For multiple least squares linear regression, can we actually specify a ratio between the coefficients as the prior? For example, for following linear model: $y = b_1*x_1 + b_2*x_2 + b_3*x_3$ ,can we specify $b1:b2:b3 = 1:2:3$?
- We don't know what exactly $b_1, b_2, b_3$ are, we only know they are expected to be proportional to each other.
- They are not exactly proportional, which means we cannot reduce it to univariate regression problem: $y = b_1*(x_1+2*x_2+3*x_3)$. The ratio between the effects are prior expectations, we want data can help adjust specific values for the coefficients.