**D**ay [or half-day] three started without firecrackers and with David Rossell (formerly Warwick) presenting an empirical Bayes approach to generalised linear model choice with a high degree of confounding, using approximate Laplace approximations. With considerable improvements in the experimental RMSE. Making feeling sorry there was no apparent fully (and objective?) Bayesian alternative! (Two more papers on my reading list that I should have read way earlier!) Then Veronika Rockova discussed her work on approximate Metropolis-Hastings by classification. (With only a slight overlap with her One World ABC seminar.) Making me once more think of Geyer’s n⁰564 technical report, namely the estimation of a marginal likelihood by a logistic discrimination representation. Her ABC resolution replaces the tolerance step by an exponential of minus the estimated Kullback-Leibler divergence between the data density and the density associated with the current value of the parameter. (I wonder if there is a residual multiplicative constant there… Presumably not. Great idea!) The classification step need be run at every iteration, which could be sped up by subsampling.

On the always fascinating theme of loss based posteriors, à la Bissiri et al., Jack Jewson (formerly Warwick) exposed his work generalised Bayesian and improper models (from Birmingham!). Using data to decide between model and loss, which sounds highly unorthodox! First difficulty is that losses are unscaled. Or even not integrable after an exponential transform. Hence the notion of improper models. As in the case of robust Tukey’s loss, which is bounded by an arbitrary κ. Immediately I wonder if the fact that the pseudo-likelihood does not integrate is important beyond the (obvious) absence of a normalising constant. And the fact that this is not a generative model. And the answer came a few slides later with the use of the Hyvärinen score. Rather than the likelihood score. Which can itself be turned into a H-posterior, very cool indeed! Although I wonder at the feasibility of finding an [objective] prior on κ.

Rajesh Ranganath completed the morning session with a talk on [the difficulty of] connecting Bayesian models and complex prediction models. Using instead a game theoretic approach with Brier scores under censoring. While there was a connection with Veronika’s use of a discriminator as a likelihood approximation, I had trouble catching the overall message…