# How to derive the GMM log-likelihood formulation in the eigenvoice modeling technique?

Given a GMM with mean $M=[M_1, M_2, ..., M_C]$ and covariance $\Sigma=[\Sigma_1, \Sigma_2, ..., \Sigma_C]$ (where $C$ is the number of mixtures),

many papers on eigenvoice modeling states that the log-likelihood of an arbitrary speech utterance $X=[X_1, X_2, ..., X_T]$ with $T$ frames is defined as

$\sum_c(-N_c\log(2\pi)^{F/2}|\Sigma_c|^{1/2}-0.5\sum_t(X_t-M_c)^T\Sigma_c^{-1}(X_t-M_c))$

where $N_c$ represents the number of frames aligned to the $c_{th}$ Gaussian mixture, and $F$ represents the dimensionality of $X_t$.

This formulation seems pretty odd to me because as far as I know, when computing the likelihood of the GMM, the weighted mixture-level likelihoods are added as follows

$\sum_c(\pi_cN(X|M_c, \Sigma_c))$

So I'm really confused why the summation on $c$ is outside the logarithm term in the first equation, and where $N_c$ came from.

Any help or opinions are appreciated.

Thanks!

PS. the paper is : P. Kenny, G. Boulianne, and P. Dumouchel, "Eigenvoice modeling with sparse training data," IEEE Transactions on Speech and Audio Processing, vol. 13, no. 3, pp. 345--354, 2005.

• You'd better site the paper where you saw the formula. It seems ok, the first term is just a common denominator which is extracted from the sum on t. N_c is strange indeed. – Nikolay Shmyrev Jan 11 '17 at 0:19
• Thanks for the reply! the paper is: P. Kenny, G. Boulianne, and P. Dumouchel, "Eigenvoice modeling with sparse training data," IEEE Transactions on Speech and Audio Processing, vol. 13, no. 3, pp. 345--354, 2005. – whkang Jan 11 '17 at 5:16

where $￼s$ ranges over all speakers in the training set, $c$￼ ranges over all mixture components and for each pair $(s, c)$ the sum over $t$ extends over all frames￼ aligned with $c$.
So actually in the formula the sum over $t$ must be written as
$$\sum_{t \in A_c} (X_t-M_c)^T\Sigma_c^{-1}(X_t-M_c)$$
Where $A_t$ is a set of frames aligned with $c$ and $|A_t| = N_c$. In that case the formula is correct and is basically an expansion of log of GMM probability.