Hi Ben,
Thanks for the brain teaser! As a sometimes believer in Occam's Razor, I
think it makes sense to assume that Xi and Xj are indepenent, unless we know
otherwise. This simplifies things, and is the "rational" thing to do (for
some definition of rational ;->). So why not construct a bayes net modeling
the distributions, with causal links only where you _know_ two variables are
dependent? For reasoning about "orphan variables" (e.g., you know nothing
at all about Xi), just assume the average of all other distributions. If
you have P(Xi|Xj), and want P(Xj|Xi), fudge something together with Bayes'
rule. This isn't a complete solution, but its how I would start... Is this
one of the things you've tried?
Cheers,
Moshe
>
> Hi,
>
> This one is for the more mathematically/algorithmically inclined people
> on the list.
>
> I'm going to present a mathematical problem that's come up in the
> Novamente development process. We have two different solutions for it,
> each with strengths and weaknesses. I'm curious if, perhaps, someone
> on this list will suggest an alternate approach. (If not, at least the
> problem itself may stimulate somebody's mind ;)
>
> I'll describe the problem here in a very simple form. Actually, inside
> Novamente, this simple problem exists in many "transformed" variants
> and takes many different guises. It is posed here in terms of simple
> conditional probabilities, but it also presents itself in other forms,
> involving n-ary relationships, complex procedures and predicates, etc.
> etc.
>
> Without further ado....
>
> Let X_i, i=1,...,n, denote a set of discrete random variables (think of
> them as concepts or percepts)
>
> Let's say we have a set of N << n^2 conditional probability
> relationships of the form
>
> P(X_j|X_i)
>
> where i, j are drawn from {1,...,n}.
>
> Let's say we also have a set of M <= n probabilities
>
> P(X_i)
>
> The problem is:
>
> * Infer the rest of the P(X_i|X_j) and P(X_i): the ones that aren't
> given
>
> * Specifically, infer cases where P(X_i|X_j) differs significantly from
> P(X_i)
>
> Clearly this is a massively "underdetermined" problem: the given data
> will generally not be enough to uniquely determine the results. This
> is what makes it interesting!
>
> As I said, we have two solutions for this, one implemented the other
> just designed; so we know the problem is approximately and
> heuristically solvable in a plausible computational timeframe. But I
> wonder if there aren't radically different solutions from the ones
> we've come up with...
>
> Any thoughts?
>
> -- Ben G
>
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