[IPTA-cs] Bayesian upper limits

[Ken Olum <kdo cosmos phy tufts edu>]

Hi, all.  There was some discussion on the call about how to set upper
limits.  Is there are some literature about this or some standard
techniques?  

I thought about these ideas once, something like this:

Suppose there is a GWB from SMBHB with unknown gamma and A, given by
some priors.  In addition there may be a cosmic string background with
some particular G mu.  The question is whether our observations are
consistent with being a realization of these backgrounds plus noise.  If
G mu is too large, we would expect a larger signal than we see.  If it
is much too large, then 95% of the time we would see a larger signalman
we do, and that is the limit we're looking for.

This depends somewhat on the priors for the SMBHB signal and the noises.
A large string signal is more consistent with observation if it is not
added to some other sources of signals and noise.  But I don't think you
need a prior for G mu here at all, because the question is about
specific values of G mu.

One might ask a different question instead: suppose we have SMBHB and
also strings and we make a PDF of the parameters.  Then we marginalize
over the SMBHB parameters, make a PDF of the G mu and find the 95%
point.  This does depend on the G mu prior.  For example, if we were
extremely confident of some large G mu beforehand, it might still have >
5% probability after unfavorable observations.

So I guess there is really a cultural question.  We would like to say
"these observations rule out G mu > ... at the 95% level".  We would
like to mean the same thing that others mean when they make similar
statements.  What is that?

Since we're not going to detect any strings in the present data, this
limit is probably the most important takeaway from our paper.  So we
should be careful about our choices of what to calculate and how to
explain what we calculated.

                                        Ken