Estimating effects using flashr (a bit like mashr)

[stephens999]

I would like to be able to fit models of the form
Bhat = N(B,SVS)
where B = LF is unknown, Bhat is observed, and S and V are considered known.

Here Bhat is p by R where maybe p is big (hundreds of thousands) and R is moderate (dozens? hundreds?) as the GTEx eQTL mapping problem.

The goal is to get (approximate) posterior distributions on elements of B (eg local false sign rates).

For simplicity let's start with V=S=I (so the Bhat are Z scores).

i) apply flash to Bhat (or S^{-1}Bhat) to get an initial L and F.
Note: Here we might want to include the R condition-specific factors to explicitly allow
condition-specific effects.

ii) use sampling (particularly of L given F?) to get approximate posterior samples from B.
See issue #16

To allow for non-diagonal V we might do an eigen-decomposition V=\sum_k lambda_k v_k v_k',
and add some fixed factors with the v_k as factors, and with loadings set to have prior g^l_k = N(0,lambda_k^2)