Suppose you have a set of N₁ measurements obtained with one detector, and another set of N₂ measurements obtained with a second detector. And let’s say you wanted something as simple as an estimate of the mean of the quantity (say the intensity) being measured. Let us further stipulate that the measurement errors of each of the points is similar in magnitude and neither instrument displays any odd behavior. How does one combine the two datasets without appealing to subjective biases about the reliability or otherwise of the two instruments?

We’ve mentioned this problem before, but I don’t think there’s been a satisfactory answer.

The simplest thing to do would be to simply pool all the measurements into one dataset with N=N₁+N₂ measurements and compute the mean that way. But if the number of points in each dataset is very different, the simple combined sample mean is actually a statement of bias in favor of the dataset with more measurements.

In a Bayesian context, there seems to be at least a well-defined prescription: define a model, compute the posterior probability density for the model parameters using dataset 1 using some non-informative prior, use this posterior density as the prior density in the next step, where a new posterior density is computed from dataset 2.

What does one do in the frequentist universe?

[Update 9/30] After considerable discussion, it seems clear that there is no way to do this without making some assumption about the reliability of the detectors. In other words, disinterested objectivity is a mirage.