E9. To be completed after lesson 25

Exercise 9.1

Discuss the relationship between these two statements.

\begin{align} &1.\quad f \sim \text{GP}(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}';\theta_k)), \\[1em] &2.\quad \mathbf{f} \sim \mathrm{MultiNorm}(\mathbf{m}(\mathbf{X}), \mathsf{K}), \\[1em] \end{align}

where the entries in \(\mathsf{K}\) are given by \(K_{ij} = k(\mathbf{x}_i, \mathbf{x}_j';\theta_k)\).

Exercise 9.2

Are Gaussian processes useful for extrapolation? That is, say we measured \(y\) values on an interval \([x_\mathrm{start}, x_\mathrm{end}]\). Could we use a Gaussian process to estimate what values of \(y\) we might get for \(x > x_\mathrm{end}\)?

Exercise 9.3

When we have a GP prior and a Normal likelihood, there are some really fortuitous consequences. What are they?

Exercise 9.4

Write down any additional questions you have.