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This question might be trivial for advanced mathematicians. Say we have two multivariate functions: $f$ over domain $A, B$ (yielding real numbers) $f(A, B) \rightarrow \mathbb{R}$, and $g$ over domain $B, C$ (also yielding real numbers) $g(B, C) \rightarrow \mathbb{R}$.

I want to define a function $h$ that is a product of $f$ and $g$. Intuitively, the domain would contain all three variables $A$, $B$, and $C$ such that $h(A, B, C) \rightarrow \mathbb{R}$. But I can't find any definition that justifies that.

Why am I bothering? In probability graphical models, probability distributions (which are in fact functions) are expressed as factors whose scope are the random variables; these factors are thus also functions over the domain containing these random variables. Many operations involve multiplying factors and in the example above, the scope of the factor product is implicitly extended to cover random variables from both factors.

Is there a mathematical explanation for this intuitive "extension" of the domain in the product of two (multivariate) functions with different domains? (Another example could be $f(A) * g(B)$ which yields a joint probability distribution of two independent random variables $A$ and $B$ -- in this case, the domain of the joint probability distribution is a union $A, B$). Or is it just trivial and can be taken for granted (by some laws of calculus)?

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So $f$ is a real-valued function on $A\times B$ and $g$ is a real-valued function on $B\times C$. Suppose you define any real-valued function $\Phi$ on $\Bbb R\times \Bbb R$. (One example might be $\Phi(u,v)=uv$.) Then you can define a real-valued function $h$ on $A\times B\times C$ by setting $$h(a,b,c) = \Phi(f(a,b),g(b,c)).$$

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  • $\begingroup$ I see your point, Ted, thanks! So it's simply a composition of two real-valued functions $f(a, b)$ and $g(b, c)$ through another function $\Phi(R, R)$; and this is left-out as implicit (and unnecessary) when defining a factor product. $\endgroup$ – John Doe Mar 5 '20 at 7:19

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