Investigate importance of cross-covariance for Cluster likelihood

[itrharrison]

There has been some discussion about including cross-covariance between Clusters and lensing and CMB etc in SOLikeT.

It seems that there is no trivial way (or indeed no way at all) to write down a joint multivariate Gaussian-Poisson likelihood as would be required to do this fully.

That would seemingly leave two options:

1. Ignore the cross-covariance but use correct likelihoods
2. Include the correct cross-covariance but approximate the likelihoods in such a way a joint one can be written down

It would be great to work through the absolute and relative sizes of any biases which may be created by either of these!

For option 2. there are probably two sub-options: a. Approximate the cluster likelihood as Gaussian. There will be a trade off between binning choices and how Gaussian the likelihood is. b. Approximate the other likelihoods as Poisson and use a multivariate Poisson likelihood (which is not a trivial likelihood, apparently).

[itrharrison]

@nbatta or @eunseongleee I don't know if you have thought about any of this already?