# Choosing values for marginal densities given non informative prior

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?

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.