Ambiguity of empty components under symmetric Dirichlet-categorical priors

Clarify the interpretation and practical implications of empty components in mixture models using symmetric Dirichlet-categorical priors when the number of components k is not known a priori, specifically determining whether a configuration with three components, one empty, is meaningfully distinct from a configuration with two non-empty components and how such representations should be handled when estimating k.

Background

Under the conventional symmetric Dirichlet-categorical prior over component assignments, component sizes can be zero, yielding empty components. When k is variable, the same partition of observations can be represented with different values of k (e.g., three components with one empty versus two non-empty), introducing ambiguity in interpretation and complicating model selection.

The authors highlight this conceptual uncertainty and address it pragmatically in their work by adopting a modified prior that forbids empty components. Nonetheless, the underlying interpretive question about the meaning and distinctness of such representations remains explicitly raised in the text.