Gaussian process regression
- Realizations from a Gaussian process correspond to random functions
- Let us first consider an unknown regression function \( \mu(x) \) that
depends on a single, continuous variable \( x \).
- The Gaussian process is written as \( \mu \sim \mathrm{GP}(m,k) \), and
is parametrized in terms of a mean function \( m(x) \) and a covariance
function \( k(x,x') \).
- The GP prior on \( \mu \) describes it as a random function for which
the values at any set of \( N \) prespecified points \( \{x_i\}_{i=1}^N \)
are a draw from a \( N \)-dimensional normal distribution
$$
$$
\mu(x_1), \ldots \mu(x_N) \sim \mathrm{N}\left( \left( m(x_1),
\ldots, m(x_N) \right), K(x_1, \ldots, x_N) \right),
$$
$$
with mean \( m \) and covariance \( K \).