Number of parameters in GMM-HMM

Question

I want to understand the use of Gaussian Mixture Models in Hidden Markov Models.

Suppose, we have speech data and we are recognizing 5 speech sounds (which are states of HMM). For example 'X' be the speech sample and O = (s,u,h,b,a) (considering characters instead of phones just for simplicity) be HMM states. Now, we use gaussian mixture model of 3 mixtures to estimate gaussian density for every state using the following equation (sorry cannot upload image because of reputation points).

P(X|O) = sum (i=1->3) w(i) * P (X|mu(i), var(i)) (considering univariate distribution)

So, we first learn the GMM parameters from the training data using EM algorithm. Then use these parameters for learning HMM parameters and once this is done, we use both of them on test data. In all we are learning 3 * 3 * 5 (weight, mean and variance for 3 mixtures and 5 states) parameters for GMM in this example. Is my understanding correct?

Answer 1

Your understanding is mostly correct, however, the number of parameters is usually larger. The mean and variance are vectors, not numbers. Variance could be matrix for rare case of full covariance GMM. Each vector usually contains 39 components for 13 cepstrum + 13 deltas + 13 delta-deltas.

So for every phone you learn

39 + 39 + 1 = 79 parameters

Total number of parameters is

79 * 5 = 395

And, usually phone is composed of 3 or so states, not from a single state. So you have 395 * 3 or 1185 parameters just for GMM. Then you need a transition matrix for HMM. Number of parameters is large thats why training requires a lot of data.