# Proof of multivariate distribution using exponential families and Hammersley Clifford Theorem

I'm reading the following seminal paper by Besag

I'm unsure how they prove on page 10 equations 4.4 and 4.5 namely that

$$A_i(.) \equiv \alpha_i+\sum_{i=1}^n \beta_{i,j} B_j(x_j)$$ and that $$G_{i,j}(x_i,x_j) = \beta_{i,j}H_i(x_i)H_j(x_j)$$

where $$H_i(x_i) = B_i(x_i) - B_i(0)$$.

The proof starts on 4.3 (page 12). I follow the first part of the proof and have worked out that $$x_1x_2G(x_1,x_2) = (A_1(0,x_2,0,\ldots,0) - A_1(0))(B_1(x_1) - B_1(0))$$ and that

$$x_1x_2G(x_1,x_2) = (A_2(x_1,0,\ldots,0) - A_2(0))(B_2(x_2) - B_2(0))$$

So i guess we need to prove that $$(A_1(0,x_2,0,\ldots,0) - A_1(0)) \propto (B_2(x_2) - B_2(0))$$ and $$(A_2(x_1,0,\ldots,0) - A_2(0)) \propto (B_1(x_1) - B_1(0))$$ which i'm struggling to do.

Things i've tried are the Direct substitution we must have $$(A_1(0,x_2,0,\ldots,0) - A_1(0)) = \frac{(A_2(x_1,0,\ldots,0) - A_2(0))}{(B_1(x_1) - B_1(0))} (B_2(x_2) - B_2(0))$$ but wasn't sure how to prove that $$\frac{(A_2(x_1,0,\ldots,0) - A_2(0))}{(B_1(x_1) - B_1(0))}$$ was a constant. I tried directly plugging in values the conditional $$\mathrm{ln}(p_i(x_i|x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n) = A_i(.)B_i(x_i)+C_i(x_i)+D_i(.)$$ for A_i and B_i (i=1,2) and when i did things didn't cancel out nicely.

I would really appreciate any help, Thanks!

The answer is down to the fact that we are looking over all combinations of $$x_1,x_2$$
We know that $$A_1(x_2)-A_1(0) = \frac{A_2(x_1)-A_2(0)}{B_1(x_1)-B_1(0)}{B_2(x_2)-B_2(0)}$$ must hold. Now keep $$x_2$$ fixed and consider varying $$x_1$$ then $$A_1(x_2)-A_1(0)$$ and $$B_2(x_2)-B_2(0)$$ are constant and so the equation only holds if $$\frac{A_2(x_1)-A_2(0)}{B_1(x_1)-B_1(0)}$$ is the same $$\forall x_1$$. We can repeat for $$x_2$$.