The Doctrine of Bayesian Statistics for Inverse Problems
G.J. Koers (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
Imagine you need to navigate through a completely dark cave. A well-known way of achieving this is echolocation, which works by making sounds and listening to how they are reflected back. The problem of determining the geometric shape of a space from a mixture of reflections of emitted sounds, is an example of an inverse problem. In many fields of science, there are situations where it is impossible to measure a parameter of interest directly and instead, the only available method is to measure a different object that is affected by the parameter of interest. These statistical problems can become challenging when it is difficult, or even impossible, to invert the observations directly into the parameter that one is interested in.
In many cases, a statistician has a belief about the true value of the parameter before even starting the experiment. The Bayesian paradigm is an attractive method of combining the new information coming from observations with this prior belief. It gives a sound mechanism, namely the posterior distribution, to update the beliefs about the truth.