  Poisson log-linear models are ubiquitous in many applications, and one of the
most popular approaches for parametric count regression. In the Bayesian
context, however, there are no sufficient specific computational tools for
efficient sampling from the posterior distribution of parameters, and standard
algorithms, such as random walk Metropolis-Hastings or Hamiltonian Monte Carlo
algorithms, are typically used. Herein, we developed an efficient
Metropolis-Hastings algorithm and importance sampler to simulate from the
posterior distribution of the parameters of Poisson log-linear models under
conditional Gaussian priors with superior performance with respect to the
state-of-the-art alternatives. The key for both algorithms is the introduction
of a proposal density based on a Gaussian approximation of the posterior
distribution of parameters. Specifically, our result leverages the negative
binomial approximation of the Poisson likelihood and the successful
P\'olya-gamma data augmentation scheme. Via simulation, we obtained that the
time per independent sample of the proposed samplers is competitive with that
obtained using the successful Hamiltonian Monte Carlo sampling, with the
Metropolis-Hastings showing superior performance in all scenarios considered.
