Robins and Wasserman

As I attended Jamie Robins’ session in Varanasi and did not have a clear enough idea of the Robbins and Wasserman paradox to discuss it viva vocce, here are my thoughts after reading Larry’s summary. My first reaction was to question whether or not this was a Bayesian statistical problem (meaning why should I be concered with the problem). Just as the normalising constant problem was not a statistical problem. We are estimating an integral given some censored realisations of a binomial depending on a covariate through an unknown function θ(x). There is not much of a parameter. However, the way Jamie presented it thru clinical trials made the problem sound definitely statistical. So end of the silly objection. My second step is to consider the very point of estimating the entire function (or infinite dimensional parameter) θ(x) when only the integral ψ is of interest. This is presumably the reason why the Bayesian approach fails as it sounds difficult to consistently estimate θ(x) under censored binomial observations, while ψ can be. Of course, if we want to estimate the probability of a success like ψ going through functional estimation this sounds like overshooting. But the Bayesian modelling of the problem appears to require considering all unknowns at once, including the function θ(x) and cannot forget about it. We encountered a somewhat similar problem with Jean-Michel Marin when working on the k-nearest neighbour classification problem. Considering all the points in the testing sample altogether as unknowns would dwarf the training sample and its information content to produce very poor inference. And so we ended up dealing with one point at a time after harsh and intense discussions! Now, back to the Robins and Wasserman paradox, I see no problem in acknowledging a classical Bayesian approach cannot produce a convergent estimate of the integral ψ. Simply because the classical Bayesian approach is an holistic system that cannot remove information to process a subset of the original problem. Call it the curse of marginalisation. Now, on a practical basis, would there be ways of running simulations of the missing Y’s when π(x) is known in order to achieve estimates of ψ? Presumably, but they would end up with a frequentist validation…