Bayesian photometric redshifts with empirical training sets
  We combine in a single framework the two complementary benefits of
chi^2-template fits and empirical training sets used e.g. in neural nets: chi^2
is more reliable when its probability density functions (PDFs) are inspected
for multiple peaks, while empirical training is more accurate when calibration
and priors of query data and training set match. We present a chi^2-empirical
method that derives PDFs from empirical models as a subclass of kernel
regression methods, and apply it to the SDSS DR5 sample of >75,000 QSOs, which
is full of ambiguities. Objects with single-peak PDFs show <1% outliers, rms
redshift errors <0.05 and vanishing redshift bias. At z>2.5, these figures are
2x better. Outliers result purely from the discrete nature and limited size of
the model, and rms errors are dominated by the instrinsic variety of object
colours. PDFs classed as ambiguous provide accurate probabilities for
alternative solutions and thus weights for using both solutions and avoiding
needless outliers. E.g., the PDFs predict 78.0% of the stronger peaks to be
correct, which is true for 77.9% of them. Redshift incompleteness is common in
faint spectroscopic surveys and turns into a massive undetectable outlier risk
above other performance limitations, but we can quantify residual outlier risks
stemming from size and completeness of the model. We propose a matched
chi^2-error scale for noisy data and show that it produces correct error
estimates and redshift distributions accurate within Poisson errors. Our method
can easily be applied to future large galaxy surveys, which will benefit from
the reliability in ambiguity detection and residual risk quantification.
