A query related to Bayesian Regression

#1
I had a query related to Bayesian Regression and I was wondering if someone had any ideas :

So in the proper Bayesian case, we have the following formulation :

For a linear line fitting (for example --> polynomials would be similar)

\(y_i = \beta_1 + \beta_2x_i + \epsilon\), which we can write in matrix form as \(Y = X \beta + \epsilon\) where \( \beta = [\beta_1 \beta_2]^T \) and x is a n by 2 matrix of observations, and n is the number of observations. Similarly Y is a n by 1 vector.

We can assume Y is varying normally, i,e. \(y_i \sim N(X \beta,\sigma^2) \)

Using standard manipulations and prior probabilities on \( \beta \), we can get the posterior for the coefficients \( \beta \).

Now, if I I am getting Y from my experiments, but for my next step I need to multiply it with k, supposedly a constant term, which shouldn't have made any difference to my regression step. But I see that 'k' itself is showing a normal distribution, hence I have to model this into my formulation. Hence, my model becomes :

\( Z = kY = \beta kX + \epsilon \) which is \( z_i = k\beta_0 + \beta_1 k x_i + \epsilon \) with \( k \sim N(\mu_k,\phi^2 \))

Does this make it a non-linear regression analysis problem ? My initial idea was to use sampling, that is sample from \( k \sim N(\mu_k,\phi^2 \)) and use that k as constant to carry on the estimation of \( \beta \). Could anyone have some ideas ?

thanks so much!