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Abstract:
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In modern applications, it is often the case that there is no likelihood function that directly ties the observable data to the unknown quantities of interest. For example, in machine learning applications, the quantities of interest are often defined as minimizers of a suitable risk function, hence no likelihood function at all. In other cases, there may be a likelihood function but it may not be accessible due to the complexity of the model. On top of there being no likelihood function, it may be that the quantity of interest is high-dimensional, e.g., a function. This talk will focus on the construction of a so-called Gibbs posterior -- a likelihood-free counterpart to the familiar Bayesian posterior -- for these high-dimensional, where known low-dimensional structure is incorporated through the prior distribution. I'll briefly present some general asymptotic properties of the Gibbs posterior, but the discussion will focus largely on applications in precision medicine and mathematical finance, along with some open questions.
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