# 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. “Multivariate Gaussian and Student-t Process Regression for Multi-Output Prediction.” Neural Computing and Applications 32 (8): 3005–28. https://doi.org/10.1007/s00521-019-04687-8.
Grigelionis, Bronius. 2013. Student’s t-Distribution and Related Stochastic Processes. SpringerBriefs in Statistics. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-31146-8.
Grosswald, E. 1976. “The Student t-Distribution of Any Degree of Freedom Is Infinitely Divisible.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 36 (2): 103–9. https://doi.org/10.1007/BF00533993.
Ismail, Mourad E. H. 1977. “Bessel Functions and the Infinite Divisibility of the Student $$t$$- Distribution.” The Annals of Probability 5 (4): 582–85. https://doi.org/10.1214/aop/1176995766.
Neal, Radford M. 1996. “Bayesian Learning for Neural Networks.” Secaucus, NJ, USA: Springer-Verlag New York, Inc. http://www.csri.utoronto.ca/~radford/ftp/thesis.pdf.
Rasmussen, Carl Edward, and Christopher K. I. Williams. 2006. Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. Cambridge, Mass: MIT Press. http://www.gaussianprocess.org/gpml/.
Shah, Amar, Andrew Wilson, and Zoubin Ghahramani. 2014. “Student-t Processes as Alternatives to Gaussian Processes.” In Artificial Intelligence and Statistics, 877–85. PMLR. http://proceedings.mlr.press/v33/shah14.html.
Tang, Qingtao, Li Niu, Yisen Wang, Tao Dai, Wangpeng An, Jianfei Cai, and Shu-Tao Xia. 2017. “Student-t Process Regression with Student-t Likelihood,” 2822–28. https://www.ijcai.org/proceedings/2017/393.
Tracey, Brendan D., and David H. Wolpert. 2018. “Upgrading from Gaussian Processes to Student’s-T Processes.” 2018 AIAA Non-Deterministic Approaches Conference, January. https://doi.org/10.2514/6.2018-1659.
Wilson, Andrew Gordon, and Zoubin Ghahramani. 2011. “Generalised Wishart Processes.” In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence, 736–44. UAI’11. Arlington, Virginia, United States: AUAI Press. http://dl.acm.org/citation.cfm?id=3020548.3020633.

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