In this note, we consider using a link function that has heavier tails than the usual exponential link function. We construct efficient Gibbs algorithms for Poisson and Multinomial models based on this link function by introducing gamma and inverse Gaussian latent variables and show that the algorithms generate geometrically ergodic Markov chains in simple settings. Our algorithms can be used for more complicated models with many parameters. 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 link function. Although less interpretable, our models are potentially more tractable or flexible from a computational point of view in some cases.

翻译：本文探讨了一种比传统指数连接函数具有更重尾部的连接函数。通过引入伽马和逆高斯潜变量，我们为基于该连接函数的泊松模型和多项模型构建了高效的吉布斯采样算法，并证明在简单设定下这些算法能生成几何遍历的马尔可夫链。我们的算法可扩展至包含多参数的复杂模型。通过将简单泊松模型应用于实际数据集，我们验证了其后验分布与传统基于指数连接函数的泊松回归模型具有相似的统计推断。尽管可解释性稍弱，但从计算视角来看，我们的模型在某些情况下可能更具可处理性或灵活性。