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? $$p(a|X)p(b|X) \propto p(X|a,b)p(a)p(b)$$