We present a new analytical method to derive the likelihood function that has the population of parameters marginalised out in Bayesian hierarchical models. This method is also useful to find the marginal likelihoods in Bayesian models or in random-effect linear mixed models. The key to this method is to take high-order (sometimes fractional) derivatives of the prior moment-generating function if particular existence and differentiability conditions hold. In particular, this analytical method assumes that the likelihood is either Poisson or gamma. Under Poisson likelihoods, the observed Poisson count determines the order of the derivative. Under gamma likelihoods, the shape parameter, which is assumed to be known, determines the order of the fractional derivative. We also present some examples validating this new analytical method.

翻译：我们提出了一种新的解析方法，用于推导贝叶斯层次模型中参数总体被边缘化后的似然函数。该方法同样适用于求解贝叶斯模型或随机效应线性混合模型中的边缘似然。该方法的核心在于：若特定的存在性与可微性条件成立，则对先验矩母函数求高阶（有时为分数阶）导数。特别地，该解析方法假设似然函数为泊松分布或伽马分布。在泊松似然下，观测到的泊松计数决定了导数的阶数；在伽马似然下，假定已知的形状参数决定了分数阶导数的阶数。我们还提供了若干验证该新解析方法的示例。