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Probabilistic metric spaces are natural extensions of metric spaces, where the function that computes the distance outputs a distribution on the real numbers rather than a single value. Such a function is called a distribution function. F-spaces are constructions for probabilistic metric spaces, where the distribution functions are built for functions that map from a measurable space to a metric space. In this paper, we propose an extension of F-spaces, called Generalized F-space. This construction replaces the metric space with a probabilistic metric space and uses fuzzy measures to evaluate sets of elements whose distances are probability distributions. We present several results that establish connections between the properties of the constructed space and specific fuzzy measures under particular triangular norms. Furthermore, we demonstrate how the space can be applied in machine learning to compute distances between different classifier models. Experimental results based on Sugeno λ-measures are consistent with our theoretical findings.

Probabilistic metric spaces are a natural generalization of metric spaces in which the function that computes the distance outputs a distribution on the real numbers rather than a single number. Such a function is called a distribution function. In this paper, we construct a distance for linear regression models using one type of probabilistic metric space called F-space. F-spaces use fuzzy measures to evaluate a set of elements under certain conditions. By using F-spaces to build a metric on machine learning models, we permit to represent more complex interactions of the databases that generate these models.

Metric spaces are defined in terms of a space and a metric, or distance. Probabilistic metric spaces are a useful extension of metric spaces where the distance is a distribution instead of a number. In this way, we can take into account uncertainty. Then, the triangle inequality is replaced by a condition based on triangle functions on the distributions. In this paper we introduce F-spaces. This is a new type of probabilistic metric spaces which is based on fuzzy measures (also known as non-additive measures and capacities). We prove some properties that describe which families of fuzzy measures are compatible with which type of triangle functions. Then, we show how we can use Sugeno, Choquet integrals, and, in general, any other fuzzy integral as a tool for building these spaces. We show how these results can be used to compute distances between functions. We illustrate the example comparing three types of means when applied to a set of databases. The example uses Sugeno λ-measures to illustrate the theoretical results presented in the paper.

Machine and statistical learning is about constructing models from data. Data is usually understood as a set of records, a database. Nevertheless, databases are not static but change over time. We can understand this as follows: there is a space of possible databases and a database during its lifetime transits this space. Therefore, we may consider transitions between databases, and the database space. NoSQL databases also fit with this representation. In addition, when we learn models from databases, we can also consider the space of models. Naturally, there are relationships between the space of data and the space of models. Any transition in the space of data may correspond to a transition in the space of models. We argue that a better understanding of the space of data and the space of models, as well as the relationships between these two spaces is basic for machine and statistical learning. The relationship between these two spaces can be exploited in several contexts as, e.g., in model selection and data privacy. We consider that this relationship between spaces is also fundamental to understand generalization and overfitting. In this paper, we develop these ideas. Then, we consider a distance on the space of models based on a distance on the space of data. More particularly, we consider distance distribution functions and probabilistic metric spaces on the space of data and the space of models. Our modelization of changes in databases is based on Markov chains and transition matrices. This modelization is used in the definition of distances. We provide examples of our definitions.