I am looking for regressions and classification examples for simple Bayesian (Probabilistic) Neural Network. Simple means neural networks that are shallow (not-deep, one hidden layers and similar).

Can you point me to these sources?

Hi, check out our Bayesian regression tutorial (part 1, part 2) and this forum post about `pyro.contrib.bnn.HiddenLayer`

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There is also a bayesian neural network example in NumPyro that makes use of HMC for inference: numpyro/bnn.py at master · pyro-ppl/numpyro · GitHub