There might be no way to “get there from here” for some priors.
That’s obviously true for priors that initially assign probability zero somewhere. But as Cosma Shalizi loves pointingout, Diaconis and Freedmanhave shown it can happen for more reasonable priors too, where the prior is “maladapted to the data generating process”.
This is of course one of those questionable cases with a lot of infinities being thrown around, and we know that applying Bayesian reasoning with infinities is not on fully solid footing. And much of the discussion is about failure to satisfy Frequentist conditions that many may not care about (though they do have a section arguing we should care). But it is still a very good paper, showing that non-zero probability isn’t quite good enough for some continuous systems.
That’s obviously true for priors that initially assign probability zero somewhere. But as Cosma Shalizi loves pointing out, Diaconis and Freedman have shown it can happen for more reasonable priors too, where the prior is “maladapted to the data generating process”.
This is of course one of those questionable cases with a lot of infinities being thrown around, and we know that applying Bayesian reasoning with infinities is not on fully solid footing. And much of the discussion is about failure to satisfy Frequentist conditions that many may not care about (though they do have a section arguing we should care). But it is still a very good paper, showing that non-zero probability isn’t quite good enough for some continuous systems.