  We present a comparison between Gaussian processes (GPs) and artificial
neural networks (ANNs) as methods for determining photometric redshifts for
galaxies, given training set data. In particular, we compare their degradation
in performance as the training set size is degraded in ways which might be
caused by the observational limitations of spectroscopy. We find that
performance with large, complete training sets is very similar, although the
ANN achieves slightly smaller root mean square errors. If the size of the
training set is reduced by random sampling, the RMS errors of both methods
increase, but they do so to a lesser extent and in a much smoother manner for
the case of GP regression. When training objects are removed at redshifts
1.3<z<1.7, to simulate the effects of the "redshift desert" of optical
spectroscopy, the GP regression is successful at interpolating across the
redshift gap, while the ANN suffers from strong bias for test objects in this
redshift range. Overall, GP regression has attractive properties for
photometric redshift estimation, particularly for deep, high-redshift surveys
where it is difficult to obtain a large, complete training set. At present,
unlike the ANN code, public GP regression codes do not take account of
inhomogeneous measurement errors on the photometric data, and thus cannot
estimate reliable uncertainties on the predicted redshifts. However, a better
treatment of errors is in principle possible, and the promising results in this
paper suggest that such improved GP algorithms should be pursued. (abridged)
