This paper describes a framework for the development of an integrative cognitive system based on probabilistic generative models (PGMs) called Neuro-SERKET. Neuro-SERKET is an extension of SERKET, which can compose elemental PGMs developed in a distributed manner and provide a scheme that allows the composed PGMs to learn throughout the system in an unsupervised way. In addition to the head-to-tail connection supported by SERKET, Neuro-SERKET supports tail-totail and head-to-head connections, as well as neural network-based modules, i.e., deep generative models. As an example of a Neuro-SERKET application, an integrative model was developed by composing a variational autoencoder (VAE), a Gaussian mixture model (GMM), latent Dirichlet allocation (LDA), and automatic speech recognition (ASR). The model is called VAE + GMM + LDA + ASR. The performance of VAE + GMM + LDA + ASR and the validity of Neuro-SERKET were demonstrated through a multimodal categorization task using image data and a speech signal of numerical digits.