Extending BEAMS to incorporate correlated systematic uncertainties
  New supernova surveys such as the Dark Energy Survey, Pan-STARRS and the LSST
will produce an unprecedented number of photometric supernova candidates, most
with no spectroscopic data. Avoiding biases in cosmological parameters due to
the resulting inevitable contamination from non-Ia supernovae can be achieved
with the BEAMS formalism, allowing for fully photometric supernova cosmology
studies. Here we extend BEAMS to deal with the case in which the supernovae are
correlated by systematic uncertainties. The analytical form of the full BEAMS
posterior requires evaluating 2^N terms, where N is the number of supernova
candidates. This `exponential catastrophe' is computationally unfeasible even
for N of order 100. We circumvent the exponential catastrophe by marginalising
numerically instead of analytically over the possible supernova types: we
augment the cosmological parameters with nuisance parameters describing the
covariance matrix and the types of all the supernovae, \tau_i, that we include
in our MCMC analysis. We show that this method deals well even with large,
unknown systematic uncertainties without a major increase in computational
time, whereas ignoring the correlations can lead to significant biases and
incorrect credible contours. We then compare the numerical marginalisation
technique with a perturbative expansion of the posterior based on the insight
that future surveys will have exquisite light curves and hence the probability
that a given candidate is a Type Ia will be close to unity or zero, for most
objects. Although this perturbative approach changes computation of the
posterior from a 2^N problem into an N^2 or N^3 one, we show that it leads to
biases in general through a small number of misclassifications, implying that
numerical marginalisation is superior.
