## A query related to Bayesian Regression

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)

, which we can write in matrix form as where 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.

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

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 :

which is with )

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

thanks so much!