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

#### 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.
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mixture models; bernoulli distribution; categorical distribution; cluster analysis; expectation-maximization algorithm; mean field games
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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]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11573/1492595`
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