I have a matrix. for each row of this matrix I made a Gaussian mixture, how can I concatenate these mixtures.
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1$\begingroup$ I'm trying to understand the question. Are you asking that how you can write down the likelihood of the bigger matrix when each of its rows is characterized by a separate Gaussian mixture model? $\endgroup$– omidiFeb 7, 2014 at 14:04
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$\begingroup$ yes that's right $\endgroup$– user38388Feb 7, 2014 at 14:14
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$\begingroup$ are the rows independent from each other? $\endgroup$– omidiFeb 7, 2014 at 14:15
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$\begingroup$ they are independent but the same type $\endgroup$– user38388Feb 7, 2014 at 14:16
1 Answer
under independence assumption, the total likelihood of a matrix $\mathbf{M}$ given that each of its rows defined by a Gaussian mixture model $f_i(X|\vec{\theta_i})$ is equal to:
\begin{equation} \mathcal{L}(\mathbf{M}) = \prod_{i=1}^{N} f_i(X_i|\vec{\theta_i}) \end{equation}
where $N$ is the number of rows in matrix $\mathbf{M}$, and $X_i$ denotes the i-th row in $\mathbf{M}$. The parameter vector $\vec{\theta_i}$ are supposed to be the mixture ratio parameters, mean and variance of the Gaussian distributions in the i-th distribution.