Using neural networks to estimate redshift distributions. An application
  to CFHTLenS
  We present a novel way of using neural networks (NN) to estimate the redshift
distribution of a galaxy sample. We are able to obtain a probability density
function (PDF) for each galaxy using a classification neural network. The
method is applied to 58714 galaxies in CFHTLenS that have spectroscopic
redshifts from DEEP2, VVDS and VIPERS. Using this data we show that the stacked
PDF's give an excellent representation of the true $N(z)$ using information
from 5, 4 or 3 photometric bands. We show that the fractional error due to
using N(z_(phot)) instead of N(z_(truth)) is <=1 on the lensing power spectrum
P_(kappa) in several tomographic bins. Further we investigate how well this
method performs when few training samples are available and show that in this
regime the neural network slightly overestimates the N(z) at high z. Finally
the case where the training sample is not representative of the full data set
is investigated. An IPython notebook accompanying this paper is made available
here: https://bitbucket.org/christopher_bonnett/nn_notebook
