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I hesitate to jump in but ...

Trevor wrote:
> Geoffrey wrote:
>> My "proof", as you were kind enough to call it, has nothing to say
>> regarding the size of the 'hat.

> I did not suggest that it had. But your proof leads to the conclusion
> that there is a 0.25 probability that the true position lies within the
> cocked hat _regardless_ of the size of that 'hat. That is
> counter-intuitive. Either there is a reason why what seems intuitive is
> wrong or else there is some error in your proof that I cannot see. I was
> wondering whether you could dismiss the latter possibility by pointing
> to the reason for disregarding my argument that the number cannot always
> be 0.25.

Once you have a specific 'hat and a specific true position there
is no matter of probability. The true position either lies in the
'hat or it doesn't. It makes no sense to talk of the probability
of the true position being in the 'hat; it's either inside or
outside.

What we can talk about is the pattern or distributions of 'hats
that get generated near a true position. As shown before, 25% of
the generated 'hats will enclose the true position and 75% will
not. There are two assumptions that have to be met for this to be
true. First, the distribution of the error must be symmetric
around 0, meaning there is no inherent bias in the errors in
either direction (toward or away).  Second, the errors must be
independent in each of the observations.

If we add the addition assumption that smaller errors are more
likely than larger errors, and we generate a large number of hats
then we'll notice several things. First, 25% of the hats will
contain the true position. Second, the hats vary quite a bit in
size but larger hats enclose the true position more often than
smaller ones. And third, there are more smaller hats than larger
ones.

More smaller hats which tend to be near but not enclosing the
true position. Larger ones are rarer but tend to surround the
true position. That should help the intuition.

But why should this be so? Try this experiment. Put down a dot
for a true position. Pick 3 values at random, these are the
magnitudes of the error. For each magnitude draw 2 parallel lines
at that magnitude distance from the point, one on each side. Use
different orientation for each pair. There will now be 8
triangles. Six are smaller and do not enclose the point. Two are
larger enclosing not only the point but 3 of the other triangles
as well. Each of these 'hats is as likely as any of the others
given that set of error magnitudes. Hence, more smaller hats than
larger; more larger hats enclose the true position than do
smaller hats.

>> Similarly, if you only have one 'hat, then placing the MPP in the centre of
>> it is the best you can do.

> That is indeed the MPP, if the 'hat is equilateral.

No. Given 3 LOPs and a 'hat, the MPP is equidistant from each side of
the hat; in the center of the inscribed circle. The mutual orientation
of the LOPs won't alter that. What is required is that the error
distribution be the same for each LOP; that the distribution be
symmetric about 0; and that smaller errors be more likely than larger
ones. Lots of "ifs" but not unreasonable.

A good analogy for finding the MPP is to take rubber bands and tie
one end of each together. Attach the other end of each to one of the
LOPs in such a way that it is free to move along the length of the
LOP. The MPP will be where the knotted end winds up. In the special
case of just 3 LOPs, equilibrium will be where the tension on all
three rubber bands is the same, each stretched the same amount. The
center of the inscribed circle.

There remains one question. Is the contour around the MPP elliptical?
For that to be true, we have to be more specific about the distribution.
If it's Gaussian then the contour is elliptical. Other distributions
might also give rise to elliptical contours as well. There are certainly
some distributions for which the contour is not ellipse. For example,
the triangular shaped distribution mentioned earlier in the thread
will give rise to a polygonal contour (at least it will at the 100%
confidence level).

But I do think we can get a feel for some more general qualities of
the contour. Our rubber band analogy is again useful. Intuitively, there
will only be one point where the knot comes to rest. I.e there is only
one "highest" peak of the contour, and no other smaller "peaks". Less
intuitive is the next assertion: that the shape of the contour is
convex. But if it weren't we could add more LOPs that cross the concavity
(indentation) of the contour and by adding enough of these we'd get a
contour with two peaks. A contradiction.

This is not a particularly satisfying argument but I suspect only
a rigorous mathematical treatment would be. And I've spent too much time
on this as is.

   

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