# Degrees of freedom for Gaussian Process

I am reading this paper on Generalised Wishart Process (GWP). It is about modelling covariance matrix of D - dimensional gaussian processes (GP) as GWP. I fail to understand interpretation of "degrees of freedom" $\nu$ for a GP.

There are quiet a few mention of degree of freedom in the paper. In section 4, they define GWP as sum of outer product of $\nu$ D-dimensional gaussian processes. Based on this definition $\nu$ seemed to be no. of observations for the gaussian processes. But then in the same section, they also mention about real-valued degrees of freedom, which rules out previous explanation that $\nu$ = no. of observations/data points.

Moving on, Section 5 Bayesian Inference says, degrees of freedom control how concentrated the prior is around the expected value $\Sigma(t)$ (This generic statement does make sense). Also, in section 5.1, they sample from posterior distribution over the gaussian processes with different degrees of freedom (which I fail to understand again). In section 5, they set $\nu$ to its minimum value D + 1 (so that the wishart distribution does not become degenerate).

I'd appreciate if you can provide me with answers or any other resources which can help me understand this.

• I've just skimmed the paper--looks very cool. My reading is that they haven't introduced real-valued $\nu$ yet, but that they plan to do so in an "upcoming" paper (Wilson and Ghahramani, 2011b). So while I have no idea if your interpretation is correct, I wouldn't say it's ruled out by this issue. – David J. Harris Apr 24 '13 at 23:20
• @DavidJ.Harris Yes, the paper seems very cool. To be specific, I want to understand how they are sampling from the posterior with a gaussian process prior (section 5.1) – steadyfish Apr 25 '13 at 14:34

I am not directly familiar with Generalised Wishart Processes, but I have recently been working with the Wishart Affine Stochastic Correlation (WASC) model and hence "ordinary" Wishart processes in continuous time. The parameter corresponding to $\nu$ is usually called $\beta$, here. In fact, $\beta$ can be a real number with $\beta > n-1$ for the model to be non-degenerate, but the integer $\beta$ is an important special case.
If $\beta$ is an integer, the Wishart process can be constructed of $\beta$ independent vectorial Ornstein-Uhlenbeck processes. I can recommend you the paper http://ssrn.com/abstract=1474728 by Gauthier and Possamai, which deals with the simulation of the Wishart model. Particularly section 6.1.2 may be of interest to you as 6.1.2a gives a simulation scheme for an integer $\beta$ and 6.1.2b for a general real $\beta$.