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I have a following problem. I am sorry if the problem is trivial...

Let us have a conditional distribution 
p(X) = \product_{i} P(L_i | R_i),
where L_i are disjoint subsets of variables from X=(X_0,...,X_{n-1}),
each X_j is in some of L_i subsets, and R_i is a subset from L_0 U ... U
L_{i-1}.

Does there for each such distribution exist a joint probability
distribution where the sets L_i has to be of cardinality 1 only? If not,
why? Is there a simple counter example?

An example of what I am saying is that for instance we have
p(ABC)=p(AB|C). Isn't p(ABC)=p(C)p(A|C)p(B|AC) equivallent to p(AB|C)?

thank you for your soon asnwer

Martin


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