Speed up single-source fits with analytic solution for free rate multiplier

[JelleAalbers]

When fitting a single source, there is a simple analytic solution for the best-fit rate multiplier at each shape parameter combination. We could use this to save the optimizer some iterations/time.

Consider a single-source fit with no prior on the rate multiplier r, and define

• s one or more shape parameters
• mu(s) the expected total events at r=1 (estimated from simulation)
• N the total observed events, and
• d_i(s) the differential rate of event i

Then log likeliood is

lnL(r, s) = -mu(s) * r + sum_i ln[ r * d_i(s) ]
          = -mu(s) * r + N ln r + sum_i ln[d_i(s)] .

Now dlnL/dr = -mu(s) + N/r, so the best-fit r = N/mu(s).

So for single-source fits, we could optimize - N ln mu(s) + sum_i ln[d_i(s)] instead of the log likelihood. Or equivalently, we could optimize the likelihood as usual but set the rate multiplier to N/mu(s) at each iteration of the fit.