Using categorical properties of probabilistic morphisms, we prove that sequential Bayesian inversions in Bayesian supervised learning models for conditionally independent (possibly not identically distributed) data, proposed by L\^e in \cite{Le2025}, coincide with batch Bayesian inversions. Based on this result, we provide a recursive formula for posterior predictive distributions in Bayesian supervised learning. We illustrate our results with Gaussian process regressions. For Polish spaces $\mathcal{Y}$ and arbitrary sets $\mathcal{X}$, we define probability measures on $\mathcal{P} (\mathcal{Y})^{\mathcal X}$, using a projective system generated by $\mathcal{Y}$ and $\mathcal{X}$. This is a generalization of a result by Orbanz \cite{Orbanz2011} for the case $\mathcal{X}$ consisting of one point. We revisit MacEacher's Dependent Dirichlet Processes (DDP) taking values on the space $\mathcal{P} (\mathcal{Y})$ of all probability measures on a measurable subset $\mathcal{Y}$ in $\mathbf{R}^n$, considered by Barrientos-Jara-Quintana \cite{BJQ2012}. We indicate how to compute posterior distributions and posterior predictive distributions of Bayesian supervised learning models with DDP priors.

翻译：利用概率态射的范畴性质，我们证明了L\^e在文献\cite{Le2025}中针对条件独立（可能非同分布）数据提出的贝叶斯监督学习模型中，序贯贝叶斯逆与批量贝叶斯逆是重合的。基于这一结果，我们给出了贝叶斯监督学习中后验预测分布的递归公式。我们通过高斯过程回归的实例来说明我们的结果。对于波兰空间$\mathcal{Y}$和任意集合$\mathcal{X}$，我们利用由$\mathcal{Y}$和$\mathcal{X}$生成的投影系统，定义了$\mathcal{P} (\mathcal{Y})^{\mathcal X}$上的概率测度。这是Orbanz \cite{Orbanz2011}在$\mathcal{X}$为单点集情形下结果的一个推广。我们重新审视了MacEacher的取值于$\mathbf{R}^n$中可测子集$\mathcal{Y}$上所有概率测度空间$\mathcal{P} (\mathcal{Y})$的依赖狄利克雷过程（DDP），该过程由Barrientos-Jara-Quintana \cite{BJQ2012}所研究。我们指明了如何计算具有DDP先验的贝叶斯监督学习模型的后验分布与后验预测分布。