A doubt on the notation of Frigola et al (2013) of Gaussian Processes(GP) for a State-Space model?

Question

The picture above is from Frigola et al (2013) - Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC.

In this paper, the authors later define $\mathbf{f}_t=f(\mathbf{x}_{t-1})$, and here lies my misunderstanding...

Let me explain better my doubt. When $f(\mathbf{x}_t)$ follows a unidimensional GP, we have for any $n\in \mathbb{N}$

$(f(\mathbf{x}_1),...,f(\mathbf{x}_n))\sim \mathcal{N}(\mu,k(X,X'))$, where $f(\cdot) \in \mathbb{R}$, and $\mu = (m(\mathbf{x}_1),...,m(\mathbf{x}_1))'$.

However, when we write $\mathbf{f}_t=f(\mathbf{x}_{t-1})$, it seems we are using a univariate function to define vector $\mathbf{f}_t$. How can this be? Any help would be appreciated.

Answer 1

Although I did not find this explicitly stated in the paper, it seems that $f$ is actually intended to be a multiple output Gaussian process (possibly with independent components) - so that a realization is a function from $\mathbb{R}^{n_x}$ to $\mathbb{R}^{n_x}$ and thus $\mathbb{f}_t = f(\mathbf{x}_t) \in \mathbb{R}^{n_x}$. Note that:

1. In the first author's PhD thesis [1], Section 3.1, similar notation is used to define a Gaussian process state-space model, and on p. 27 the following remark appears:

Note that the covariance function $\kappa(\cdot, \cdot)$ returns a matrix of the same size as the state in an analogous manner to multi-output Gaussian processes (Rasmussen and Williams, 2006).

2. In an earlier paper considering Gaussian process state-space models, Turner et al. [2] (reference [2] of the case paper), "independent GPs are used for each target dimension of $f$" (p. 869, last paragraph of the left-hand-side column). Which is a special case of a multiple-output GP.

References

[1] Roger Frigola-Alcalde (2015) Bayesian Time Series Learning with Gaussian Processes. PhD thesis, University of Camridge. http://www.rogerfrigola.com/doc/thesis.pdf

[2] Ryan Turner, Marc Deisenroth, and Carl Rasmussen (2010). "State-space inference and learning with Gaussian processes." AISTATS 2010, http://proceedings.mlr.press/v9/turner10a/turner10a.pdf.

Answer 2

You need to understand the notation here. When the author say \begin{align} x_{t+1}| f_t \sim \mathcal{N}(x_{t+1}|f_t, Q), \end{align} this implies that $x_{t+1}$ is distributed with mean $f_t$ and covariance $Q$, where $f_t$ is obtained from the state evolution equation given in (1a), i.e., \begin{align} x_{t+1} = f(x_t,u_t) + v_t. \end{align} As $f$ depends on $t$, they denote the mean of $x_{t+1}$ as $f_t$ which is nothing but $\mathbb{E}(x_{t+1}) = \mathbb{E}(f(x_t,u_t)) = f_t$ if $v_t \sim \mathcal{N}(0,Q)$. Therefore $x_{t+1}| f_t \sim \mathcal{N}(x_{t+1}|f_t, Q)$.

An easy way to understand this by assuming $f(x_t,u_t)$ takes the form \begin{align} f(x_t, u_t) = A x_t + B u_t \end{align}