Comment by fjkdlsjflkds | Hacker News Reader

Comment by fjkdlsjflkds | Hacker News Reader

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> For one, Bayesian inference and UQ fundamentally depends on the choice of the prior, but this is rarely discussed in the Bayesian NN literature and practice, and is further compounded by how fundamentally hard to interpret and choose these priors are (what is the intuition behind a NN's parameters?).

I agree that, computationally, it is hard to justify the use of Bayesian methods on large-scale neural networks when stochastic gradient descent (and friends) is so damn efficient and effective.

On the other hand, the fact that there's a dependence on (subjective) priors is hardly a fair critique: non-Bayesian training of neural networks also depends on the use of (subjective) loss functions with (subjective) regularization terms (in fact, it can be shown that, mathematically, the use of priors is precisely equivalent to adding regularization to a loss function). Non-Bayesian training of neural networks is not "a failed approach" just because someone can arbitrarily choose L1 regularization (i.e., a Laplacian prior) over L2 regularization (i.e., a Gaussian prior).

Furthermore, we do have some intuition over NN parameters (particularly when inputs and outputs are properly scaled): a value of 10^15 should be less likely than a value of 0. Note that, in Bayesian practice, people often use weakly-informative priors (see, e.g., http://www.stat.columbia.edu/~gelman/presentations/weakprior...) to encode such intuitive statements while ensuring that (for all practical purposes) the data will effectively overwhelm the prior (again, this is equivalent to adding a minimal amount of regularization to a loss function, to make a problem well-posed when e.g. you have more parameters than data points).

Non-Bayesian NN training does indeed use regularizers that are chosen subjectively —- but they are then tested in validation, and the best-performing regularizer is chosen. Thus the choice is empirical, not subjective.

A Bayesian could try the same thing: try out several priors, and pick the one that performs best in validation. But if you pick your prior based on the data, then the classic theory about “principled quantification of uncertainty” doesn’t apply any more. So you’re left using a computationally unwieldy procedure that doesn’t offer theoretical guarantees.

You can, in fact, do that. It's called (aptly enough) the empirical Bayes method. [1]

[1] https://en.wikipedia.org/wiki/Empirical_Bayes_method

Empirical Bayes is exactly what I was getting at. It's a pragmatic modelling choice, but it loses the theoretical guarantees about uncertainty quantification that pure Bayesianism gives us.

(Though if you have a reference for why empirical Bayes does give theoretical guarantees, I'll be happy to change my mind!)

> Non-Bayesian NN training does indeed use regularizers that are chosen subjectively —- but they are then tested in validation, and the best-performing regularizer is chosen. Thus the choice is empirical, not subjective.

I'd argue the choice is still subjective, since you are still only testing over a limited (subjective) set of options. If you are doing this properly (i.e., using an independent validation set), then you can apply the same approach to a Bayesian method and obtain the same type of information ("when I use prior A vs. prior B, how does that change the generalization/out-of-bag error properties of my model?"), without violating any properties or theoretical guarantees of "Bayesianism".

> A Bayesian could try the same thing: try out several priors, and pick the one that performs best in validation. But if you pick your prior based on the data, then the classic theory about “principled quantification of uncertainty” doesn’t apply any more.

If you subjectively define a set of possible priors (i.e., distributions and parameters) to test in a validation setting, then you are not picking your prior based on the data (again, assuming that you have set up a leakage-free partition of your data in training and validation data), and you are not doing empirical Bayes, so you are not violating any supposed "principled quantification of uncertainty" (if you believe that applying a standard subjective Bayesian approach provides you with "principled quantification of uncertainty").

My point was that, in practice, there are ways of choosing (subjective) priors such that they provide sufficient regularization while ensuring that their impact on the results is minimized, particularly when you can assume certain things about the scale of data (and, in the context of neural networks, you often can, due to things like "normalization layers" and prior scaling of inputs and outputs): "subjective" doesn't have to mean "arbitrary".

> So you’re left using a computationally unwieldy procedure that doesn’t offer theoretical guarantees.

I won't argue about the fact that training NN using Bayesian approaches is computationally unwieldy. I just don't see how evaluating a modelling decision (be in Bayesian or non-Bayesian modelling), using a proper validation process, would violate any specific theoretical guarantees.

If you can explain to me how evaluating the generalization properties of a Bayesian training recipe on an independent dataset violates any specific theoretical guarantees, I would be thankful (note: as far as I am concerned, "principled quantification of uncertainty" is not a specific theoretical guarantee).

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