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2.7. GAUSSIAN PROCESSES 41
where ǫ ∼ N(0, β −1 ). We are interested in encoding our prior knowledge about
the relationship in a distribution over f. For regression we usually have a prefer-
ence to functions varying smoothly over X,
limxi→xj+ |f(xi) − f(xj)| =
= limxi→−xj |f(xi) − f(xj)| = 0,
∀xj ∈ X. This assumption can be encoded by the GP through the choice of
covariance function k(x,x ′ ). The covariance function encodes how we expect
variables to vary together,
k(x,x ′ ) = E ((f(x − µ(x)))(f(x ′ ) − µ(x ′ ))),
this means that we can encode the smoothness behavior over X by choosing a
covariance function which is smooth over the same domain. The mean function
µ(x) = E(f(x)) encodes the expected value of f. By translating the observed
data to be centered around zero the mean function can, for simplicity, be chosen
as the constant function zero.
2.7.1 Prediction
Having specified a prior distribution encoding our knowledge (and preference)
about the relationship between X and Y we are interested in inferring the lo-
cations y∗ corresponding to a previously unobserved point x∗ ∈ X. The joint
distribution of the observed data (y,x) and the unobserved point (y∗,x∗) can be