e. not safe) and yield consideration of the health of the people surrounding it. These computations are performed using Monte-Carlo codes or deterministic tools, and yield one number that is expected to be less than 1. Since material assemblies generally have many different constituents and have physical measurements with some uncertainties, the exercise is always bound to be a sensitivity analysis which can be very cumbersome. Some people try to develop models so that one can get rid of the multidimensional sensitivity studies but always end up having to convince other folks in the nuclear engineering community that their model is the better. Maybe this bayesian approach could help in raising the belief of others.
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