Supervised learning with probabilistic morphisms and kernel mean embeddings

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 $\mathcal{Y}$, both of which are measurable spaces. A correct loss function in a generative model of supervised learning must accurately measure the discrepancy between elements of a hypothesis space $\mathcal{H}$ of possible predictors and the supervisor operator, even when the supervisor operator does 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 an additional characterization of a regular conditional probability measure $\mu_{\mathcal{Y}|\mathcal{X}}$ as a minimizer of mean square error on the space of Markov kernels, referred to as probabilistic morphisms, from $\mathcal{X}$ to $\mathcal{Y}$. This characterization utilizes kernel mean embeddings. Building upon these results and employing inner measure to quantify the generalizability of a learning algorithm, I extend a result due to Cucker-Smale, which addresses the learnability of a regression model, to the setting of a conditional probability estimation problem. Additionally, I present a variant of Vapnik's regularization method for solving stochastic ill-posed problems, incorporating inner measure, and showcase its applications.