  We propose two neural network based mixture models in this article. The
proposed mixture models are explicit in nature. The explicit models have
analytical forms with the advantages of computing likelihood and efficiency of
generating samples. Computation of likelihood is an important aspect of our
models. Expectation-maximization based algorithms are developed for learning
parameters of the proposed models. We provide sufficient conditions to realize
the expectation-maximization based learning. The main requirements are
invertibility of neural networks that are used as generators and Jacobian
computation of functional form of the neural networks. The requirements are
practically realized using a flow-based neural network. In our first mixture
model, we use multiple flow-based neural networks as generators. Naturally the
model is complex. A single latent variable is used as the common input to all
the neural networks. The second mixture model uses a single flow-based neural
network as a generator to reduce complexity. The single generator has a latent
variable input that follows a Gaussian mixture distribution. We demonstrate
efficiency of proposed mixture models through extensive experiments for
generating samples and maximum likelihood based classification.
