  We calculate photometric redshifts from the Sloan Digital Sky Survey Main
Galaxy Sample, The Galaxy Evolution Explorer All Sky Survey, and The Two Micron
All Sky Survey using two new training-set methods. We utilize the broad-band
photometry from the three surveys alongside Sloan Digital Sky Survey measures
of photometric quality and galaxy morphology. Our first training-set method
draws from the theory of ensemble learning while the second employs Gaussian
process regression both of which allow for the estimation of redshift along
with a measure of uncertainty in the estimation. The Gaussian process models
the data very effectively with small training samples of approximately 1000
points or less. These two methods are compared to a well known Artificial
Neural Network training-set method and to simple linear and quadratic
regression. Our results show that robust photometric redshift errors as low as
0.02 RMS can regularly be obtained. We also demonstrate the need to provide
confidence bands on the error estimation made by both classes of models. Our
results indicate that variations due to the optimization procedure used for
almost all neural networks, combined with the variations due to the data
sample, can produce models with variations in accuracy that span an order of
magnitude. A key contribution of this paper is to quantify the variability in
the quality of results as a function of model and training sample. We show how
simply choosing the "best" model given a data set and model class can produce
misleading results.
