I have the following model.
$$ z(k) = a(k) e^{i \psi(k)} + n(k)$$
The distribution of $a$ is known to be a Rayleigh and $\psi$ is known to be uniformly distributed. The noise $n$ is a white Gaussian random noise with variance $\sigma_n^2$.
$$ a \sim \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}} $$ $$ \psi \sim \mathcal{U}[-\pi, \pi] $$ $$ n \sim \mathcal{N}(0, \sigma_n^2) $$
When there are not many observations of $z$, my question is how to estimate the parameter $\sigma$. However, I do not know how to write the likelihood of $\sigma$ given $z$ and $\sigma_n$; that is $p(z|\sigma)$. Any ideas ?