Comments on "When Bayesian Inference Shatters"?

This is an interesting paper, thanks for linking it (posting here to remind myself to read it carefully later).

For those wanting to see the proofs of the authors theorems, they are in this other paper of theirs.

My immediate reaction to the notched distributions that they use to exemplify their results is that it's cheating -- as indeed say all, including the authors. The priors giving pathological posteriors are chosen in response to the data. Any measure of closeness that puts these distributions close to the un-notched distribution is a silly measure of their suitability as priors. However, I don't have a mathematical expression of what the right measure would be, and no-one that I've seen commenting has explicitly set out a reason for dismissing these "posterior priors", although Entsophy of course does dismiss them on the blog page Cyan linked. (Whatever happened to Entsophy, BTW?) In fact, the authors defend these priors against the charge of disreputability, by arguing that varying the priors in response to the data is exactly what is done in Bayesian sensitivity analysis.

If instead of illustrating their theorems, I imagine a real-world scenario of someone using one of these notched distributions, I get something like this:

I pray to God to show himself by a miracle, then use a quantum mechanical device to generate a string of one million random digits R. I look at these digits and construct a prior such that P(God|R) is high, while P(God|R') is low for all other R'. This prior is very close by various measures to the one that assigns uniformly low probability to the existence of God whatever string of digits I got.

Something is going wrong here, but I don't think it's Bayesian inference. Russell's teapot seems relevant.

The authors write here:

We do not think that this is the end of Bayesian Inference

but go on to argue that it should always be subservient to non-Bayesian reasoning.

I call forth the mighty Cyan!

Here's a discussion of the paper by the authors. For a sort of critical discussion of the result, see the comments in this blog post.

I don't know much about Bayesian Inference, but I am familiar with the well-posedness problem the paper seems to allude to. The authors seem to claim that in the continuous limit the inference problem is ill-posed, specifically, that the solution's behavior does not change continuously with the initial conditions. "Chaotic" is the corresponding popular meme. If true, it means that the continuous version of the Aumann's agreement theorem is unstable: a tiny difference in priors may result in a complete disagreement. Which is very interesting and has direct applications to the FAI research.

EDIT: the relevant quote (emphasis mine):

This brittleness persists beyond the discretization of continuous systems and suggests that Bayesian inference is generically ill-posed in the sense of Hadamard when applied to such systems: if closeness is defined in terms of the total variation metric or the matching of a finite system of moments, then (1) two practitioners who use arbitrarily close models and observe the same (possibly arbitrarily large amount of) data may reach diametrically opposite conclusions; and (2) any given prior and model can be slightly perturbed to achieve any desired posterior conclusions.

My impression is that implicitly relying on arbitrary precision of a prior can give updates that are diametrically opposed to the ones you'd get with different, but arbitrarily similar priors.

I'm not sure what the "precision of a prior" means. A prior is an expression of the knowledge you have before obtaining the data. It is not something that is a measurement of something else, which it could be a more or less precise measurement of.

Has anyone produced a scenario in which the brittleness phenomenon arises in realistic practice?

I refuse to entertain the idea that bayesian should be called what it is and not inverse probability. Bayesians ego's are overinflated on their own hot air. Kolmogorov comes first.

Protocols of Anti-Bayes:

http://vserver1.cscs.lsa.umich.edu/~crshalizi/weblog/601.html

http://projecteuclid.org/euclid.aos/1176349830

http://vserver1.cscs.lsa.umich.edu/~crshalizi/notebooks/bayesian-consistency.html

http://link.springer.com/article/10.1007%2FBF00535479

and how could I forget this masterpiece?

http://repository.cmu.edu/cgi/viewcontent.cgi?article=1382&context=philosophy