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I am unable to refute the various arguments for the 25% (or less)
probability of the cocked hat containing the MPP. However, this note
from Chuck resolves a great deal of my puzzlement over how to improve on
the practice of placing my assumed position at sea within a reasonably
sized cocked hat from a reliable set of sites. It makes sense to me that
while the probability of being within the hat may only be 25%, that 25%
probability is better than I can do by placing the MPP in any of the
regions outside the hat.

The only additional observation I would make is that I don't believe
that for a typical set of star sights taken at sea the error around each
of the sites will be random and independent error.  The horizon is
bright on one side, dim on the other, cloud cover intervenes
intermittently, and posture and stance vary for the different sites.  I
frequently feel I know which sites I nailed, and which I bobbled, and
those judgments factor into my determination of where I place the MPP.


-----Original Message-----
From: Navigation Mailing List
[mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of Chuck Taylor
Sent: Monday, April 22, 2002 3:46 AM
To: NAVIGATION-L@LISTSERV.WEBKAHUNA.COM
Subject: Re: Still on LOP's

Trevor wrote:

> But, unless I am mistaking something (which is by no means
> impossible!), the probability that the cocked hat lies over the true
> position is numerically equal to the probability that that true
position
> lies within the cocked hat.

From the point of view of probability theory, it is precisely the
probability
that the cocked hat lies over the true position that we are talking
about. All
the uncertainty lies in the measurements and plots of the bearings and
hence in
the resulting cocked hat.  There is no uncertainty in the true position,
only in
our knowledge of the true position, as represented by the cocked hat.

Trevor then went on to mention "the best estimate of the MPP".  Even if
we grant
that the probability is 0.25 that the cocked hat lies over the true
position, I
would argue that the Most Probable Position (MPP) is still the center of
the
cocked hat.  Notice that the various possible cocked hats in Geoffrey's
diagram
lie symmetrically about the true position. Even if you believe (with
probability
0.75) that your particular cocked hat does not contain the true
position, you
have no idea in which direction the true position lies relative to your
cocked
hat, so you can do no better than select the center of your cocked hat
as the
best estimate of your position given available information.  Can anyone
refute
that argument?

Chuck Taylor
Everett, WA, USA

   

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