  Penalized and robust regression, especially when approached from a Bayesian
perspective, can involve the problem of simulating a random variable
$\boldsymbol z$ from a posterior distribution that includes a term proportional
to a sum of powers, $\|\boldsymbol z \|^q_q$, on the log scale. However, many
popular gradient-based methods for Markov Chain Monte Carlo simulation from
such posterior distributions use Hamiltonian Monte Carlo and accordingly
require conditions on the differentiability of the unnormalized posterior
distribution that do not hold when $q \leq 1$ (Plummer, 2023). This is
limiting; the setting where $q \leq 1$ includes widely used sparsity inducing
penalized regression models and heavy tailed robust regression models. In the
special case where $q = 1$, a latent variable representation that facilitates
simulation from such a posterior distribution is well known. However, the
setting where $q < 1$ has not been treated as thoroughly. In this note, we
review the availability of a latent variable representation described in
Devroye (2009), show how it can be used to simulate from such posterior
distributions when $0 < q < 2$, and demonstrate its utility in the context of
estimating the parameters of a Bayesian penalized regression model.
