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?


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