Consider spatially distributed data $\boldsymbol x = \{x(t_1),x(t_2),\dots,x(t_n)\}$ which is described by a multivariate Gaussian distribution with mean $\mu$, standard deviation $\sigma$ and stationary correlation function which depends on the absolute distance $\tau = |t_1-t_2|$ between points:
$$\rho(\tau) = \exp\left(-\frac{2\tau}{\theta}\right)$$
Which produces a correlation matrix $\boldsymbol \rho(\theta)$.
$$\boldsymbol x \sim \mathcal{N}(\mu,\sigma^2 \boldsymbol \rho)$$
Given samples of $\boldsymbol x$ the aim is to predict $\mu$, $\sigma$ and $\theta$. Under the Bayesian approach I believe that a joint prior distribution must be specified. Since I want to use Markov Chain Monte Carlo to sample from the posterior I believe the prior should be proper. My approach is to specify marginal distributions $P(\mu)$, $P(\sigma)$, $P(\theta)$ and take the joint prior to be given by:
$$P(\mu,\sigma,\theta) = P(\mu)P(\sigma)P(\theta)$$
Is it reasonable to take the priors to be independent?
Can I take the priors to be uniform if am confident the parameter values fall in certain intervals i.e.
$$\mu \sim U[-100,100]$$ $$\sigma \sim U[0,1000]$$ $$\theta \sim U[0,1000]$$
Or should I use a Cauchy for $\mu$ and the folded-Cauchy for $\sigma$, $\theta$?