Thursday, February 20, 2003, 4:25:24 AM, Jonathan Standley wrote:
JS> a challenge! cool :) but let me try to put it in less-math terms
JS> for myself and others who are not math-types
BG>Let X_i, i=1,...,n, denote a set of discrete random variables
JS> X_i is the set of all integers between i and n, initial value for i is 1?
JS> or is i any member of the set X?
JS> or does i function only as a lower bound to set X?
Jonathan, I'll try to restate it. Note that I'm not a mathematician
either; so this could be off. I'd appreciate further clarification
from other list members.
X_i is a set of variables X_1, X_2, X_3 ... X_n -- you can think of
these as a set of statements. They're "random" in that there's no
discernable pattern. It's not a series or anything, think of a set
of possibilities.
P(X_i|X_j) is the probability of X_i GIVEN X_j
[ i.e. say that X_i is "You test positive for cancer"
and X_j is "You have cancer"
and the test is 90% accurate
then P(X_i|X_j) is 0.9
i.e. IF you have cancer THEN the chance of testing positive is
0.9
conversely,
say that X_i is "You have cancer"
and X_j is "You test positive for cancer"
then P(X_i|X_j) DEPENDS very strongly on the overall
probability of having cancer, because some % of the
people who don't have cancer will test positive. ]
BG> Let's say we have a set of N << n^2 conditional probability relationships of
BG> the form
BG>
BG> P(X_j|X_i)
BG>
BG> where i, j are drawn from {1,...,n}.
i.e. we know a number far less (<<) than the total number of
the conditional probabilities i.e. only a few of them.
BG> Let's say we also have a set of M <= n probabilities
BG>
BG> P(X_i)
BG>
i.e. we also know a number of, maybe all, the basic probabilities.
BG> The problem is:
BG>
BG> * Infer the rest of the P(X_i|X_j) and P(X_i): the ones that aren't given
BG>
BG> * Specifically, infer cases where P(X_i|X_j) differs significantly from
BG> P(X_i)
i.e. how can we infer the rest.
This is where my confusion sets in...
- Is the sum of all conditional probabilities P(X_i|X_j) for all i
given a specific j equal to 1?
- Is the sum of all probabilities P(X_i) for all i equal to 1?
If not, what sort of relationships hold? I have the feeling I'm
missing some basic assumption here.
Thanks
--
Cliff
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