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## Combination of several measurements

Let us imagine making a second set of measurements of the physical quantity, which we assume unchanged from the previous set of measurements. How will our knowledge of change after this new information? Let us call and the new average and standard deviation of the average ( may be different from of the sample of measurements), respectively. Applying Bayes' theorem a second time we now have to use as initial distribution the final probability of the previous inference:

 (5.17)

The integral is not as simple as the previous one, but still feasible analytically. The final result is

 (5.18)

where
 (5.19) (5.20)

One recognizes the famous formula of the weighted average with the inverse of the variances, usually obtained from maximum likelihood. There are some comments to be made.
• Bayes' theorem updates the knowledge about in an automatic and natural way.
• If (and is not too far'' from ) the final result is only determined by the second sample of measurements. This suggests that an alternative vague a priori distribution can be, instead of uniform, a Gaussian with a large enough variance and a reasonable mean.
• The combination of the samples requires a subjective judgement that the two samples are really coming from the same true value . We will not discuss this point in these notes5.3, but a hint on how to proceed is to: take the inference on the difference of two measurements, , as explained at the end of Section  and judge yourself whether is consistent with the probability density function of .

Next: Measurements close to the Up: Normally distributed observables Previous: Final distribution, prevision and   Contents
Giulio D'Agostini 2003-05-15