Parameter inference in a probabilistic model using clustered data

We propose a method to infer the parameters of a probabilistic model from given data samples. Our method is based on the pseudolikelihood and composite likelihood methods. We cluster the given data samples and apply the clustered data samples to the pseudolikelihood and composite likelihood methods. From an expansion of the pseudolikelihood method around the mean of a cluster, the mean-field and Thouless-Anderson-Palmer equations are derived. Likewise, from an expansion of the composite likelihood method around the mean of a cluster, a method that is similar to the Bethe approximation is derived. We then perform numerical simulations using our method. We find that our method gives an accurate estimate in the range of weak coupling parameters but has an inferior accuracy compared to the pseudolikelihood and composite likelihood methods in the range of strong coupling parameters. In the range of strong coupling parameters, as the number of clusters increases, the inference accuracy of our method improves. Compared to the pseudolikelihood and composite likelihood methods, our method reduces the number of computational tasks for the estimation, therefore, sacrificing the inference accuracy.