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Say I have a (multivariate) Gaussian mixture model $$p(x)=\sum_k\pi_iN(\mu_i,\Sigma_i),$$ of which the $\boldsymbol\mu$ and $\boldsymbol\Sigma$ are known, so the likelihood function of the coefficients $\boldsymbol\pi$ is $p(x|\boldsymbol\pi)$.

What would be a proper conjugate prior $p(\boldsymbol\pi)$ for $p(x|\boldsymbol\pi)$?

Thanks!

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The likelihood function $$\prod_{i=1}^n \sum_{k=1}^K \pi_k \varphi(x_i|\mu_k,\Sigma_k)$$expands into a mixture of Dirichlet-type terms: $$\sum_{(n_1,\ldots,n_K)} \omega(n_1,\ldots,n_K) \pi_1^{n_1}\cdots \pi_K^{n_K}$$where the sum is taken over all partitions of $n$ terms into $K$ groups, with $n_1+\ldots+n_K=n$.

Therefore the set of conjugate priors on $\mathbf{\pi}=(\pi_1,\ldots,\pi_K)$ is made of mixtures of Dirichlet distributions: $$\sum_{(m_1,\ldots,m_K)} \alpha(m_1,\ldots,m_K) \pi_1^{m_1}\cdots \pi_K^{m_K}$$where the sum is taken on an arbitrary collection of integers $(m_1,\ldots,m_K)$. Despite its conjugate nature, this is not a convenient prior in that the posterior cannot be easily exploited.

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