# t-processes, t-distributions 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.

## 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 $\boldsymbol{X}_2 \mid \boldsymbol{X}_1 \sim \boldsymbol{t}_{p_2}\left(\boldsymbol{\mu}_{2 \mid 1}, \frac{v+d_1}{v+p_1} \boldsymbol{\Sigma}_{22 \mid 1}, \boldsymbol{v}+p_1\right).$

## t-process 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.

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.

## 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. ### No comments yet. Why not leave one?

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