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I've been away, and just finished wading through all the good
discussions.  I wanted to respond to this post of Rodney
Myrvaagnes, even though it might be superseded by what came
later.

> a counterexample that covers practically all plausible cases:
>
> Taking George's simplifying assumption of a rectangular probability 
distribution, let's be > sure we know what that means. Any
> probability distribution must sum to 1 over all possible cases. Or, the integral from
> minus infinity to plus infinity of [P(x) dx] =1. Hence,
> the probability of the obsevation falling in any intervalof x within the non=zero range
> is exactly the fraction of that range the interval
> takes up.
>
> With three lines of position, all independently measured by the same method, we must
> assign the observations the same probability
> distribution.
>
> One possibility is that they are just wide enough to make all three 
observations possible. > That results in the traditronal center point of
> the triangle being the only non-zero probability.

NO, NO, NO!  You can't assign the probabilities after the measurement!
If the (assumed rectangular) distribution were of this size, then a
number of different triangles could result, and this is only one (the
largest possible).  Of those triangles, 25% would enclose the true
position.  If we knew beforehand how big the maximum error was, we
could know whether a particular triangle should be considered "big"
(more likely to enclose the true position) or "small" (less likely to
enclose the true position), but we've mostly been assuming the error
distribution to be unknown (but balanced left/right).

If I misunderstood, I apologize...

   

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