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I'm trying to use Gibbs sampling to get the following target distribution:

$$ p(a,b,c \lvert x, z) $$

Where $z = f(x,a,b,c)$ and the rest are independent. I know the following conditional probabilities and can sample from them:

$$ p(a\lvert x, z, b, c), \, p(b\lvert x, z, a, c), \, p(c\lvert x, z, b, a)$$

My question is how to express the target distribution I need using the ones above.

My attempt was to start with the following:

$$ p(a,b,c \lvert x, z) = \frac{p(a,b,c,x,z)}{p(x,z)} $$

But I'm not sure how to continue (or if it's the right direction)

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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 condition.

Theorem (Besag, 1974) Under the above positivity condition, the joint distribution $g$ satisfies $$ g(y_1,\ldots,y_p) \propto \prod_{j=1}^p \; {g_{\ell_j}(y_{\ell_j}|y_{\ell_1}, \ldots,y_{\ell_{j-1}},y_{\ell_{j+1}}^\prime,\ldots,y_{\ell_p}^\prime) \over g_{\ell_j}(y_{\ell_j}^\prime|y_{\ell_1},\ldots,y_{\ell_{j-1}}, y_{\ell_{j+1}}^\prime,\ldots,y_{\ell_p}^\prime)} $$ for every permutation $\ell$ on $\{1,2,\ldots,p\}$ and every $y' \in \mathfrak Y$.

The conditioning on $(x,z)$ mentioned in the question is irrelevant in that case, since every distribution appearing above is then conditional on $(x,z)$.

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    $\begingroup$ I really appreciate the answer, but I'm not sure how to use the theorem for my case yet $\endgroup$ Mar 19, 2021 at 18:39

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