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How to estimate parameters of a Gaussian mixture model with a mix of categorical and continuous data using log-likelihood?

Indeed, I have a set of data consisting of categorical and continuous data. I want to do clustering using the mixture model. For the categorical part, I use a multinomial model and the continuous part, a Gaussian model. Basically here is the function under the assumption that continuous and categorical features are independent conditionally to $Z=k$

$$f_{k}(X) = \prod_{j=1}^{c}\prod_{h=1}^{m_j}(\alpha_{k}^{jh})^{X_j^h} \times \frac{1}{(2\pi)^{(p-c)/2}|\Sigma_k|^{1/2}}\exp\left(-\frac{1}{2}(\tilde {X} -\mu_k)^{t}\Sigma_k^{-1}(\tilde{X}-\mu)\right) $$

I have a bit of trouble understanding and I don't have a high level of statistics.

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  • $\begingroup$ This is the conditional density of one data point, conditional on the cluster indicator $Z$. The likelihood is then the product of these expressions over the whole sample and it can be maximised on the parameters. $\endgroup$ – Xi'an Feb 16 at 7:21

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