Tom L: if you’re reading this, you really should’ve published those notes

The short version: the optimal estimator for the photon index is the maximum
likelihood estimator. If the data are binned, and the total number of counts over
all bins is not fixed (i.e., is a random variable), then the likelihood function is

\Prod_{i=1}^k \frac{ (np_i)^{y_i} }{ y_i ! } e^{-np_i}

where y_i are the number of counts in bin i, n = \sum_i y_i, and p_i is the probability
that a count would be recorded in bin i (and this depends on the distribution
parameter, in this case the photon index).

You can derive the \chi^2 function from this using Stirling’s approximation
(np_i \gtrsim 5 in each bin) and then a Taylor series expansion. Depending on
how you do that expansion, you can derive either \chi^2 with model variance
or \chi^2 with data variance. It’s the combination of Stirling’s approximation
with how you cut off the Taylor expansion that creates the bias term.

Tom Loredo wrote this all up years and years ago, so I take no credit for the
explanation.