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    Re: That darned old cocked hat
    From: George Huxtable
    Date: 2010 Dec 10, 19:04 -0000

    John Karl writes-
    "Next we acquire three LOPs.  They do not intersect at a common point.  We 
    don’t know where the true fix is."
    Let's say we do. Take an anchored vessel at a precisely known location, say 
    by GPS.
    Measuring the altitude of a star does not alter that location. The 
    probability of the observation being towards or away depends only on the 
    random scatter of the observation itself, being precisely 50:50. Instead of 
    actually measuring it, me might just as well simulate the observation by 
    tossing a coin. And we could simulate the observations of the other two 
    stars in exactly the same way.
    In which case the probability of the three observations being in the same 
    direction, TTT or AAA, is exactly the same as the probability of getting 
    three coins the same (all-heads or all-tails), which is 1 in 4.
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    ----- Original Message ----- 
    From: "John Karl" 
    Sent: Friday, December 10, 2010 6:21 PM
    Subject: [NavList] Re: That darned old cocked hat
    George Huxtable wrote:
    Three separate observations are made, one of each star. By our prior 
    condition, the  probability of each being toward or away is 50:50. There is 
    no correlation between them. They could have been made by three separate 
    observers, each putting his observation into a sealed envelope, to be 
    opened together after the event. Then the probability of any combination, 
    of the 8 possible, is one in 8. How can it be different?
     Now I think we’ll back up a little.  We’ll take the same approach that I 
    used in my discussion of the “long-run” running fix – the Bayes (and E.T. 
    Jaynes) logic of inference, used in the face of incomplete knowledge:  Now 
    say we’ve made a lot of measurements with the device that locates our LOPs 
    and have found that random errors occur with a normal distribution of 
    standard deviation, sigma.
    Next we acquire three LOPs.  They do not intersect at a common point.  We 
    don’t know where the true fix is.  We do know that each LOP represents a 
    sample from a normal distribution, each with the same sigma.  Lacking any 
    other information, we are forced to assume that each LOP locates the center 
    of a linear normal distribution in the x-y plane (see the first attachment 
    below).  This is a very poor assumption, but it’s the best we can do.  Are 
    we to assume the LOPs are located somewhere less likely in their 
    distributions?  (Here we see a major difference with another logic that 
    might start by considering the three probability distributions of LOPs 
    intersecting at precisely the same point, the true fix location.  But this 
    contradicts the Bayesian logic because, in fact, we don’t know the true fix 
    location.  This might be Georges logic.)
    Using our logic, the three distributions are uncorrelated (in fact they are 
    in disagreement).  And the probability of point in the x-y plane being a 
    fix from all three normal distributions is the product of the three 
    distributions centered on the three LOPs.  This gives a plot of the 
    probability per unit area P(x,y) that the fix is at a given x-y point, as 
    shown in the second attachment.  We see from this figure that, because of 
    the shape of the normal distribution, there is one single point of MP 
    (maximum probability).  The probability that the fix is inside any given 
    area is the integral of P(x,y) over that area.  (I’ve numerically 
    calculated some of these probabilities, such as the 84% and 94% that I’ve 
    posted earlier.)  We see that areas of different shapes, different 
    locations, and different sizes can have very different probabilities that 
    the fix is inside them.  This explains George’s question above, how 
    different probabilities arise.  George seems to be counting types of 
    regions, rather then calculating the different probabilities that the fix 
    is inside them.
    Since P(x,y) is a probability density – a probability per unit area -- we 
    don’t talk about the probability that the fix is at a specific point.  That 
    would be zero since a point encloses zero area.  Rather we do calculate and 
    plot the value of P(x,y) -- the density.  Those are the values on the 
    contours I’ve posted. For example a value of P = 0.013 means that the 
    probability that the fix is inside a unit area surrounding P(x,y) is 1.3%. 
    (The 100x100 plot, representing 10x10 nm had 100x100 data points.  So the 
    unit area is 0.1x0.1 nm.)  The values of P(x,y) are small because the total 
    area is so large, 10,000 data points.
    It is true that as the cocked hat becomes smaller the MP inside it 
    increases, but it never can exceed one,  So the probability that the fix is 
    inside the hat decreases to zero as the hat’s area decreases to zero.  Thus 
    the navigator’s “desired” result of three LOPs intersecting at one point 
    means, that in all probability, the true fix is not there.  But of course, 
    the probability that the fix is somewhere in the x-y plane is always 100%. 
    It follows that the probability that the fix is outside the hat is 100% 
    minus the probability the fix is inside.  Since the probability that the 
    fix is inside goes to zero with the hat size going to zero, the probability 
    that the fix is outside goes to 100%.  There can be no constant 75% of the 
    fix being outside.
    Finally from considering the form of P(x,y), as I’ve point out, the 
    location of the MP minimizes the sum of the squared distances to the three 
    sides of the cocked hat.  Now Herbert Prince (and Frank) tells us that this 
    point is called the symmedian of the cocked hat, as explained by Villarceau 
    in 1877.   Have we navigators been lost all this time??
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