The evidence lower bound (ELBO) appears in many algorithms for maximum likelihood estimation (MLE) with latent variables because it is a sharp lower bound of the marginal log-likelihood. For neural latent variable models, optimizing the ELBO jointly in the variational posterior and model parameters produces state-of-the-art results. Inspired by the success of the ELBO as a surrogate MLE objective, we consider the extension of the ELBO to a family of lower bounds defined by a Monte Carlo estimator of the marginal likelihood. We show that the tightness of such bounds is asymptotically related to the variance of the underlying estimator. We introduce a special case, the filtering variational objectives (FIVOs), which takes the same arguments as the ELBO and passes them through a particle filter to form a tighter bound. FIVOs can be optimized tractably with stochastic gradients, and are particularly suited to MLE in sequential latent variable models. In standard sequential generative modeling tasks we present uniform improvements over models trained with ELBO, including some whole nat-per-timestep improvements.