Uncertain Photometric Redshifts
  Photometric redshifts play an important role as a measure of distance for
various cosmological topics. Spectroscopic redshifts are only available for a
very limited number of objects but can be used for creating statistical models.
A broad variety of photometric catalogues provide uncertain low resolution
spectral information for galaxies and quasars that can be used to infer a
redshift. Many different techniques have been developed to produce those
redshift estimates with increasing precision. Instead of providing a point
estimate only, astronomers start to generate probabilistic density functions
(PDFs) which should provide a characterisation of the uncertainties of the
estimation. In this work we present two simple approaches on how to generate
those PDFs. We use the example of generating the photometric redshift PDFs of
quasars from SDSS(DR7) to validate our approaches and to compare them with
point estimates. We do not aim for presenting a new best performing method, but
we choose an intuitive approach that is based on well known machine learning
algorithms. Furthermore we introduce proper tools for evaluating the
performance of PDFs in the context of astronomy. The continuous ranked
probability score (CRPS) and the probability integral transform (PIT) are well
accepted in the weather forecasting community. Both tools reflect how well the
PDFs reproduce the real values of the analysed objects. As we show, nearly all
currently used measures in astronomy show severe weaknesses when used to
evaluate PDFs.
