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In a Gaussian mixture model, the labels assigned to the data points are often called auxiliary variables, whereas the cluster means and covariances are called latent variables. Since both types of variables are hidden (only the data is observed), they both are technically latent. How does one distinguish between what is an auxiliary variable and what is a latent variable? How do you do this for models in general (i.e. not mixture models)?

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According to section 10.2 of Bishop's Pattern Recognition and Machine Learning, latent variables grow with the size of the data (cluster labels in the GMM setting). Variables that have the same count regardless of the size of the data are called model parameters.

To be consistent with the question posed, auxiliary variables in my question are latent variables according to Bishop, and latent variables in my question are model parameters according to Bishop.

The distinction between latent and auxiliary variables (or equivalently, by Bishop's definitions, model parameters and latent variables) is important when constructing approximations to a joint posterior distribution through variational inference.

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