# Linear regression: right-hand tail of $\sigma$ marginal posterior

Suppose we're doing plain vanilla linear regression for $$y$$.

The likelihood: $$y_i \sim {\cal N}(\mu, \sigma^2)$$, $$i=1:N$$.

Priors: $$\mu \sim {\cal N}(0,1)$$, $$\sigma \sim {\rm HalfNormal}(1)$$.

As noted in Chapter 4.3 of Richard McElreath's Statistical Rethinking book, the marginal posterior $$\sigma | y_{1:N}$$ tends to have a long right-hand tail, for "complex, deep reasons". He gives only an intuitive argument: if $$\sigma >0$$ is small enough, there's not a lot of "room" for uncertainty in the left tail of the PDF.

What would the "real" mathematical reasons be? Is there a closed analytical form for the posterior? Can anyone point me to a good reference?