t-processes, t-distributions

November 10, 2021 — November 24, 2021

Figure 1

Stochastic processes with Student-t marginals. Much as Student-\(t\) distributions generalise Gaussian distributions, \(t\)-processes generalise Gaussian processes. Another useful member of the family of elliptically contoured distributions.

1 Multivariate Student-\(t\)

The multivariate \(t\) (MVT) distribution \(\boldsymbol{X} \sim \boldsymbol{t}_p(\boldsymbol{\mu}, \boldsymbol{\Sigma}, \boldsymbol{v})\), with location \(\boldsymbol{\mu}\), scale matrix \(\boldsymbol{\Sigma}\), and degrees of freedom \(v\), has the probability density function \[ f(\boldsymbol{x})=\frac{\Gamma\{(\boldsymbol{v}+p) / 2\}}{\Gamma(\boldsymbol{v} / 2)(\boldsymbol{v} \pi)^{p / 2}|\boldsymbol{\Sigma}|^{1 / 2}}\left\{1+\boldsymbol{v}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})^{\top} \boldsymbol{\Sigma}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})\right\}^{-(v+p) / 2} . \] There is a cool relationship to the multivariate normal: \[ \boldsymbol{X}=\boldsymbol{\mu}+\boldsymbol{\Sigma}^{1 / 2} \boldsymbol{Z} / \sqrt{q}, \] where \(Z\) follows a \(p\) dimensional standard normal distribution, \(q \sim \chi_v^2 / v\), and \(Z\) is independent of \(q\). ( \(W \sim \chi_b^2 / c\) denotes the scaled \(\chi^2\) distribution, with density proportional to \(w^{b / 2-1} e^{-c w / 2}\).) It differs from the multivariate normal distribution \(\mathscr{N}_p(\boldsymbol{\mu}, \boldsymbol{\Sigma})\) only by the random scaling factor \(\sqrt{q}\).

Ding (2016) uses this latter property to show that the conditional distribution of \(\boldsymbol{X}_2\) given \(\boldsymbol{X}_1\) is \[ \boldsymbol{X}_2 \mid \boldsymbol{X}_1 \sim \boldsymbol{t}_{p_2}\left(\boldsymbol{\mu}_{2 \mid 1}, \frac{v+d_1}{v+p_1} \boldsymbol{\Sigma}_{22 \mid 1}, \boldsymbol{v}+p_1\right). \]

2 t-process regression

There are a couple of classic cases in ML where \(t\)-processes arise, e.g. in Bayes NNs (Neal 1996) or GP literature (9.9 Rasmussen and Williams 2006). Recently there has been an uptick in actual applications of these processes in regression (Chen, Wang, and Gorban 2020; Shah, Wilson, and Ghahramani 2014; Tang et al. 2017; Tracey and Wolpert 2018). See Wilson and Ghahramani (2011) for a Generalized Wishart Process construction that may be helpful? This prior is available in GPyTorch. Recent papers (Shah, Wilson, and Ghahramani 2014; Tracey and Wolpert 2018) make it seem fairly straightforward.

At first blush it looks like it might be a more robust regression model than Gaussian process regression. However, I am not so sure. As Ding (2016) points out, the conditional distribution of \(\boldsymbol{X}_2\) given \(\boldsymbol{X}_1\) jointly $t4-distributed grows eventually linearly in the number of observations sites, which means that it is essentially just Gaussian for even small problems.

Some papers discuss t-process regression in term of inference using Inverse Wishart distribuitons.

3 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.

Figure 2

4 References

Aste. 2021. Stress Testing and Systemic Risk Measures Using Multivariate Conditional Probability.”
Bånkestad, Sjölund, Taghia, et al. 2020. The Elliptical Processes: A Family of Fat-Tailed Stochastic Processes.”
Chen, Wang, and Gorban. 2020. Multivariate Gaussian and Student-t Process Regression for Multi-Output Prediction.” Neural Computing and Applications.
Ding. 2016. On the Conditional Distribution of the Multivariate \(t\) Distribution.”
Fang, Kotz, and Ng. 2017. Symmetric Multivariate and Related Distributions.
Grigelionis. 2013. Student’s t-Distribution and Related Stochastic Processes. SpringerBriefs in Statistics.
Grosswald. 1976. The Student t-Distribution of Any Degree of Freedom Is Infinitely Divisible.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete.
Ismail. 1977. Bessel Functions and the Infinite Divisibility of the Student \(t\)- Distribution.” The Annals of Probability.
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.
Neal. 1996. Bayesian Learning for Neural Networks.”
Rasmussen, and Williams. 2006. Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning.
Roth. 2012. On the Multivariate t Distribution.
———. 2013. Kalman Filters for Nonlinear Systems and Heavy-Tailed Noise.”
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.
Tang, Niu, Wang, et al. 2017. Student-t Process Regression with Student-t Likelihood.”
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.