<p>In this note, we consider using new link functions for Poisson and multinomial Bayesian generalized linear models. The new links are based on replacing the exponential function that appears in the usual canonical inverse links with a heavier tailed positive and strictly increasing function. We construct efficient Gibbs algorithms for Poisson and multinomial models based on the link functions by introducing gamma and inverse Gaussian latent variables and show that the algorithms generate geometrically ergodic Markov chains in simple settings. We fit our simple Poisson model to a real dataset and confirm that the posterior distribution has similar implications to those under the usual Poisson regression model based on the exponential inverse link function. Also, our method is compared with the Pólya-gamma method in a multinomial setting. Although less interpretable, our models are potentially more tractable or flexible from a computational point of view in some cases.</p>