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.

本文提出了一种在监督学习中正确的损失函数概念，并通过条件概率测度和马尔科夫核的最小化来解决一般化问题。提出了解决随机不适定问题的新方法，并探讨其应用。