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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