# Problem Description

I have a posterior distribution $$p(\theta\mid y) \propto p(y \mid \theta) p(\theta)$$ with a uniform prior $$p(\theta)= \mathcal{U}([a, b]^n)$$, which is bounded. However, for my particular application, it's easier to work with an unbounded prior so I would like to reparametrize the problem to work with a multivariate standard normal distribution $$z\sim\mathcal{N}(0, I_m).$$ How does the posterior distribution become after the reparametrization? I would like to find an expression for $$p(z\mid y) \propto p(y\mid z) p(z)$$ where prior and likelihood are in terms of the original ones and so that this posterior distribution is equivalent.

# Reparametrizing the Prior

Suppose I can find a diffeomorphism $$f:\mathbb{R}^m\to\mathbb{R}^m$$ such that $$z = f(\theta)$$. Then by the change of variables formula $$p_z(z) = p_\theta(f^{-1}(z)) |J_{f^{-1}}(z)|.$$ Luckily, it is known that $$\theta = a + (b - a) F_z(z).$$

# Reparametrizing the Likelihood

I have no idea how to do this.

• that's the easy part friend, you just plug in the transformed value to the likelihood. Jul 28 at 14:20
• @JohnMadden So are you saying that the posterior is this? $$p(z\mid y) \propto p(y \mid f^{-1}(z)) p_\theta(f^{-1}(z)) |J_{f^{-1}}(z)|$$ How come though? Jul 28 at 14:26