# Does the posterior necessarily follow the same conditional dependence structure as the prior?

One of the assumptions in a model is the conditional dependence between random variables in the joint prior distribution. Consider the following model, $$p(a,b|X) \propto p(X|a,b)p(a,b)$$

Now suppose an independence assumption for the prior $$p(a,b) = p(a)p(b)$$.

Does this assumption imply the posterior has the following conditional dependence as well? $$p(a|X)p(b|X) \propto p(X|a,b)p(a)p(b)$$

• If $p(X \mid a,b)=f(X\mid a)\,g(X\mid b)$ for some $f,g$ then perhaps this might give posterior independence – Henry Jun 21 at 14:35

Your question can also be stated as: "$$X$$ is dependent on $$a$$ and $$b$$. And $$a$$ and $$b$$ are independent. Does this imply that $$a$$ and $$b$$ are conditionally independent given $$X$$?"

The answer is no. We just need a counter-example to show it isn't the case. Suppose $$X = a + b$$.

Then, once we know $$X$$'s value, $$a$$ and $$b$$ are dependent (information about one tells us what the other will be). For example, suppose $$X=5$$. Then, if $$a=3$$, it tells us that $$b=2$$. Similarly, if $$b=4$$, it tells $$a=1$$.

No, it doesn't: Under the assumption that $$a \ \bot \ b$$, the right-hand-side of your last equation is:

$$p(x|a,b) \cdot p(a) \cdot p(b) = p(x,a,b) \overset{a,b}{\propto} p(a,b|x).$$

Thus, you are effectively asking whether or not:

$$a \ \bot \ b \quad \quad \implies \quad \quad p(a|x) \cdot p(b|x) \propto p(a,b|x).$$

That is, you are asking whether prior independence of $$a$$ and $$b$$ implies posterior independence of these random variables. Generally speaking, no it doesn't --- many statistical models involve data $$x$$ that give information about both prior variables, in such a way that they exhibit statistical dependence a posteriori.