Non-Gaussian Bayesian functional regression

2019-10-09 — 2023-05-25

Wherein an investigation is presented into regression with non-Gaussian random fields, attention being paid to higher moments and the use of sparse stochastic process priors as a practical alternative

Hilbert space
kernel tricks
PDEs
regression
spatial
stochastic processes
time series
Figure 1

Regression using non-Gaussian random fields. 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, via GP quantile regression, or some variational GP approximation or non-Gaussian likelihood over Gaussian latents? Presumably if we suspect higher moments than the second are important, or that there is some actual stochastic process that we know matches our phenomenon, we might bother with this, but oh my it can get complicated.

TO: example, maybe using sparse stochastic process priors, Neural process regression Singh et al. (2019); is that distinct ? See also Q-EP and elliptical processes…

1 References

Bostan, Kamilov, Nilchian, et al. 2013. “Sparse Stochastic Processes and Discretization of Linear Inverse Problems.” IEEE Transactions on Image Processing.
Louizos, Shi, Schutte, et al. 2019. “The Functional Neural Process.” In Advances in Neural Information Processing Systems.
Singh, Yoon, Son, et al. 2019. “Sequential Neural Processes.” arXiv:1906.10264 [Cs, Stat].
Unser, M. 2015. “Sampling and (Sparse) Stochastic Processes: A Tale of Splines and Innovation.” In 2015 International Conference on Sampling Theory and Applications (SampTA).
Unser, Michael A., and Tafti. 2014. An Introduction to Sparse Stochastic Processes.
Unser, M., Tafti, Amini, et al. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes - Part II: Discrete-Domain Theory.” IEEE Transactions on Information Theory.
Unser, M., Tafti, and Sun. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes—Part I: Continuous-Domain Theory.” IEEE Transactions on Information Theory.