A recent trend in Bayesian research has been revisiting generalizations of the likelihood that enable Bayesian inference without requiring the specification of a model for the data generating mechanism. This paper focuses on a Bayesian nonparametric extension of Wedderburn's quasi-likelihood, using Bayesian additive regression trees to model the mean function. Here, the analyst posits only a structural relationship between the mean and variance of the outcome. We show that this approach provides a unified, computationally efficient, framework for extending Bayesian decision tree ensembles to many new settings, including simplex-valued and heavily heteroskedastic data. We also introduce Bayesian strategies for inferring the dispersion parameter of the quasi-likelihood, a task which is complicated by the fact that the quasi-likelihood itself does not contain information about this parameter; despite these challenges, we are able to inject updates for the dispersion parameter into a Markov chain Monte Carlo inference scheme in a way that, in the parametric setting, leads to a Bernstein-von Mises result for the stationary distribution of the resulting Markov chain. We illustrate the utility of our approach on a variety of both synthetic and non-synthetic datasets.

翻译：近年来贝叶斯研究的一个趋势是重新审视似然函数的广义化形式，使得贝叶斯推断无需指定数据生成机制的模型。本文聚焦于韦德伯恩拟似然的贝叶斯非参数扩展，采用贝叶斯加性回归树对均值函数进行建模。在此框架中，分析者仅需设定结果变量的均值与方差之间的结构关系。我们证明该方法为将贝叶斯决策树集成模型扩展到许多新场景（包括单纯形取值数据和强异方差数据）提供了统一且计算高效的框架。同时，我们提出了推断拟似然离散参数的贝叶斯策略——该任务的复杂性在于拟似然本身不包含该参数的信息；尽管存在这些挑战，我们仍能将离散参数的更新注入马尔可夫链蒙特卡罗推断方案中，且在参数化设定下，该方案能使所得马尔可夫链的平稳分布满足伯恩斯坦-冯·米塞斯定理。我们通过多组合成与非合成数据集验证了所提方法的实用性。