**T**oday I attended Persi Diaconis’ de Finetti’s ISBA Lecture and not only because I was an invited discussant, by all means!!! Persi was discussing his views on Bayesian numerical analysis. As already expressed in his 1988 paper. Which now appears as a foundational precursor to probabilistic numerics. And which is why I had a very easy time in preparing my discussion as I mostly borrowed from my NIPS slides. With some degree of legitimacy since I was already a discussant there. Anyway, here is the most novel slide in the discussion, built upon my realisation that the principle behind nested sampling is fairly generic for integral approximation, rather than being restricted to marginal likelihood approximation.

Among many interesting things, Persi’s talk made me think anew about infinite variance importance sampling. And about the paper by Souraj Chatterjee and Persi that I discussed a few months ago. In that some regularisation of those “useless” importance estimates can stem from prior modelling. Not as an aside, let me add I am very grateful to the ISBA 2016 organisers and to the chair of the de Finetti lecture committee for their invitation to discuss this talk!