What is Half-Inverse-Student-t distribution? In the paper of Vanhatalo et al 2018, section 5.3 page 15, they use Half-Inverse-Student-t distribution as a prior:

I have never heard of such distribution and I have problems finding it. What is it?
 A: Let $T_\nu$ have the central $t_\nu$ distribution, then the Half-Inverse-t must be the distribution of $| 1/T |$, where $|\cdot|$ is the absolute value.  See https://en.wikipedia.org/wiki/Folded-t_and_half-t_distributions.
Plot of the density with some simulations:

Note that while the $t$ distribution have heavier tail with small df, the opposite is the case for the inverse-$t$ and half-inverse-$t$. 
Code for simulation and plot (R):
N <- 1000000
nu <- 4
set.seed(7*11*13) # My public seed

x <- 1/abs( rt(N, nu) )

# The inverse-half-t density: 

dinvhalft <- function(x, nu=1) 2*dt(1/x, nu)*(1/x)^2 

hist(x, breaks="FD", prob=TRUE, xlim=c(0, 15), main="Simulation from dinvhalft,  df=4", ylim=c(0, 0.55), col="skyblue")
plot(function(x)dinvhalft(x, nu=4), from=0, to=15, add=TRUE, col="red")

The question also asks for the half-$t$
 distribution. That is simpler, if $T$ have a $t_\nu$-distribution with density $f_\nu$, then the half-$t$ distribution is the distribution of $|T|$. It will have density (by symmetry about 0) given by $f_{\text{half-$t_\nu$}}(x)= 2 f_\nu(x)\cdot \mathcal{I}(x \ge 0)$, which can be coded in R as
dhalft <- function(x,nu) ifelse(x >= 0, 2*dt(x,nu), 0)

