# 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,
$$p(a,b|X)∝p(X|a,b)p(a,b)p(a,b|X) \propto p(X|a,b)p(a,b)$$

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

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

Your question can also be stated as: “$$XX$$ is dependent on $$aa$$ and $$bb$$. And $$aa$$ and $$bb$$ are independent. Does this imply that $$aa$$ and $$bb$$ are conditionally independent given $$XX$$?”
The answer is no. We just need a counter-example to show it isn’t the case. Suppose $$X=a+bX = a + b$$.
Then, once we know $$XX$$‘s value, $$aa$$ and $$bb$$ are dependent (information about one tells us what the other will be). For example, suppose $$X=5X=5$$. Then, if $$a=3a=3$$, it tells us that $$b=2b=2$$. Similarly, if $$b=4b=4$$, it tells $$a=1a=1$$.