Stochastic processes with Student-t marginals. Much as Student- distributions generalize Gaussian distributions, -processes generalize Gaussian processes, useful in the regression setting. Another useful member of the family of elliptically contoured distributions. Another fun generalization of the Gaussian process is the q-exponential process.
I was exploring the implications of -processes in various contexts, particularly in regression settings. My conclusion is that this is essentially a curiosity, as it converges very rapidly to the Gaussian process as the number of observations increases, so you may as well use a Gaussian. The other elliptic extension might be more intersting, or a variational Gaussian process with a non-Gaussian likelihood, depending on the application.
Multivariate Student-
The multivariate (MVT) distribution , with location , scale matrix , and degrees of freedom , has the probability density function There is a cool relationship to the multivariate normal: where follows a dimensional standard normal distribution, , and is independent of . ( denotes the scaled distribution, with density proportional to It differs from the multivariate normal distribution only by the random scaling factor .
Ding (2016) uses this latter property to show that the conditional distribution of given is
Markov t-process
Process with -distributed increments is in fact a Lévy process, which follows from the fact that the Student- distribution is divisible. As far as I can see here Grigelionis (2013) is the definitive collation of results on that observation.
References
Grosswald. 1976.
“The Student t-Distribution of Any Degree of Freedom Is Infinitely Divisible.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete.
Kibria, and Joarder. 2006. “A Short Review of Multivariate t-Distribution.” Journal of Statistical Research.
Lange, Little, and Taylor. 1989.
“Robust Statistical Modeling Using the t Distribution.” Journal of the American Statistical Association.
Rasmussen, and Williams. 2006.
Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning.
Shah, Wilson, and Ghahramani. 2014.
“Student-t Processes as Alternatives to Gaussian Processes.” In
Artificial Intelligence and Statistics.
Song, Park, and Kim. 2014.
“A Note on the Characteristic Function of Multivariate t Distribution.” Communications for Statistical Applications and Methods.
Tracey, and Wolpert. 2018.
“Upgrading from Gaussian Processes to Student’s-T Processes.” 2018 AIAA Non-Deterministic Approaches Conference.
Wilson, and Ghahramani. 2011.
“Generalised Wishart Processes.” In
Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence. UAI’11.