The space of models in machine learning: using Markov chains to model transitions

Vicenç Torra*, Mariam Taha, Guillermo Navarro-Arribas

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review

1 Citation (Scopus)


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.

Original languageEnglish
Pages (from-to)321-332
Number of pages12
JournalProgress in Artificial Intelligence
Issue number3
Publication statusPublished - Sept 2021


  • Hypothesis space
  • Machine and statistical learning models
  • Probabilistic metric spaces
  • Space of data
  • Space of models


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