Fair and unbiased machine learning is an important and active field of research, as decision processes are increasingly driven by models that learn from data. Unfortunately, any biases present in the data may be learned by the model, thereby inappropriately transferring that bias into the decision making process. We identify the connection between the task of bias reduction and that of isolating factors common between domains whilst encouraging domain specific invariance. To isolate the common factors we combine the theory of deep latent variable models with information bottleneck theory for scenarios whereby data may be naturally paired across domains and no additional supervision is required. The result is the Nested Variational AutoEncoder (NestedVAE). Two outer VAEs with shared weights attempt to reconstruct the input and infer a latent space, whilst a nested VAE attempts to reconstruct the latent representation of one image, from the latent representation of its paired image. In so doing, the nested VAE isolates the common latent factors/causes and becomes invariant to unwanted factors that are not shared between paired images. We also propose a new metric to provide a balanced method of evaluating consistency and classifier performance across domains which we refer to as the Adjusted Parity metric. An evaluation of NestedVAE on both domain and attribute invariance, change detection, and learning common factors for the prediction of biological sex demonstrates that NestedVAE significantly outperforms alternative methods.