Creating neural networks which infer whole probability densities their predictions, (usually approximately) rather than point estimates. Or part, at least, of the density estimating problem, accomplished with neural nets, in a Bayesian setting. Prediction uncertainties, approximate model averaging etc would all fit in this category.

AFAICT this usually boils down to doing variational inference, in which case the neural network is a big approximate PDGM. Apparently you can also do simulation-based inference here, somehow using gradients? Must look into that. Also, Gaussian Processes can be made to fit into this framing.

To learn:

- natural gradient
- how does this work outside of KL-divergence?
- marginal likelihood in model selection: how does it work with many optima?

## Backgrounders

Radford Neal's thesis (Neal96) is a foundational asymptotically-Bayesian use of neural netwroks. Yarin Gal's PhD Thesis (Gal16) summarizes some implicit approximate approaches (e.g. the Bayesian interpretation of dropout). Diederik P. Kingma's thesis is the latest blockbuster in this tradition.

Alex Graves did a nice poster of his paper (Grav11) of a simplest prior uncertainty thing for recurrent nets - (diagonal Gaussian weight uncertainty) There is a half-arsed implementation.

## Reparameterisation

## Practicalities

Blei Lab's software tool: Edward (source) Tensorflow indeed comes with a contributed Bayesian library called BayesFlow (Which is not the same as the cytometry library of the same name) which by contrast has documentation so perfunctory that I can't imagine it not being easier to reimplement it than to understand it.

Thomas Wiecki, Bayesian Deep Learning shows how to some variants with PyMC3.

Christopher Bonnett: Mixture Density Networks with Edward, Keras and TensorFlow.

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