In this paper I propose a concept of a correct loss function in a generative model of supervised learning for an input space $\mathcal{X}$ and a label space $\mathcal{Y}$, which are measurable spaces. A correct loss function in a generative model of supervised learning must correctly measure the discrepancy between elements of a hypothesis space $\mathcal{H}$ of possible predictors and the supervisor operator, which may not belong to $\mathcal{H}$. To define correct loss functions, I propose a characterization of a regular conditional probability measure $\mu_{\mathcal{Y}|\mathcal{X}}$ for a probability measure $\mu$ on $\mathcal{X} \times \mathcal{Y}$ relative to the projection $\Pi_{\mathcal{X}}: \mathcal{X}\times\mathcal{Y}\to \mathcal{X}$ as a solution of a linear operator equation. If $\mathcal{Y}$ is a separable metrizable topological space with the Borel $\sigma$-algebra $ \mathcal{B} (\mathcal{Y})$, I propose another characterization of a regular conditional probability measure $\mu_{\mathcal{Y}|\mathcal{X}}$ as a minimizer of a mean square error on the space of Markov kernels, called probabilistic morphisms, from $\mathcal{X}$ to $\mathcal{Y}$, using kernel mean embedding. Using these results and using inner measure to quantify generalizability of a learning algorithm, I give a generalization of a result due to Cucker-Smale, which concerns the learnability of a regression model, to a setting of a conditional probability estimation problem. I also give a variant of Vapnik's method of solving stochastic ill-posed problem, using inner measure and discuss its applications.

翻译：本文提出了一种在监督学习生成式模型中定义正确损失函数的概念，其中输入空间 $\mathcal{X}$ 和标签空间 $\mathcal{Y}$ 均为可测空间。在监督学习的生成式模型中，正确损失函数必须准确衡量假设空间 $\mathcal{H}$（由可能的预测器构成）与可能不属于 $\mathcal{H}$ 的监督算子之间的差异。为定义正确损失函数，本文提出将概率测度 $\mu$（定义于 $\mathcal{X} \times \mathcal{Y}$ 上）关于投影映射 $\Pi_{\mathcal{X}}: \mathcal{X}\times\mathcal{Y}\to \mathcal{X}$ 的正则条件概率测度 $\mu_{\mathcal{Y}|\mathcal{X}}$ 表征为某个线性算子方程的解。若 $\mathcal{Y}$ 是具备 Borel $\sigma$-代数 $ \mathcal{B} (\mathcal{Y})$ 的可分度量拓扑空间，本文还提出利用核均值嵌入，将正则条件概率测度 $\mu_{\mathcal{Y}|\mathcal{X}}$ 表征为从 $\mathcal{X}$ 到 $\mathcal{Y}$ 的马尔可夫核空间（即概率态射）中均方误差的最小化元。基于上述结果，并采用内测度量化学习算法的泛化能力，本文将Cucker-Smale关于回归模型可学习性的结论推广至条件概率估计问题框架。此外，本文还利用内测度给出了Vapnik求解随机不适定问题方法的一个变体，并讨论了其应用。