The answer appears to be no because the distribution of X is defined conditionally by θ which is also assumed to have a distribution as opposed to be a constant. Essentially, the formulation of the probability distribution function of X is explicitly dependent on θ so how could X and θ be independent of each other.
However, I am confused by an equation in the proof of Bayes’ Theorem for random variables:
Suppose that the n random variables X1, . . . , Xn form a random sample from a distribution for which the p.d.f. or the p.f. is f (x | θ). Suppose also that the value of the parameter θ is unknown and the prior p.d.f. or p.f. of θ is ξ(θ). Then the posterior p.d.f. or p.f. of θ is
ξ(θ|x) =[f(x1|θ) . . . f(xn|θ)ξ(θ)]/gn(x)
for θ ∈ Ω,
where gn is the marginal joint p.d.f. or p.f. of X1, . . . , Xn.
In the proof of the theorem it is stated that the joint distribution of x (vector of observed X's) and θ--f(x,θ)--is equal to the product of the conditional joint distribution of x given θ--fn(x|θ)--and the marginal distribution of θ--ξ(θ).
I know that if X and θ are independent random variables, then the joint distribution of X and θ, f(x,θ)=g(x)ξ(θ). The above looks similar except for, of course, the p.d.f. of X is conditional upon θ. So, I wanted to make sure I was not missing the implication that X and θ are actually independent of each other.
Thanks in advance for your help. It seems straightforward, but I couldn't find an explicit answer on the website or in my readings...perhaps because it is so obvious.