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 $\mathcal{Y}$, 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 embeddings. 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 regularization method for solving stochastic ill-posed problems, using inner measure, and present its applications.

Epidemic Dynamics via Wavelet Theory and Machine Learning, with Applications to Covid-19

We introduce the concept of epidemic-fitted wavelets which comprise, in particular, as special cases the number $I(t)$ of infectious individuals at time $t$ in classical SIR models and their derivatives. We present a novel method for modelling epidemic dynamics by a model selection method using wavelet theory and, for its applications, machine learning based curve fitting techniques. Our universal models are functions that are finite linear combinations of epidemic-fitted wavelets. We apply our method by modelling and forecasting, based on the John Hopkins University dataset, the spread of the current Covid-19 (SARS-CoV-2) epidemic in France, Germany, Italy and the Czech Republic, as well as in the US federal states New York and Florida.