How to use Gibb's sampling when the conditional probability doesn't depend on the observations I have a model that looks like this
$$ x(k) = \sum_{m}^{M} e^{i (U_m k + \beta_m)} + n(k)$$
Where $U_m$ has a Gaussian distribution with parameters $\mu$ and $\sigma^2$.
$$ U_m \sim \mathcal{N}(\mu, \sigma^2)  $$ and $\beta_m$ has a uniform distribution
$$ \beta_m \sim \mathcal{U}(0, 2\pi) $$
The $n$ is an additive zero mean complex Gaussian noise
$$ \Re(n) \sim \frac{1}{\sqrt{2}} \mathcal{N}(0, \sigma_n^2) $$
$$ \Im(n) \sim \frac{1}{\sqrt{2}} \mathcal{N}(0, \sigma_n^2) $$
I want to estimate $\mu$ and $\sigma$ when $M$ is a large number. I wanted to use Gibbs's sampling. However, when I tried to find the conditional probabilities, I found that they are not dependent on the data anymore when $M$ is large and $\sigma_n$ is known.
$$ p(\mu | \sigma, x) = p(\mu | \sigma) $$
and
$$ p(\sigma | \mu, x) = p(\sigma | \mu) $$
It is explained here.
If I think of my application, these prior probabilities may not depend on each other a lot either. So,
$$ p(\mu | \sigma) = p(\mu) $$ and
$$ p(\sigma | \mu) = p(\sigma) $$
What I don't understand is how this will work. I don't have an intuitive prior for both these parameters (Any recommendation is appreciated). I just have intuitive initial points for these two parameters. If there is no dependence on data, how can I estimate these two parameters based on Gibbs's sampling? Should I consider something else in this case?
 A: Assuming $\sigma_n$ is a known parameter, the question is about simulating from the posterior distribution of $(\mu,\sigma^2)$ given a sample of $x(k)'s$ (e.g., $x(1),\ldots,x(K)$)
$$p(\mu,\sigma|x(1),\ldots,x(K))\propto p(\mu,\sigma)\times f(x(1),\ldots,x(K)|\mu,\sigma)$$
As expressed in the question, this joint distribution density $$f(x(1),\ldots,x(K)|\mu,\sigma)$$is not easily computed and hence an MCMC algorithm is difficult to implement, incl. the specific case of the Gibbs sampler.
One traditional approach in simulation (and found in one of the earliest Gibbs samplers, see Tanner & Wong, 1987) is to complement the vector to be simulated, $(\mu,\sigma)$, with latent or auxiliary variables in order to achieve a manageable joint density. We call this method demarginalisation in our book and it simply means that simulating from the marginal density $p(\cdot)$ can proceed by (i) constructing a manageable but otherwise arbitrary joint density $q(\cdot,\cdot)$ such that $p$ is its marginal:
$$p(u)=\int q(u,v)\,\text dv$$
and (ii) simulate from this joint $q$ a sample of $(u_i,v_i)_{i=1,\ldots,I}$'s, because the $u_i$'s will then be distributed from $p$.
In the current setting, the latent variables are naturally available as made of the $\mathbf U=U_1,\ldots,U_M$ and $\mathfrak B=\beta_1,\ldots,\beta_M$ since the original model can be decompose as the hierarchy of distributions
\begin{align}
x(k)|\mathbf U,\mathfrak B &\sim \mathcal N\left(\sum_{m}^{M} \exp\{i (U_m k + \beta_m)\},\sigma_n^2\right)\\
U_m|\mu,\sigma &\sim \mathcal N\left(\mu,\sigma^2\right)\\
\beta_m &\sim \mathcal U(0,2\pi)\\
\mu,\sigma &\sim p(\mu,\sigma)
\end{align}
and the associated demarginalisation is
$$p(\mu,\sigma|x(1),\ldots,x(K)) = \int p(\mu,\sigma,\mathbf U,\mathfrak B|x(1),\ldots,x(K))\,\text d(\mathbf U,\mathfrak B)$$
Furthermore, thanks to the hierarchical decomposition
$$p(\mu,\sigma,\mathbf U,\mathfrak B|x(1),\ldots,x(K))\propto
p(\mu,\sigma)\times\prod_m \mathfrak u(\beta_m)\times\prod_m \varphi(U_m|\mu,\sigma)\times\prod_{k=1}^K\varphi(x(k)|\mathbf U,\mathfrak B)$$
the joint distribution can be easily simulated by a Gibbs sampler based on the full conditional. Obviously, if $M$ and/or $K$ is large, this sampler may be time consuming and slowly mixing.
