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This is the Hammersley-Clifford(-Besag) theorem:. Here is the version presented in our book.
Definition: Let $(Y_1, Y_2, \ldots, Y_p) \sim g(y_1,\ldots,y_p)$, where $g^{(i)}$
denotes the marginal distribution of $Y_i$. If $g^{(i)}(y_i)> 0$
for every $i=1, \ldots, p$, implies that $$g(y_1,\ldots,y_p) > 0$$
then $g$ is said to satisfy the positivity ...
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