  Unnormalised latent variable models are a broad and flexible class of
statistical models. However, learning their parameters from data is
intractable, and few estimation techniques are currently available for such
models. To increase the number of techniques in our arsenal, we propose
variational noise-contrastive estimation (VNCE), building on NCE which is a
method that only applies to unnormalised models. The core idea is to use a
variational lower bound to the NCE objective function, which can be optimised
in the same fashion as the evidence lower bound (ELBO) in standard variational
inference (VI). We prove that VNCE can be used for both parameter estimation of
unnormalised models and posterior inference of latent variables. The developed
theory shows that VNCE has the same level of generality as standard VI, meaning
that advances made there can be directly imported to the unnormalised setting.
We validate VNCE on toy models and apply it to a realistic problem of
estimating an undirected graphical model from incomplete data.
