How Do I choose parameters of prior on regression coefficients in a Bayesian linear model?

I'm trying to perform a linear regression in a Bayesian way.

The response is normal,the prior I would like to put over $\mathbf{\beta}$ (vector of regression coefficients) and $\Sigma^2$ (variance of the error term) is a Normal inverse-gamma one: Pi(Beta,Sig^2)=P(Beta|Sig^2)*P(Sig^2)

$P(\mathbf{\beta}|\Sigma^2) \sim N_p(b,B)$

$P(\Sigma^2) \sim InvGamma(u,U)$

My problem regards the choice of the parameters of the prior($b,B,u,U$).

Thank you for your help!

• If you're in doubt about the parameters, why not just put hyper priors on them? – demodw Feb 10 '16 at 9:57
• Actually that seems to me like postponing the problem.. I have an idea on how to centre the Beta|Sig prior (I thought about using the MLE for Beta) , but no clue about its variance.. And no idea on the Inv Gamma parameters, – Tommaso Guerrini Feb 10 '16 at 10:05
• Postponing? why? Assigning (hyper)priors is the normal approach in any Bayesian analysis if you are not sure about the values. Obviously, the prior you assign need a clear formulation. For variance, some people use a half-normal, gamma or half-cauchy. As for the inverse-gamma, I would try to plot the density function using different values for alpha and beta, and see which one makes most sense in terms of your prior knowledge. – demodw Feb 10 '16 at 10:11
• Thank you for your help, I'll get into it a lot more and try out your suggestions.. – Tommaso Guerrini Feb 10 '16 at 10:22
• See elicitation. – Scortchi Feb 10 '16 at 11:37