# Variance estimation problem using MCMC

I have the following mode

$y_k \sim \mathcal{N}( a_0+a_1 x_k+a_2 x_k^2 , \sigma^2 )$

I have a dataset $\mathcal{D} = \{y_k , x_k\}_1^N$. I am using Metropolis-hasting MCMC to estimate the model parameters $a_0, a_1, a_2$ and $\sigma^2$. I am getting a good estimation for $a_0, a_1, a_2$ but not equally good for the variance. The proposal distribution for $a$'s is multivariate Gaussian

$a^{(k+1)} = a^{(k)} + \sigma_a \mathcal{N}(0,I)$

and for $1/\sigma^2$ is Gamma distribution. I can have better estimation of the variance using the WLS estimator! How can I use the results of the WLS to improve the MCMC estimation?