Conjugate prior is useful and beautiful in theory, for instance, Dirichlet distribution can serves as the conjugate prior for multinomial distribution. But I don't find any actual applications based on that.
Can you provide some?
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Thinking specifically of the Dirichlet-Multinomial conjugate pair, have a look at the popular latent Dirichlet allocation (LDA) topic model. LDA models topics as multinomial distributions over words with a Dirichlet prior, and models documents as multinomial distributions over topics with a Dirichlet prior. Due to conjugacy, we are able to build a collapsed Gibbs sampler for LDA (ie we integrate out the parameters we don't necessarily care about, keeping just the latent topic assignments for each word), making inference tractable. Without conjugacy, we could not collapse the model, much less make a Gibbs sampler (although we could do Metropolis-Hastings), so inference would be much more costly at run time.
