Spatially Embedded Neural Networks: Verifying Single-Neuron Approximate Bayesian Updating in 3D

This research note verifies that one can apply Bayesian updates to a 3D Gaussian PDF to optimise a spatially embedded node’s coordinates. A spatially embedded neural network is a computational graph whose coordinates lie in the Euclidean RD space. The author applies Bayes’ rule to obtain an evidence-dependent analytical update for the neuron’s medial node PDF. Updating the medial node PDF allows weight training in spatially conditioned networks. The mean-squared error loss and a Gaussian prior are used.