1
$\begingroup$

I am trying to understand how to choose values for marginal densities of the paramters given a non informative prior in a Bayesian linear regression model s.th.:

$$Y_n|X_{nk}, \beta_k, \sigma^2 \sim N(X\beta, \sigma^2I)$$

Given a non-informative prior $p(\beta, \sigma^2) \propto 1 / \sigma^2$ and the likelihood functions from above, the marginal distribution for the linear regression model are given by:

$$\sigma^2 \sim InvGamma(df/2, df*s2/2)$$ with $df = n - k$, and $$\beta \sim MvNormal(\beta, \sigma^2(X'X)^{-1})$$

Using an artificial data generated by:

$$ y = X\beta + \varepsilon\quad \varepsilon \sim N(0,1)$$ with $$X \sim U(0,1), \quad\beta = (2,2,0.1)' $$

Now I came across a few tutorials this or this where the authors use $\beta_i \sim N(0, 10^4)$ and $\sigma^2 \sim IG(1 / 10^4, 1 / 10^4)$.

My question is what is their reasoning for choosing those values?

$\endgroup$
1
$\begingroup$

I suspect those values were chosen as they yield suitably vague priors.

A normal distribution centered at 0 with variance $10^4$ spreads the majority of its probability density (say, 95%) over the interval $(-200, 200)$, which is reasonably vague for many applications. The inverse gamma case is similar.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.