# Solved – Bayesian Weighted Linear regression

I am currently reading the following paper which formulates the weighted linear regression in a Bayesian setting. In classic weighted LS, we minimise the following:

\$\$
sum_{i=1}^{N} w_i (beta^Tx_i – y_i)
\$\$

In this paper, they try and have a Bayesian formulation of the WLS. So, it makes the following modelling choices about the probability distributions of the random variables:

\$\$
y_i sim N(beta^tx_i, sigma^2/w_i)
\$\$

So, here we are modelling each of the \$y_i\$ to have variance which can be weighted by their individual weight. There is a normal prior also over the regression parameters \$beta\$.

\$\$
beta sim N(beta_0, Sigma_{beta, 0})
\$\$

There is a Gamma prior over the weights \$w_i\$.

\$\$
w_i sim Gamma(a_i, b_i)
\$\$

Now, my question is that the regression problem is basically:

\$\$
y_i = beta^T x_i + epsilon_i
\$\$

My question is why is there no prior on \$epsilon\$? In this paper, they estimate \$sigma^2\$ through some standard regression formula (Apologies as I have not gone far to derive it yet). However, to me it seems that \$sigma^2\$ is also an unknown parameter in the model and if we follow Bayesian statistical modelling, we should specify a prior for it.

If anyone is curious, the paper is here: