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We have a weather station somewhere in the country, with the aid of satelitte information and it gives us the probability of rain every day.

Suppose it is an average value p, for the season.

At some other place there is an Indian chief who used to do the job for us for hundreds of years until such time as he was overtaken by modern technology and suppose he assigns the probability q to rain.

We know that the modern weather station is more reliable so we are likely to discard the value q and we say that p is the correct probability.

But is that altogether true ?

Could it be that the correct probability in such situations is some function f(p,q) ? And if so how do we express it ?

Here is a short treatment I found:

http://www.mathpages.com/home/kmath267.htm

The author of mathpages investigates the issue to some depth.

But what happens if we have correlations also present ?

Any more references ?

mathpages suggests the answer:

f(p,q) = p*q / (p*q+(1-p)*(1-q))

on the assumption that the separate predictions are uncorrelated.

But some correlation is likely to exist in such situations, which in the case of modern weathermen v. Indian chief might be the identicality of certain methods used by both.