A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Brad Morris
Date: 2013 Mar 18, 08:26 -0400
I agree with the premise that there will be a bias when drawing the third LOP towards the intersection of the first two.
But which LOP comes last is arbitrary. Repeat the exercise 3 times, with a different LOP last.
All three biases are now towards the vertices of the opposing two LOPs.
Now where is the Most Probable Position?
At 03:04 18/03/2013, you wrote:
Now that the Monty Hall Problem is dispensed with, we can return to the original point of your post, Bayesian Decision Theory.Â I fondly remember that class.Â You wrote:
This statement of P(x,y) assumes that all three LOPs are independent and the that probability function for each LOP is symmetrical about each LOP. But consider: Suppose you make your first two sightings and you plot the resulting LOPs out on a chart. You will get the two LOPs crossing at some point, which will become your most probable position. Now, suppose you then take your third sighting and on plotting that LOP on the chart, you see that the most probable position from the first two sightings is to the left of the this third LOP. Is it still correct that the probability of the true position being to the left of the third LOP is exactly the same of it being to the right of that third LOP, as assumed in John Karl's probability function? Or should the probability distribution around the third LOP be informed by the previous data? This is the basis of what is called Bayesian statistics, which is the fasted growing area of statistical theory today
The point is that probabilities can change depending on the history of the situation. That is the essence of Bayesian statistics.
So in the case of the cocked hat, for the first LOP you draw on the chart, the probability that the true position is on one side of the LOP is clearly exactly the same as it is for the other side. We have no information to tell us otherwise. You can construct a Gaussian probability distribution centered on that LOP indicating how far away from that LOP the true position might be.
Suppose the second LOP is orthogonal to the first LOP. Once again you can construct a Gaussian probability distribution centered on that LOP, indicating how far away from that LOP the true position might be. The likelyhood that the true position is on one side of this second LOP is exactly the same as for the other side. At the point where the two LOPs cross, there will be a point where the added probability distributions maximise and form a "Most Probable position" for our true position.
So far, so good.
But, wherever you draw the third LOP, you can no longer assume that the likelyhood of the true position being on one side of the third LOP is exactly the same as on the other side. You cannot create a probability distribution that is centered on that LOP. Why? Because we now have some history indicating where the true position might be, and that position will be off to one side or other of the third LOP. The probability distribution around the third LOP must change to take that into account and that distribution will no longer be centered on the third LOP.
So, this Bayesian approach to the cocked hat problem will give a different probability distribution to the "frequentist" approach used by John Karl and others.
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