Consistency of the marginals

I have random variables $U,V_1,V_2,V_3$, and I want to specify the following distributions (the "sum notation" means a mixture): $$U\sim\mathrm{Gamma}(2,3)$$ $$V_1 \mid U=u,V_2=v_2,V_3=v_3 \sim \frac{1}{2} \mathrm{Poisson}(1+u \cdot v_2) + \frac{1}{2} \mathrm{Poisson}(1+u \cdot v_3)$$ $$V_2 \mid U=u,V_1=v_1,V_3=v_3 \sim \frac{1}{2} \mathrm{Poisson}(1+u \cdot v_1) + \frac{1}{2} \mathrm{Poisson}(1+u \cdot v_3)$$ $$V_3 \mid U=u,V_1=v_1,V_2=v_2 \sim \frac{1}{2} \mathrm{Poisson}(1+u \cdot v_1) + \frac{1}{2} \mathrm{Poisson}(1+u \cdot v_2)$$

Two questions:

1. How dow I know (and prove) that the joint distribution of $U,V_1,V_2,V_3$ is consistently specified by these marginals?

2. How can I compute the distribution of $U\mid V_1=v_1,V_2=v_2,V_3=v_3$? If there is no way to compute it, can I draw random samples from this distribution by any method?

Thank you very much.

• Hello, and welcome ! Do you know whether it works in the case when there are only three r.v. $U$, $V_1$, $V_2$ ? Jun 8 '13 at 8:58

Here is a way to investigate the simplified case when there are only three random variables $U$, $V_1$ and $V_2$. I have not tried to generalize it to your case with four random variables but I think this exercise is the way to go.

We are asking for the existence of three random variables $U$, $V_1$ and $V_2$ such that :

• $U$ has a given distribution

• the conditional distributions $(V_1 \mid U=u, V_2=v_2)$ are given and defined on $\mathbb{N}$

• the conditional distributions $(V_2 \mid U=u, V_1=v_1)$ are given and defined on $\mathbb{N}$

For each $u$, lemma below (whose proof is currently let as an exercise) gives a criterion for the existence of the conditional law $(V_1, V_2 \mid U=u)$ and the expression of this law in case when it exists. If these conditional laws exist for all $u$ then this obviously gives the joint distribution of $(U,V_1,V_2)$.

Lemma. Let $E,F \subset \mathbb{N}$. For each $j \in F$ let $p(\cdot \mid j)$ be a probability on $E$ and for each $i \in E$ let $p'(\cdot \mid i)$ be a probability on $F$. We assume that $p(i \mid j)>0$ and $p'(j \mid i)>0$ for every $i \in E$ and $j \in F$. Let $a\in E$ and $b \in F$ be two arbitrarely fixed points. If there exists a random pair $(X,Y)$ such that $p(i \mid j) = P(X=i \mid Y=j)$ and $p'(j \mid i) = P(Y=j \mid X=i)$ for every $(i,j) \in E\times F$, then its law is uniquely defined by $$P(X=i, Y=j) = P(X=a, Y=b)\frac{P(Y=j| X=i)P(X=i| Y=b)}{P(Y=b| X=i)P(X=a| Y=b)}.$$ Thus, the existence of (the law of) $(X,Y)$ holds if and only if the above expression defines a probability on $E \times F$.

• ... and similarly you can drop $U$ in your problem. Thus the question is about the existence of the joint distribution with specified "full conditional distributions" in the language of Gibbs sampling theory. Jun 8 '13 at 9:34

1. The marginal and conditionals that you have specified define their joint distribution. This can be justified by using the Rosenblatt transformation

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177729394

• I think you're talking about the case when we specify the law $X_1$, then the conditional law $(X_2 \mid X_1)$, then the conditional law $(X_3 \mid X_2, X_1)$, and so on; here that does not solve the question at all. Jun 8 '13 at 8:43