# Why do we use Gamma($\epsilon, \epsilon$) as non-informative prior for precision and Normal prior for betas in Linear Regression

Suppose my regression model is $$Y_i = \beta_0 + \beta_1X_{i1} + \epsilon_i$$ In most books I am seeing that the prior used for precision $\tau = 1/\sigma^2$ is $Gamma(\epsilon, \epsilon)$. However I am unable to understand how is that found given that the beta's are unknown too? Similarly I am not getting how is Normal prior used for the beta's.

• Betas could be negative. Precisions must be positive. – Nick Cox Apr 8 '15 at 14:13
• but why exactly these distributions? – Durin Apr 8 '15 at 14:15

A simple linear regression model is $$Y_i = \beta_0 + \beta_1 X_{i1} + \epsilon_i, \epsilon_i \stackrel{ind}{\sim} N(0,\sigma^2).$$ A frequently used prior is $$p(\beta_0,\beta_1,\sigma^2) \propto 1/\sigma^2$$ Notice that this is a joint prior for $\beta_0$, $\beta_1$ and $\sigma^2$. Marginally the priors for $\beta_0$ and $\beta_1$ are uniform over the real line and the marginal prior for $\sigma^2$ is the limit as $\epsilon \to 0$ of an inverse gamma distribution with shape and scale both $\epsilon$ (or equivalently a gamma distribution on the precision $\tau=1/\sigma^2$ with the shape and rate both $\epsilon$).