# t-processes

Stochastic processes with Student-t marginals. Much as Student-$$t$$ distributions generalise Gaussian distributions, $$t$$-processes generalise Gaussian processes.

## t-processes regression

There are a couple of classic cases in ML where $$t$$-processes arise, e.g. in Bayes NNs or GP literature . Recently there has been an uptick in actual applications of these processes in regression . See Wilson and Ghahramani (2011) for a Generalized Wishart Process construction that may be helpful? This prior is available in GPyTorch. Recent papers make it seem fairly straightforward.

I am interested in seeing if these can be pressed into service as a model for mis-specification in Gaussian process regression.

Some papers discuss this in term of inference using Inverse Wishart

## Markov t-process

Process with t-distributed increments is in fact a Lévy process, which follows from the fact that the Student-$$t$$ distribution is divisible. As far as I can see here Grigelionis (2013) is the definitive collation of results on that observation.

## References

Chen, Zexun, Bo Wang, and Alexander N. Gorban. 2020. Neural Computing and Applications 32 (8): 3005–28.
Grigelionis, Bronius. 2013. Student’s t-Distribution and Related Stochastic Processes. SpringerBriefs in Statistics. Berlin, Heidelberg: Springer Berlin Heidelberg.
Grosswald, E. 1976. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 36 (2): 103–9.
Ismail, Mourad E. H. 1977. The Annals of Probability 5 (4): 582–85.
Neal, Radford M. 1996. Secaucus, NJ, USA: Springer-Verlag New York, Inc.
Rasmussen, Carl Edward, and Christopher K. I. Williams. 2006. Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. Cambridge, Mass: MIT Press.
Shah, Amar, Andrew Wilson, and Zoubin Ghahramani. 2014. In Artificial Intelligence and Statistics, 877–85. PMLR.
Song, Dae-Kun, Hyoung-Jin Park, and Hyoung-Moon Kim. 2014. Communications for Statistical Applications and Methods 21 (1): 81–91.
Tang, Qingtao, Li Niu, Yisen Wang, Tao Dai, Wangpeng An, Jianfei Cai, and Shu-Tao Xia. 2017. 2822–28.
Tracey, Brendan D., and David H. Wolpert. 2018. 2018 AIAA Non-Deterministic Approaches Conference, January.
Wilson, Andrew Gordon, and Zoubin Ghahramani. 2011. In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence, 736–44. UAI’11. Arlington, Virginia, United States: AUAI Press.

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