Probabilistic neural nets

Bayesian and other probabilistic inference in overparameterized ML

Inferring densities and distributions in a massively parameterised deep learning setting.

This is not intrinsically a Bayesian thing to do but in practice much of the demand to do probabilistic nets comes from the demand for Bayesian posterior inference for neural nets. Bayesian inference is, however, not the only way to do uncertainty quantification.

Neural networks are very far from simple exponential families where conjugate distributions might help, and so typically rely upon approximations or luck to approximate our true target of interest.

Closely related: Generative models where we train a process to generate a (possibly stochastic) phenomenon of interest.


Jospin et al. (2022) is a modern high-speed intro and summary of many approaches.

Radford Neal’s thesis (Neal 1996) is a foundational Bayesian use of neural networks in the wide NN and MCMC sampling settings. Diederik P. Kingma’s thesis is a blockbuster in the more recent variational tradition.

Alex Graves’ poster of his paper (Graves 2011) of a simplest prior uncertainty thing for recurrent nets - (diagonal Gaussian weight uncertainty) I found elucidating. (There is a 3rd party quick and dirty implementation.)

One could refer to the 2019 NeurIPS Bayes deep learning workshop site which will have some more modern positioning. There was a tutorial in 2020: by Dustin Tran, Jasper Snoek, Balaji Lakshminarayanan: Practical Uncertainty Estimation & Out-of-Distribution Robustness in Deep Learning.

Generative methods are useful, e.g. the variational autoencoder and affiliated reparameterization trick. Likelihood free methods seems to be in the air too.

We are free to consider classic neural network inference as sort-of a special case of Bayes inference. Specifically, we interpret the loss function \(\mathcal{L}\) of a net \(f:\mathbb{R}^n\times\mathbb{R}^d\to\mathbb{R}^k\) in the likelihood setting \[ \begin{aligned} \mathcal{L}(\theta) &:=-\sum_{i=1}^{m} \log p\left(y_{i} \mid f\left(x_{i} ; \theta\right)\right)-\log p(\theta) \\ &=-\log p(\theta \mid \mathcal{D}). \end{aligned} \]

Obviously a few things are different from the point-estimate case; the parameter vector \(\theta\) is not interpretable, so what do posterior distributions over it even mean? What are sensible priors? Choosing priors over by-design-uninterpretable parameters such as NN weights is a whole fraught thing in ways we will mostly ignore for now. Usually a prior is by default something like \[ p(\theta)=\mathcal{N}\left(0, \lambda^{-1} I\right) \] for want of a better idea. This ends up being equivalent to the “weight decay” regularisation in the sense that Bayesian priors and regularisations often are.

With that basis e could do the usual stuff for Bayes inference, like considering the predictive posterior \[ p(y \mid x, \mathcal{D})=\int p(y \mid f(x ; \theta)) p(\theta \mid \mathcal{D}) d \theta \]

Usually this turns out to be intractable to calculate in the very high dimension parameters spaces of NNs, so we choose something simpler. We could summarise our posterior update by simple maximum a posteriori estimate \[ \theta_{\mathrm{MAP}}:=\operatorname{argmin}_{\theta} \mathcal{L}(\theta). \] In this case we have recovered the classic training of non-Bayes nets with some ad hoc regularisation which we claimed was a prior. But we have no notion of predictive uncertainty if we stop there.

Usually the model will possess many optima, and this will lead suspicion that we have not found a good global one. How do we maximise model evidence here in any case?

Somewhere between the full belt-and-braces Bayes approach and the MAP point estimate there are various approximations to Bayes inference we might try. What follows is an non-exhaustive smörgåsbord of options to do probabilistic inference in neural nets with different trade-offs.

🏗 To discuss: so many options for predictive uncertainty, but fewer for inverse uncertainty.

Natural Posterior Network

borchero/natural-posterior-network (Charpentier et al. 2022): some kind of reparameterization uncertainty.

MC sampling of weights by low-rank Matheron updates

Needs a shorter names but looks cool (Ritter et al. 2021).

microsoft/bayesianize: Bayesianize: A Bayesian neural network wrapper in pytorch. This also leverages Laplace approximations.

Mixture density networks

Nothing to say for now but here are some recommendations I received about this classic (C. Bishop 1994) method.

Variational autoencoders

See variational autoencoders.

Sampling via Monte Carlo

TBD. For now, if the number of parameters is smallish see Hamiltonian Monte Carlo.

Stochastic Gradient Descent as MC inference

I have a vague memory that this argument is leveraged in Neal (1996)? But see the version in Mandt, Hoffman, and Blei (2017) for a highly developed modern take:

Stochastic Gradient Descent with a constant learning rate (constant SGD) simulates a Markov chain with a stationary distribution. With this perspective, we derive several new results.

  1. We show that constant SGD can be used as an approximate Bayesian posterior inference algorithm. Specifically, we show how to adjust the tuning parameters of constant SGD to best match the stationary distribution to a posterior, minimizing the Kullback-Leibler divergence between these two distributions.
  2. We demonstrate that constant SGD gives rise to a new variational EM algorithm that optimizes hyperparameters in complex probabilistic models.
  3. We also propose SGD with momentum for sampling and show how to adjust the damping coefficient accordingly.
  4. We analyze MCMC algorithms. For Langevin Dynamics and Stochastic Gradient Fisher Scoring, we quantify the approximation errors due to finite learning rates. Finally,
  5. we use the stochastic process perspective to give a short proof of why Polyak averaging is optimal. Based on this idea, we propose a scalable approximate MCMC algorithm, the Averaged Stochastic Gradient Sampler.

A popular recent version of this is the Stochastic Weight Averaging family (Izmailov et al. 2018, 2020; Maddox et al. 2019; Wilson and Izmailov 2020), which I am interested in. See Andrew G Wilson’s web page for a brief description of the sub methods, since he seems to have been involved in all of them.

Laplace approximation

See Laplace approximations

Via random projections

I do not have a single paper about this, but I have seen random projection pop up as a piece of the puzzle in other methods. TBC.

In Gaussian process regression

See kernel learning.

Via measure transport

See reparameterization.

Via infinite-width random nets

See wide NN.


How does this work? He, Lakshminarayanan, and Teh (2020).

Ensemble methods

Deep learning has its own variants model averaging and bagging: Neural ensembles. Yarin Gal’s PhD Thesis (Gal 2016) summarizes some implicit approximate approaches (e.g. the Bayesian interpretation of dropout) although dropout as he frames it has become highly controversial these days as a means of inference.


The computational toolsets for “neural” probabilistic programming and vanilla probabilistic programming are converging. See the tool listing under probabilistic programming.


Dustin Tran’s uncertainty layers [1812.03973] Bayesian Layers: A Module for Neural Network Uncertainty:

In our work, we extend layers to capture “distributions over functions”, which we describe as a layer with uncertainty about some state in its computation — be it uncertainty in the weights, pre-activation units, activations, or the entire function. Each sample from the distribution instantiates a different function, e.g., a layer with a different weight con- figuration.…

While the framework we laid out so far tightly integrates deep Bayesian modelling into existing ecosystems, we have deliberately limited our scope. In particular, our layers tie the model specification to the inference algorithm (typically, variational inference). Bayesian Layers’ core assumption is the modularization of inference per layer. This makes inference procedures which depend on the full parameter space, such as Markov chain Monte Carlo, difficult to fit within the framework.


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