NB: this is not current; I am doing too much research in the area to summarise it well, and it is large area.
Probabilistic neural networks are loosely, speaking, recipes for creating neural networks which plausibly solve some approximation to inference of a whole probability density of prediction, rather than a mere best point prediction. Sometimes this term seems to be used to mean the more related problem of finding some manner of Bayesian justification for a neural network. The mathematics comes out similar either way.
An alternative emphasis from where I am sitting is this:
Learning problems involve composition of differentiating and integrating various terms that measure various properties of how well you have approximated the state of the world. Probabilistic neural networks find combinations of integrals that we can solve by Monte carlo, and derivatives that we can solve via automatic differentiation, and which are both fast on modern hardware, and then use those cunning combination to find approximate solutions that we would have probably phrased in terms of specific integrals that are in practice completely intractable. The result is machine learning in strange and wonderful places where we could not have solved those integrals and derivatives in the traditional manner. Although… There is something odd about that setup. From this perspective the generative models (such as GANs and autoencoders) are solving an intractable integral by simulating samples probabilistically from them, in lieu of processing the continuous, unknowable, intractable integral that we actually wish to solve. But that continuous intractable integral was in any case a contrivance, a thought experiment imagining a world populated with such weird Platonic objects as integrals-over-possible-states-of-the-world which only mathematicians would consider reasonable. The world we live in has, as far as I know, no such thing. We do not have a world where the things we observe are stochastic samples from an ineffable probability density, but rather the observations themselves are the phenomena, and the probability density over them is an improbable abstraction. It must look deeply weird from the outside when we to talk about how we are solving integrals by looking at data, instead of solving data by looking at integrals.
- marginal likelihood in model selection: how does it work with many optima?
Radford Neal’s thesis (Neal 1996) is a foundational asymptotically-Bayesian use of neural networks. Yarin Gal’s PhD Thesis (Gal 2016) 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 poster of his paper (Graves 2011) of a simplest prior uncertainty thing for recurrent nets - (diagonal Gaussian weight uncertainty) 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.
The computational toolsets for “neural” probabilistic programming and vanilla probabilistic programming are converging. See the tool listing under probabilistic programming.
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