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Non-Gaussian Bayesian functional regression

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

Random fields with non-Gaussian marginals. Generalised Gaussian process regression.

Is there ever an actual need for this? Or can we just use mostly-Gaussian process with some non-Gaussian distribution marginal and pretend? Presumably if we suspect higher moments than the second are important we might bother with this, but oh my there will be some bad scaling and ugly tensor mathematics.