A Sober Look at Bayesian Neural Networks

My summary:

Neural nets have massive "capacity" which means that, in the face of finite data sets, they can both (a) reasonably represent generalizable and non-generalizable functions and (b) can take on priors which do not distinguish between those classes of functions. The upshot is that after training, the posterior weight of robust/generalizable models will equal that of fragile/non-generalizable ones.

We need to believe that the priors that we actually use don't have that property if we're to believe in the posteriors produced by BNNs. Should we?

Today, priors in networks arise mostly out of network topology since initialization methods are somewhat constrained by practicalities in training. The article criticizes those who would assert that network topology (+ initialization) leads to a reasonable prior in the space of effective input -> output functions as realized by the network.

To put that in different terms, you might imagine an argument saying that network topologies are biologically inspired and thus represent a decent approximation of the space of "achievable" implementations of functions in the given task. But does an argument like this say anything about the generalization capability of functions favored by this prior? You might characterize this as "Easy" versus "Correct".

I'm not trying to actually represent argumentation that neural network topologies actually are reasonable in shaping "uninformative" Gaussian priors in the weight space into "uninformative" and "generalizable" priors in the function space. There may be some really good arguments out there. But, if we're going to understand NNs as reasonable Bayesian processes, then that question needs to be interrogated.

Hi. Yes, I see- I misunderstood this. My apologies for the hasty reading of your post.

But, in that case, there does exist a very good generalisation prior on function space that is well known and well understood: the simplest hypothesis (e.g. the one with the smallest minumum description length) is always better (because it results in a reduction of the hypothesis search space with a corresponding reduction to the size of the error on unseen data while keeping the number of examples constant).

See:

Occam's Razor (Blumer and friends):

https://www.sciencedirect.com/science/article/pii/0020019087...

Quoting from the abstract:

We show that a polynomial learning algorithm, as defined by ["A theory of the learnable", Valiant 1984], is obtained whenever there exists a polynomial-time method of producing, for any sequence of observations, a nearly minimum hypothesis that is consistent with these observations.

Would that begin to address your concerns?

Agree. Throwing prior weights into hyperparameter sets is not helpful. If each hyperparameter or architecture decision were rigorously tied to an interpretable event or process, we'd have much better explainability and traceability. Its a step in the right direction to try.

Shouldn’t the priors be updated and improved each step to be closer to a good prior, and that’s why inaccurate priors may be acceptable? (Do BNNs not iteratively update the (next step’s) prior with the previous step’s posterior?) I haven’t worked on BNNs, but since Bayesian are always talking about updating their priors I thought this would be the case.

I am going to be obtuse and say that since Bayes theorem is a non controversial rule about conditional probability, if you interpret your NN as a probability distribution over outputs that is updated by data, it is always able to be interpreted as "Bayesian" and can be a helpful way to examine what your implict priors are (i.e. via architecture or regularization terms) to see if they are reasonable for the problem at hand. It is no surprise that weak uninformative priors are sort of useless. Explicit priors shine when you know something about the actual problem (say some moments or some invariances in the problem set up).