Finite mixture models are an important tool in the statistical analysis of data, for example in data clustering. The optimal parameters of a mixture model are usually computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm. We propose an alternative approach based on the theory of Mean Field Games, a class of differential games with an infinite number of agents. We show that the solution of a finite state space multi-population Mean Field Games system characterizes the critical points of the log-likelihood functional for a Bernoulli mixture. The approach is then generalized to mixture models of categorical distributions. Hence, the Mean Field Games approach provides a method to compute the parameters of the mixture model, and we show its application to some standard examples in cluster analysis.

A mean field games model for finite mixtures of Bernoulli and categorical distributions / Aquilanti, Laura; Cacace, Simone; Camilli, Fabio; De Maio, Raul. - In: JOURNAL OF DYNAMICS AND GAMES. - ISSN 2164-6066. - 8:1(2021), pp. 35-59. [10.3934/jdg.2020033]

A mean field games model for finite mixtures of Bernoulli and categorical distributions

Aquilanti, Laura;Cacace, Simone;Camilli, Fabio
;
De Maio, Raul
2021

Abstract

Finite mixture models are an important tool in the statistical analysis of data, for example in data clustering. The optimal parameters of a mixture model are usually computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm. We propose an alternative approach based on the theory of Mean Field Games, a class of differential games with an infinite number of agents. We show that the solution of a finite state space multi-population Mean Field Games system characterizes the critical points of the log-likelihood functional for a Bernoulli mixture. The approach is then generalized to mixture models of categorical distributions. Hence, the Mean Field Games approach provides a method to compute the parameters of the mixture model, and we show its application to some standard examples in cluster analysis.
2021
mixture models; bernoulli distribution; categorical distribution; cluster analysis; expectation-maximization algorithm; mean field games
01 Pubblicazione su rivista::01a Articolo in rivista
A mean field games model for finite mixtures of Bernoulli and categorical distributions / Aquilanti, Laura; Cacace, Simone; Camilli, Fabio; De Maio, Raul. - In: JOURNAL OF DYNAMICS AND GAMES. - ISSN 2164-6066. - 8:1(2021), pp. 35-59. [10.3934/jdg.2020033]
File allegati a questo prodotto
File Dimensione Formato  
Aquilanti_MeanField_2020.pdf

solo gestori archivio

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 561.87 kB
Formato Adobe PDF
561.87 kB Adobe PDF   Contatta l'autore

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1492595
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact