[nritchie]

A handful of the comments are skeptical of the utility of this method. I can tell you as a physical scientist, it is common to make the same measurement with a number of measuring devices of differing precision. (e.g. developing a consensus standard using a round-robin.) The technique Cook suggests can be a reasonable way to combine the results to produce the optimal measured value.

[sfpotter]

I'm not a physical scientist, but I spend a lot of time assessing the performance of numerical algorithms, which is maybe not totally dissimilar to measuring a physical process with a device. I've gotten good results applying Simple and Stupid statistical methods. I haven't tried the method described in this article, but I'm definitely on the lookout for an application of it now.

[shoo]

I wonder if this minimum variance approach of averaging the measurements agrees with the estimate of the expected value we'd get from a Bayesian approach, at least in a simple scenario, say a uniform prior over the thing we're measuring and assume that our two measuring devices have unbiased errors described by normal distributions.

[jampekka]

At least in the mathematically simpler scenario of a gaussian prior and gaussian observations, the posterior mean is computed by weighing by the the inverses of variances (aka precisions) just like this.

https://en.wikipedia.org/wiki/Conjugate_prior

[ogogmad]

To add, for anyone who's followed the link - that's entries numbers 1 and 2, or "Normal with known variance σ²" and Normal with known precision τ", under "When likelihood function is a continuous distribution".

Also, note that the "precision" τ is defined as 1/σ².

[ninja3925]

This seems to be incorrect. The correct way to combine measurements with various degree of precision is to use the inverse variance weighting law

[kgwgk]

Unless I’m missing something that’s exactly what is proposed:

t_i Var [X_i]] = t_j Var [X_j]

[geon]

Like a Kalman filter?