In this paper, we are interested in unsupervised (unknown noise) audio-visual speech enhancement based on variational autoencoders (VAEs), where the probability distribution of clean speech spectra is simulated using an encoder-decoder architecture. The trained generative model (decoder) is then combined with a noise model at test time to estimate the clean speech. In the speech enhancement phase (test time), the initialization of the latent variables, which describe the generative process of clean speech via decoder, is crucial, as the overall inference problem is non-convex. This is usually done by using the output of the trained encoder where the noisy audio and clean visual data are given as input. Current audio-visual VAE models do not provide an effective initialization because the two modalities are tightly coupled (concatenated) in the associated architectures. To overcome this issue, inspired by mixture models, we introduce the mixture of inference networks variational autoencoder (MIN-VAE). Two encoder networks input, respectively, audio and visual data, and the posterior of the latent variables is modeled as a mixture of two Gaussian distributions output from each encoder network. The mixture variable is also latent, and therefore the inference of learning the optimal balance between the audio and visual inference networks is unsupervised as well. By training a shared decoder, the overall network learns to adaptively fuse the two modalities. Moreover, at test time, the visual encoder, which takes (clean) visual data, is used for initialization. A variational inference approach is derived to train the proposed generative model. Thanks to the novel inference procedure and the robust initialization, the proposed MIN-VAE exhibits superior performance on speech enhancement than using the standard audio-only as well as audio-visual counterparts.