When we want to combine several estimates of a physical quantity into a more precise one that accounts for all in the correct way, we routinely rely to the weighted average: we take each of the N independent, Gaussian-distributed measurements x_i +- σ_i (i=1,...,N) and compute with them the quantity
to which we associate the error computed as
The recipe above is simply the result of applying the method of least squares, or -equivalently- of a likelihood maximization.
It turns out that in many cases we are confronted with estimates of the same physical quantity that appear to be incompatible with each other: say 10+-4 and 30+-5. If we take the weighted average we do get a result in-between, and closer to the determination with the smallest uncertainty as our intuition would want it: x_wa = (10/16+30/25)/(1/16+1/25)=17.80. However, the error on the average is σ_wa=1/(1/16+1/25)=3.12 and so we are in the embarassing situation that our result, 17.8+-3.1, is very far from either of the two inputs.
I have described elsewhere of the way the Particle Data Group computes the error on a weighted average of N independent, Gaussian-distributed estimates of the same physical quantity. In short, what the PDG does when confronted with the problem of obtaining a meaningful error for the weighted average of determinations that are incompatible with each other, is to "scale up" the variance of each independent estimate by the same factor S, which is computed as the reduced chisquared of the N determinations: S^2 = χ^2/(N-1).
The symbol χ^2, for those of you who are not too familiar with basic statistics, is just a sum of the squares of the differences between the average and each input, divided by the corresponding variance:
So the PDG recipe is to replace the weighted average error σ_wa with a scaled version, σ'=σ_wa*S. The introduction of the "Review of Particle Properties" explains how this is a democratic way of handling the situation: being unable to decide which, among a set of N independent determinations, is the likely cause of the large χ^2 (read: the incompatibility of the input measurements), the most sensible choice is to scale each of them by the same amount. By doing that, the central value of the average remains unchanged (since all weights, which are inverse variances, get scaled up by the same factor 1/S^2).
But is it really the case that the PDG recipe is a "know nothing" approach to the problem ? I argue otherwise. For let us consider the likelihood function of the N determinations: before the application of a scale factor, we would write this as
Now, if we replace all variances σ_i^2 with their scaled versions S^2*σ_i^2, we get the modified likelihood
Note that in this second expression I have inserted a factor k(S) which, for reasons not yet to be disclosed, I will refer to as the "prior for S". If we now take the logarithm of the likelihood, we obtain the following expression:
Taking the derivative with respect to S of the above quantity, and equating the result to zero, allows us to determine the value of S which maximizes the likelihood. In so doing we obtain:
Observe now that th
