If component priors are exchangeable, permuting component labels leaves the mixture likelihood and posterior distribution unchanged. A sampling chain can visit these equivalent modes, making naive component-by-component averages misleading. Predictive densities and other permutation-invariant summaries avoid the issue; component-specific interpretation requires an explicit identification or relabelling rule. Nonexchangeable weight priors can break exact symmetry but do not remove every multimodality problem.
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