Re: probabilistic selector functions
- From: "justinkdavis@xxxxxxxxx" <justinkdavis@xxxxxxxxx>
- Date: Fri, 31 Aug 2007 14:04:04 -0000
On Aug 29, 8:18 pm, quasi <qu...@xxxxxxxx> wrote:
Let X be a nonempty set and let p:X->[0,1] be an arbitrary function.
The function p induces a (random) subset Y of X as follows ...
For each x in X, p(x) is the probability that x is in Y.
The difficulty here is that p _may_ be used to construct such a subset
Y, but the creation of Y is an entirely different process with
apparently no impact on the structure of p itself. The "selection" bit
isn't a necessary part of the function, since the properties of p may
be fully considered before the selection begins.
For instance, you say that p is an arbitrary function. If X is
countable, then it may be a good exercise to consider p as a sequence
of random variables. As it stands, however, we can't say much about p
other than p:X->[0,1], though it is a simultaneous presentation of a
number of sample spaces.
You might have more success by beginning with p:2^X->[0,1], which
could be built to give you the probability that the randomly chosen
subset is Y. Ultimately, after all your work, this is probably what
you'd end up with anyway.
Thus, if X = {x_1, x_2, x_3, ... }, the function p can be regarded as
a sequence p_1, p_2, p_3, ... where p_n is the probability that x_n is
in Y. The set Y can then be built recursively, using p to decide
membership, as follows:
Y_0 = the empty set.
For n>0, define Y_n by
Y_n = Y_(n-1) union {x_n} with probability p_n, otherwise Y_(n-1)
Then let Y = union of the the sets Y_n, for n = 0, 1, 2, 3, ...
This construction, rather than the function above, is the really
interesting part. You could examine the full process by generating a
sequence of random variables {y_n} and checking the probability that
y_n > p(x_n). If the probabilities were random variables themselves,
this can get really messy. If everything is IID and nicely distributed
though, you might get some sensible properties from it.
Ultimately, it really depends on what you want to get out of this, but
I think it all will boil down to finding p:2^X -> [0,1], the
probability measure on subsets of X. Fernando above gives one example
of the process.
.
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