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!