Accurate photometric redshift probability density estimation - method
  comparison and application
  We introduce an ordinal classification algorithm for photometric redshift
estimation, which significantly improves the reconstruction of photometric
redshift probability density functions (PDFs) for individual galaxies and
galaxy samples. As a use case we apply our method to CFHTLS galaxies. The
ordinal classification algorithm treats distinct redshift bins as ordered
values, which improves the quality of photometric redshift PDFs, compared with
non-ordinal classification architectures. We also propose a new single value
point estimate of the galaxy redshift, that can be used to estimate the full
redshift PDF of a galaxy sample. This method is competitive in terms of
accuracy with contemporary algorithms, which stack the full redshift PDFs of
all galaxies in the sample, but requires orders of magnitudes less storage
space.
  The methods described in this paper greatly improve the log-likelihood of
individual object redshift PDFs, when compared with a popular Neural Network
code (ANNz). In our use case, this improvement reaches 50\% for high redshift
objects ($z \geq 0.75$).
  We show that using these more accurate photometric redshift PDFs will lead to
a reduction in the systematic biases by up to a factor of four, when compared
with less accurate PDFs obtained from commonly used methods. The cosmological
analyses we examine and find improvement upon are the following: gravitational
lensing cluster mass estimates, modelling of angular correlation functions, and
modelling of cosmic shear correlation functions.
