Estimate a number within two (or more) overlapping ranges of number

[lost&found]

Hi everyone,

Let's say a simulation outputs a range of estimates with various accuracy level, ranging from 30% to 100%, with the range given at 30% being widest and the range at 100% being the narrowest. The ultimate goal of this simulation is to arrive at a single number within the estimated range.

For example, 30% estimate is 185 to 285 and 100% estimate is 219 to 249. (There are intermediate estimated ranges with accuracy ranging from 31% to 99%, but I've left out for now.)

The result of the simulation: 235. Not sure what the algorithm was for the simulation (I was leaning toward RNG), but given the information of the different sets of ranges, can we predict this number reliably and if so, how does such method work? Will having more sets of ranges give a more reliable prediction?

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I've been told that if we model each range as a linear function as followed:

f(x)= (285-185)x + 185
g(x)= (249-219)x + 219

Then finding the intersection of the two lines will yield the answer:

Set f(x)=g(x), solve for x, and sub it back into f(x), yielding f(34/70)=233.4

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233.4 is so really close to the actual 235. This method mirrors linear interpolation isn't it? So same questions as above (if this is how we predict the number, how/why does it work? and can we improve on our prediction, perhaps with more ranges?), if someone could kindly answers, thanks!

 

[fresh_42]

There is a lot of guesswork in this question. What immediately came to mind was to weight all intervals with their according level of confidence, which basically is the expectation value. But this is only one measure which doesn't distinguish possible preferences or correlations among the intervals.