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A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions

  • * Corresponding author: Fabio Camilli

    * Corresponding author: Fabio Camilli 
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  • 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.

    Mathematics Subject Classification: 62H30, 60J10, 49N80, 91C20.

    Citation:

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  • Figure 1.  Samples of hand-written digits from the MNIST database

    Figure 2.  Different samples of hand-written digits from the MNIST database

    Figure 3.  Clusterization histogram for digits $ \mathbf{1},\mathbf{3} $ and the corresponding Bernoulli parameters

    Figure 4.  Clusterization histogram for digits $ \mathbf{3},\mathbf{5} $ and the corresponding Bernoulli parameters

    Figure 5.  Clusterization histogram for even digits and the corresponding Bernoulli parameters

    Figure 6.  Samples of fashion products from the Fashion-MNIST database

    Figure 7.  Averaged categorical distributions for the Fashion-MNIST database

    Figure 8.  Clusterization histogram for types T-shirt, Trouser and the corresponding categorical parameters

    Figure 9.  Clusterization histogram for types Dress, Sneaker, Bag, Boot and the corresponding categorical parameters

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