It is proposed to cluster an observed p-dimensional random sample y1, . . . , yn, of size n into g clusters by fitting a mixture model with g multivariate normal components with mean μi and covariance matrix Σi (i = 1, . . . , g) in proportions π1, . . . , πg. In order to reduce the number of parameters in the component-covariance matrices Σi a factor model is to be adopted for the ith-component distribution (i = 1, . . . , g) of Y j (j = 1, . . . , n Specify the component distribution of Y j under the so-called MFA model

The Mixture of Factor Analyzers (MFA) model is a probabilistic model that is used for clustering high-dimensional data. It is a generalization of the Gaussian Mixture Model (GMM) where each component of the mixture is modeled by a Factor Analysis model.

Under the MFA model, the component distribution of Yj (j = 1, ..., n) is specified as follows:

1. The observed data Yj is assumed to be generated from a mixture of g multivariate normal distributions. Each of these distributions corresponds to a cluster.

2. The ith component of the mixture (i = 1, ..., g) is modeled by a Factor Analysis model. This means that the covariance matrix Σi of the ith component is decomposed into a lower-dimensional factor loading matrix Λi and a diagonal matrix Ψi of unique variances.

3. The distribution of Yj given that it belongs to the ith component is then a multivariate normal distribution with mean μi + Λi * ηij and covariance matrix Σi = Λi * Λi' + Ψi, where ηij is a q-dimensional vector of factor scores.

4. The factor scores ηij are assumed to follow a standard multivariate normal distribution.

5. The mixing proportions π1, ..., πg are the probabilities that an observation Yj belongs to each of the g components.

In summary, the MFA model reduces the number of parameters in the component-covariance matrices Σi by decomposing them into factor loading matrices and unique variance matrices. This makes the model more suitable for high-dimensional data.