Consider a heteroskedastic model of the form, $y_i|x_i \sim \mathcal{N}\left(x_i, \text{exp}\{\boldsymbol\beta^\top\boldsymbol{x}\}\right)$ where $\boldsymbol{\beta}=\left[\beta_0,\beta_1\right]$ and $\boldsymbol{x}=\left[1,x_i\right]$. $x_i$ and $\boldsymbol{\beta}$ must be estimated from the data.
We set a multivariate normal prior on $\boldsymbol{\beta}, \boldsymbol{\beta}\sim \mathcal{N}\left(0,I\right)$ and try to derive the posterior for $\boldsymbol{\beta}, p\left(\boldsymbol{\beta}|y,\boldsymbol{x}\right)$ for the purpose of sampling. It is not possible to get a posterior in closed form, so we must resort to Metropolis-Hastings. For M-H to work we sample a new value $\boldsymbol{\beta}^*$ from a multivariate normal proposal density of the form $\boldsymbol{\beta}^*\sim\mathcal{N}\left(\boldsymbol{\beta}^{t},\Sigma\right)$, where $\boldsymbol{\beta}^t$ is the value from the current iteration.
The question I have is, how does one choose $\Sigma$, the proposal variance, given that there are no predictors only latent variables.
This problem is similar to Poisson regression except that unlike Poisson regression problem, we have unobserved or latent variables instead of predictors. Hence, proposal variance cannot be estimated from the data.
Any references are appreciated.