Abstract
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.
Highlights
Machine and statistical learning can be seen as a search problem
We have proposed the use of Markov chains and transition matrices to model transitions between databases, and used them to define a probabilistic metric space for models
Our goal is to better understand the relationship between data and models. This requires a metric space on the space of models that reflects the relationships between the databases that can generate these models
Summary
Machine and statistical learning can be seen as a search problem. That is, we have a state space corresponding to possible. In [14], the authors proposed the use of probabilistic metric spaces [9] for modeling the similarity between models These spaces are defined in terms of distance distribution functions. We propose the use of Markov chains and transition matrices to represent, respectively, sequences of changes in databases and the probability of changes taking place This representation permits the definition of probabilistic metric spaces on the space of data. We use them later to define distance distribution functions for the space of models in terms of the databases that have generated them. This is a much simpler approach than the one introduced in [14]. The paper finishes with a discussion and lines for future work
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