Learning meaningful representations of data that can address challenges such as batch effect correction and counterfactual inference is a central problem in many domains including computational biology. Adopting a Conditional VAE framework, we show that marginal independence between the representation and a condition variable plays a key role in both of these challenges. We propose the Contrastive Mixture of Posteriors (CoMP) method that uses a novel misalignment penalty defined in terms of mixtures of the variational posteriors to enforce this independence in latent space. We show that CoMP has attractive theoretical properties compared to previous approaches and we prove counterfactual identifiability of CoMP under additional assumptions. We demonstrate state of the art performance on a set of challenging tasks including aligning human tumour samples with cancer cell-lines, predicting transcriptome-level perturbation responses, and batch correction on single-cell RNA sequencing data. We also find parallels to fair representation learning and demonstrate that CoMP is competitive on a common task in the field.
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