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Imagine that I have some observable value predicted with a theory for some process to be: $1+a x + b x^2$ and observed value is 1.3 with an error 0.2; a and b are some numerical constants. I also have another observable influenced by the same x to be $1 + cx+ dx^2$ and the measured value 1.2 with an error 0.6, c and d are some numerical constants.

My question is: How to calculate value of x that explains both measurements the best? I guess this is called benchmark point, but how to get it and how to quantify it?

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I take it a, b, c, and d are all known, is that correct? Otherwise we'll be looking at an under-determined system. If you had a set of (x,y) data for both cases, it would look like something a linear least squares method would work for, but your problem appears to be more simple. –  AlanSE Jun 1 '12 at 0:25
    
yes, a,b,c,d are all known. –  Newman Jun 1 '12 at 0:30
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Don't be so abstract. The precise answer will depend on things like what the range of x really is, how close to Gaussian can we assume/approximate, etc. Generically, I would say: define the likelihood function and apply Bayes' theorem. –  genneth Jun 1 '12 at 0:43
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