Help with problem
Bhiksha Raj
First, let me apologize if this post is inappropriate for this
newsgroup. I dont konw where else to post it.
I am looking for refs/pointers/any kind of help with the following
problem.
We have N known density functions of the form
P1(X), P2(X),P3(X)....PN(X)
Note that not all densities need have the same parametric form.
No two densities are identical.
For any vector X, we can compute a sequence of density values
Y1 = P1(X), Y2 = P2(X) and so on.
The densities are, of course, many to 1. So knowing Y1 does not
tell us the value of X. However, it does localize X to lie in the
region/domain
X1(Y1), where all elements in X1(Y1) return the density value Y1.
Similaraly regions X2(Y2), X3(Y3) etc can be defined.
If we know the Yi values from all N densities, this localizes X to lie
in
the region X1(Y1). intersect. X2(Y2). .... . intersect XN(YN).
What I want to be able to say is that as the number of densities
for which Yi is known increases, i.e. as N increases, this increasingly
localizes X. More specifically, I want to be able to make some statement
about what classes of densities/ what conditions on the densities ensure
that as N->infinity, the intersection of the various Xi(Yi)s tends to X
in probability (if not absolutely). Consequently, that for these classes
of densities, as if we know all Yis, as N tends to infinity we (almost
certainly)
know X.
Im sure there is some theorem/paper/standard textbook that analyses
this, but I havent been able to locate it after much searching.
I would be greatly obliged for any pointers/help/guidance.
Thank you very much for your patience,
-Bhiksha
Robert Israel
>We have N known density functions of the form
>P1(X), P2(X),P3(X)....PN(X)
>Note that not all densities need have the same parametric form.
>No two densities are identical.
>For any vector X, we can compute a sequence of density values
>Y1 = P1(X), Y2 = P2(X) and so on.
>The densities are, of course, many to 1. So knowing Y1 does not
>tell us the value of X. However, it does localize X to lie in the
>region/domain
>X1(Y1), where all elements in X1(Y1) return the density value Y1.
>Similaraly regions X2(Y2), X3(Y3) etc can be defined.
>If we know the Yi values from all N densities, this localizes X to lie
>in
>the region X1(Y1). intersect. X2(Y2). .... . intersect XN(YN).
>What I want to be able to say is that as the number of densities
>for which Yi is known increases, i.e. as N increases, this increasingly
>localizes X. More specifically, I want to be able to make some statement
>about what classes of densities/ what conditions on the densities ensure
>that as N->infinity, the intersection of the various Xi(Yi)s tends to X
>in probability (if not absolutely). Consequently, that for these classes
>of densities, as if we know all Yis, as N tends to infinity we (almost
>certainly)
>know X.
I don't know what you mean by "in probability" here, because you haven't
said anything about X being chosen from some probability distribution.
If there are two x values x1 and x2 such that P_i(x1) = P_i(x2)
for all i, there is no localization. Or, if you intended X to be chosen
from some continuous distribution, so that X = x1 or X = x2 will have
probability 0, how about if all the P_i have the same symmetry, e.g.
P_i(-x) = P_i(x)?
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
Bhiksha Raj
The overall density of X is a weighted combination of the individual
densities P1(X), P2(X) etc.
The weights follow the condition that
0<= wi <= 1 (wi is the weight of Pi(X))
and
Sum( wi) = 1;
One may place any additional constraints on the wis as needed.
The precise constraints I would need will probably be a part of
the solution to the problem. Im not sure what form that will take.
Thanks,
-Bhiksha