  In pursuit of explainability, we develop generative models for sequential
data. The proposed models provide state-of-the-art classification results and
robust performance for speech phone classification. We combine modern neural
networks (normalizing flows) and traditional generative models (hidden Markov
models - HMMs). Normalizing flow-based mixture models (NMMs) are used to model
the conditional probability distribution given the hidden state in the HMMs.
Model parameters are learned through judicious combinations of time-tested
Bayesian learning methods and contemporary neural network learning methods. We
mainly combine expectation-maximization (EM) and mini-batch gradient descent.
The proposed generative models can compute likelihood of a data and hence
directly suitable for maximum-likelihood (ML) classification approach. Due to
structural flexibility of HMMs, we can use different normalizing flow models.
This leads to different types of HMMs providing diversity in data modeling
capacity. The diversity provides an opportunity for easy decision fusion from
different models. For a standard speech phone classification setup involving 39
phones (classes) and the TIMIT dataset, we show that the use of standard
features called mel-frequency-cepstral-coeffcients (MFCCs), the proposed
generative models, and the decision fusion together can achieve $86.6\%$
accuracy by generative training only. This result is close to state-of-the-art
results, for examples, $86.2\%$ accuracy of PyTorch-Kaldi toolkit [1], and
$85.1\%$ accuracy using light gated recurrent units [2]. We do not use any
discriminative learning approach and related sophisticated features in this
article.
