In this paper, we describe an adaptive integration method for an audio-visual speech recognition system that uses not only the speaker's audio speech signal but visual speech signals like lip images. Human beings communicate with each other by integrating multiple types of sensory information such as hearing and vision. Such integration can be applied to automatic speech recognition, too. In the integration of audio and visual speech features for speech recognition, there are two important issues, i.e., (1) a model that represents the synchronous and asynchronous characteristics between audio and visual features, and makes the best use of a whole database that includes uni-modal, audio only, or visual only data as well as audio-visual data, and (2) the adaptive estimation of reliability weights for the audio and visual information. This paper mainly investigates two issues and proposes a novel method to effectively integrate audio and visual information in an audio-visual Automatic Speech Recognition (ASR) system. First, as the model that integrates audio-visual speech information, we apply a product of hidden Markov models (product HMM), the product of an audio HMM and a visual HMM. We newly propose a method that re-estimates the product HMM using audio-visual synchronous speech data so as to train the synchronicity of the audio-visual information, while the original product HMM assumes independence from audio-visual features. Second, for the optimal audio-visual information reliability weight estimation, we propose a Gaussian mixture model (GMM) based-MCE-GPD (minimum classification error and generalized probabilistic descent) algorithm, which enables reductions in the amount of adaptation data and amount of computations required for the GMM estimation. Evaluation experiments show that the proposed audio-visual speech recognition system improves the recognition accuracy over conventional ones even if the audio signals are clean.