VB calibration to improve the interface between phone recognizer and i-vector extractor

The EM training algorithm of the classical i-vector extractor [1, 2] is often incorrectly described as a maximum-likelihood method. The i-vector model is however intractable—the likelihood itself and the hidden-variable posteriors needed for the EM algorithm cannot be computed in closed form. We show here that the classical i-vector extractor recipe is actually a mean-field variational Bayes (VB) solution. This theoretical VB interpretation turns out to be of further use, because it also offers an interpretation of the newer phonetic i-vector extractor recipe, thereby unifying the two flavours of extractor. More importantly, the VB interpretation is also practically useful—it suggests ways of modifying existing i-vector extractors to make them more accurate. In particular, in existing methods, the approximate VB posterior for the GMM states is fixed, while only the parameters of the generative model are adapted. Here we explore the possibility of also mildly adjusting (calibrating) those posteriors, so that they better fit the generative model. In what follows, we introduce notation by summarizing the i-vector model. Then we interpret the classical recipe to deal with this intractable model as a mean-field VB algorithm. Next, we do the same with the newer phonetic i-vector recipe. Finally, we extend the phonetic recipe by allowing calibration of the phone posteriors.