This is an interesting counterexample, and I agree with Larry that using priors which depend on pi(x) is really no Bayesian solution at all. But if this example is really so problematic for Bayesian inference, can one give an explicit example of some function theta(x) for which no reasonable Bayesian prior is consistent? I would guess that only extremely pathological and unrealistic examples theta(x) would cause trouble for Bayesians. What I notice about many of these “Bayesian non-consistency” examples is that they require consistency over very large function classes: hence they shouldn’t really scare a subjective Bayesian who knows that any function you might encounter in the real world would be much better behaved.
In terms of practicality, it’s certainly inconvenient to have to compute a non-parametric posterior just to do inference on a single real parameter phi. To me, the two practical aspects of actually specifying priors and actually computing the posterior remain the only real weakness of the subjective Bayesian approach (or the Likelihood principle more generally.)
PS: Perhaps it’s worth discussing this example as its own thread.
This is an interesting counterexample, and I agree with Larry that using priors which depend on pi(x) is really no Bayesian solution at all. But if this example is really so problematic for Bayesian inference, can one give an explicit example of some function theta(x) for which no reasonable Bayesian prior is consistent? I would guess that only extremely pathological and unrealistic examples theta(x) would cause trouble for Bayesians. What I notice about many of these “Bayesian non-consistency” examples is that they require consistency over very large function classes: hence they shouldn’t really scare a subjective Bayesian who knows that any function you might encounter in the real world would be much better behaved.
In terms of practicality, it’s certainly inconvenient to have to compute a non-parametric posterior just to do inference on a single real parameter phi. To me, the two practical aspects of actually specifying priors and actually computing the posterior remain the only real weakness of the subjective Bayesian approach (or the Likelihood principle more generally.)
PS: Perhaps it’s worth discussing this example as its own thread.