The_Future_of_Employment

Williams, 2006). Such classifiers model the latent function f with a Gaussian
process (GP): a non-parametric probability distribution over functions.
A GP is defined as a distribution over the functionsf: X → R such that the
distribution over the possible function values on any finite subset ofX (such as
X) is multivariate Gaussian. For a functionf(x), the prior distribution over its
valuesf on a subsetx ⊂ X are completely specified by a covariance matrixK
(4)
p(f | K) = N(f;0,K) =
1
√
det2πK
exp ( − 1 2 f⊺ K −1 f ) .
The covariance matrix is generated by a covariance function κ : X ×X ↦→ R;
that is, K = κ(X,X). The GP model is expressed by the choice of κ; we consider
the exponentiated quadratic (squared exponential) and rational quadratic.
Note that we have chosen a zero mean function, encoding the assumption that
P(y ∗ = 1) = 1 sufficiently far from training data.
2
Given training dataD, we use the GP to make predictions about the function
valuesf ∗ at inputx ∗ . With this information, we have the predictive equations
(5)
p(f ∗ | x ∗ ,D) = N ( f ∗ ;m(f ∗ | x ∗ ,D),V(f ∗ | x ∗ ,D) ) ,
where
(6)
(7)
m(f ∗ | x ∗ ,D) = K(x ∗ ,X)K(X,X) −1 y
V(f ∗ | x ∗ ,D) = K(x ∗ ,x ∗ )−K(x ∗ ,X)K(X,X) −1 K(X,x ∗ ).
Inferring the label posterior p(y ∗ | x ∗ ,D) is complicated by the non-Gaussian
form of the logistic (3). In order to effect inference, we use the approximate
Expectation Propagation algorithm (Minka, 2001).
We tested three Gaussian process classifiers using the GPML toolbox (Rasmussen
and Nickisch, 2010) on our data, built around exponentiated quadratic,
rational quadratic and linear covariances. Note that the latter is equivalent to
logistic regression with a Gaussian prior taken on the weights w. To validate
these classifiers, we randomly selected a reduced training set of half the available
dataD; the remaining data formed a test set. On this test set, we evaluated
how closely the algorithm’s classifications matched the hand labels according
to two metrics (see e.g. Murphy (2012)): the area under the receiver operat-
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