  We present a new method to estimate redshift distributions and galaxy-dark
matter bias parameters using correlation functions in a fully data driven and
self-consistent manner. Unlike other machine learning, template, or correlation
redshift methods, this approach does not require a reference sample with known
redshifts. By measuring the projected cross- and auto- correlations of
different galaxy sub-samples, e.g., as chosen by simple cells in
color-magnitude space, we are able to estimate the galaxy-dark matter bias
model parameters, and the shape of the redshift distributions of each
sub-sample. This method fully marginalises over a flexible parameterisation of
the redshift distribution and galaxy-dark matter bias parameters of sub-samples
of galaxies, and thus provides a general Bayesian framework to incorporate
redshift uncertainty into the cosmological analysis in a data-driven,
consistent, and reproducible manner. This result is improved by an order of
magnitude by including cross-correlations with the CMB and with galaxy-galaxy
lensing.
  We showcase how this method could be applied to real galaxies. By using
idealised data vectors, in which all galaxy-dark matter model parameters and
redshift distributions are known, this method is demonstrated to recover
unbiased estimates on important quantities, such as the offset $\Delta_z$
between the mean of the true and estimated redshift distribution and the 68\%
and 95\% and 99.5\% widths of the redshift distribution to an accuracy required
by current and future surveys.
