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The learning problem for Factorial Hidden Markov Models with discrete and multi-variate latent variables remains a challenge. Inference of the latent variables required for the E-step of Expectation Minimization algorithms is usually computationally intractable. In this paper we propose a variational learning algorithm mimicking the Baum-Welch algorithm. By approximating the filtering distribution with a variational distribution parameterized by a recurrent neural network, the computational complexity of the learning problem as a function of the number of hidden states can be reduced to quasilinear instead of quadratic time as required by traditional algorithms such as Baum-Welch whilst making minimal independence assumptions. We evaluate the performance of the resulting algorithm, which we call Variational BOLT, in the context of unsupervised end-to-end energy disaggregation. We conduct experiments on the publicly available REDD dataset and show competitive results when compared with a supervised inference approach and state-of-the-art results in an unsupervised setting.