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I understand that you can use the elbow method to plot LCA solutions vs log likelihood to figure out, at which k, it is no longer worth it to add more clusters. And I will resort to this if need be. But I think it would be much more disciplined to infer the k from some Bayesian inference or maximum likelihood method.

I know that there are latent class clustering methods with unspecified k, such as the Hierarchical Dirichlet Process (unspecified-k version of Latent Dirichlet Allocation). However, LCA is much closer to what I need than LDA or HDP.

Is there a technique that assumes no k and uses a similar model process to LCA to output the same parameters (mixture components, item probability matrix)? Or do I have to resort to methods that force me to run LCA multiple times for different k and compare?

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Yes, Dirichlet Process Mixture Models do this. With MCMC, for modest $n$ you get an item similarity matrix (O(n^2) so you will likely run out of space for this output before your inference algorithm stops working). You also get a particular clustering with each sample, and the distribution of $k$. The interpretation of the results is likely to be different than any finite $k$ methods, e.g. you're almost assured to get a mean $k$ which is non-integer.

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