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Subject: Re: Still on LOP's
From: Michael Wescott (wescott_mike@XXX.XXX)
Date: Tue May 07 2002 - 11:48:29 EDT
WSMurdoch@XXX.XXX said:
> Mike Wescott wrote:
>> Usual assumptions apply: no "systemic errors", equally probable that
>> error is + or -. If both are plus, they're both on one side of you.
>> If both are - then they're both on the other side of you. If #1 is +
>> and #2 is - then one is one each side. Likewise, if #1 is - and #2 is
>> +. Four equiprobable possibilities and 2 of the four have you between
>> the POPs: 50% and 1 in four (25%) for each of the other two
>> possiblities.
> This is where I 'fell off the train'. If we stand to the side and
> watch the tight rope walker, we see along the rope from left to
> right POP#1, tight rope walker, and POP#2.
Why in that order?
You've just placed POP#1 and POP#2 in relation to each other, not to
mention the tight rope walker. And our tight rope walker's postion
with respect to POP#2 is no longer independent of his relationship to
POP#1.
> It is just as likely that the tight rope walker is to the left or
> the right of POP#1, and it is also equally likely that he is to the
> left or the right of POP#2.
No. Now that you've ordered the placement of the POPs with respect to
each other these probabilities are no longer independent of each
other. Hence the contradiction in the analysis below. You can still do
this mathematically but it's more complex to take all the conditional
probabilities into account.
> If he is to the left of both, he is to the left of POP#1. If he is
> to the right of both, he is to the right of POP#2. If he is to the
> right of POP#1 and to the left of POP#2, he is between the two POPs.
> If he is to the left of POP#1 and to the right of POP#2, he is not
> on the rope. I understand + +, - -, and + -. I do not understand -
> +. Or, am I missing much more?
What you have done here is ordered the relationship of the POPs and
destroyed the independence assumption. POP#2 can't have the full range
of possible values since it can't be greater than POP#1. If you let
POP#2 be to the left of POP#1 the fourth possibility is no longer "not
on the rope" but between the two POPs.
Think of it the other way around. Our guy, perch high above, takes a
sight. The sight has a small amount of random error in it. The POP
derived (POP#1) falls a short distance from him and is either in front
(+) of him to our left, or behind (-) to our right. By our assumption
these have equal probability. The same happens with POP#2. Now all
four possibilities appear: ++, +-, -+, and --; only two of which
straddle our guy. We don't say anything about whether POP#2 is to
the right or left of POP#1.
We are allowed to order the POPs by time because the random error is
assumed NOT to be a function of time. When they are ordered by
distance the we've put additional constraints on the second that
weren't on the first. The second is no longer independent of the
first. We've changed the range of values that the second can have.
That's always, for me, been the hard part about probability and
statistics: making sure that the analysis hasn't introduced additional
constraints or assumptions.
--
Mike Wescott
Wescott_Mike@XXX.XXX
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