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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.

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  • $\begingroup$ that's the easy part friend, you just plug in the transformed value to the likelihood. $\endgroup$ Jul 28 at 14:20
  • $\begingroup$ @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? $\endgroup$ Jul 28 at 14:26

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