# Prior on a non identifiable parameter-MCMC integration

To introduce the problem I will explain the Projected normal distribution.

Let $\mathbf{z}_i=(z_{i1},z_{i2})$ be a bivariate vector distributed as a bivariate normal with vector mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$.

If we make the following transformation $x = \text{atan2}(z_{i2},z_{i1})$ (the the definition of the $\text{atan2}$ can be seen at http://en.wikipedia.org/wiki/Atan2), $x_i$ is said to be distributed as a projected normal variable with parameter $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$.

Since $\text{atan2}(z_{i2},z_{i1})=\text{atan2}(cz_{i2},cz_{i1})$ for every $c>0$, the parameter of the projected normal distribution are not identifiable unless we use an identification constraint.

Generally the variance of the second component of $\mathbf{z}$ is assumed to be $1$, and we let $\mathbf{V}$ be $\boldsymbol{\Sigma}$ with the constraint and with $\boldsymbol{\mu}^*$ the associated vector of $\boldsymbol{\mu}$, from $(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ we can always compute $(\boldsymbol{\mu}^*, \mathbf{V})$.

Consider now to have $n$ observations $x_i, i=1,..., n$ and we want to estimate the posterior distribution $\boldsymbol{\mu}^*, \mathbf{V}| x_1,...,x_n$. I have to choose prior distributions, for $\boldsymbol{\mu}^*$ I use a normal distribution, but I have some problems with the prior on $\mathbf{V}$.

In this simple case I can parameterize $\mathbf{V}$ in terms of variance and correlation and put priors on the variance of the fist component and the correlation coefficient but in some cases $x_i$ can be multidimensional and then $\mathbf{V}$ can be, for example, a $10 \times 10$ matrix and a parametrization in term of variance and correlation cannot be feasible.

This is my question/idea: I can estimate the following posterior $\boldsymbol{\mu}, \boldsymbol{\Sigma}| x_1,...,x_n$ and then map the posterior samples to the posterior samples of $\boldsymbol{\mu}^*, \mathbf{V}| x_1,...,x_n$. I am not sure that I can do this but I think that it is just an MCMC integration, something like: $$f(\boldsymbol{\mu}^*, \mathbf{V}| x_1,...,x_n) = \int_{(\boldsymbol{\Sigma}, \boldsymbol{\mu})\rightarrow(\mathbf{V}, \boldsymbol{\mu}^*)} f(\boldsymbol{\mu}, \boldsymbol{\Sigma}| x_1,...,x_n) d(\boldsymbol{\Sigma}, \boldsymbol{\mu})$$ where $f()$ is the pdf and the integral $\int_{(\boldsymbol{\Sigma}, \boldsymbol{\mu})\rightarrow(\mathbf{V}, \boldsymbol{\mu}^*)} d (\boldsymbol{\Sigma}, \boldsymbol{\mu})$ must be intended as the multiple integral over all the values of $\boldsymbol{\Sigma}$ and $\boldsymbol{\mu}$ that can be mapped into $\mathbf{V}$ and $\boldsymbol{\mu}^*$ (I am not sure that the integral is well written).

Since $\boldsymbol{\Sigma}$ is a covariance matrix, am I right in thinking I can use a Normal Inverse Wishart distribution on $\boldsymbol{\mu}, \boldsymbol{\Sigma}$ and the problem is solved? Moreover, how can i interpret the parameters of the Normal Inverse Wishart distribution

Forget about using $V$, you don't need a constraint, it's just a convenient parametrization.
You can indeed use the normal inverse-Wishart distribution which will draw $\mu$ and $\Sigma$ for you.
The parameter $\mu_0$ is straighforward, $\lambda$ controls the influence of your covariance matrix on the uncertainty around the mean vector. A good way to think about $\nu$ and $\mathbf{\Psi}$ is that they represent the distribution of the inverse of the empirical covariance matrices estimated from $\nu-1$ samples distributed according to $\mathbf{\Psi}^{-1}$