Assume you have noisy measurements X1, X2, X3 of physical quantities Y1, Y2, Y3 respectively; variables 1, 2, and 3 are independent; X2 is much noisier than the others; and you want a point-estimate of Y = Y1+Y2+Y3. Then you shouldn’t use either X1+X2+X3 or X1+X3. You should use E[Y1|X1] + E[Y2|X2] + E[Y3|X3]. Regression to the mean is involved in computing each of the conditional expectations. Lots of noise (relative to the width of your prior) in X2 means that E[Y2|X2] will tend to be close to the prior E[Y2] even for extreme values of X2, but E[Y2|X2] is still a better estimate of that portion of the sum than E[Y2] is.
Assume you have noisy measurements X1, X2, X3 of physical quantities Y1, Y2, Y3 respectively; variables 1, 2, and 3 are independent; X2 is much noisier than the others; and you want a point-estimate of Y = Y1+Y2+Y3. Then you shouldn’t use either X1+X2+X3 or X1+X3. You should use E[Y1|X1] + E[Y2|X2] + E[Y3|X3]. Regression to the mean is involved in computing each of the conditional expectations. Lots of noise (relative to the width of your prior) in X2 means that E[Y2|X2] will tend to be close to the prior E[Y2] even for extreme values of X2, but E[Y2|X2] is still a better estimate of that portion of the sum than E[Y2] is.