  Accurate estimation of the states of a nonlinear dynamical system is crucial
for their design, synthesis, and analysis. Particle filters are estimators
constructed by simulating trajectories from a sampling distribution and
averaging them based on their importance weight. For particle filters to be
computationally tractable, it must be feasible to simulate the trajectories by
drawing from the sampling distribution. Simultaneously, these trajectories need
to reflect the reality of the nonlinear dynamical system so that the resulting
estimators are accurate. Thus, the crux of particle filters lies in designing
sampling distributions that are both easy to sample from and lead to accurate
estimators. In this work, we propose to learn the sampling distributions. We
put forward four methods for learning sampling distributions from observed
measurements. Three of the methods are parametric methods in which we learn the
mean and covariance matrix of a multivariate Gaussian distribution; each
methods exploits a different aspect of the data (generic, time structure, graph
structure). The fourth method is a nonparametric alternative in which we
directly learn a transform of a uniform random variable. All four methods are
trained in an unsupervised manner by maximizing the likelihood that the states
may have produced the observed measurements. Our computational experiments
demonstrate that learned sampling distributions exhibit better performance than
designed, minimum-degeneracy sampling distributions.
