Outline:

Particle Filters are a class of methods which allow us to conduct approximate Bayesian inference in time series models, online, as observations arise. They have been extensively studied and used since their introduction in 1993 in the seminal paper of Gordon, Salmond and Smith (1993). However, there remain interesting theoretical questions and there is scope for further methodological development to further increase the class of problems which can be satisfactorily addressed using these methods. For example, the iterative scheme of Guarniero, Johansen and Lee (2016+) could be readily adapted to such online settings and the question of how to specify good proposal distributions
in the context of block sampling methods (Doucet, Briers and Sénécal, 2006) remains largely unanswered.

References:

1. A. Doucet, M. Briers, and S. Sénécal (2006). “Efficient Block Sampling Strategies for Sequential Monte Carlo Methods”. Journal of Computational and Graphical Statistics 15.3, pp. 693–711.
2. N. J. Gordon, S. J. Salmond, and A. F. M. Smith (1993). “Novel approach to nonlinear/non-Gaussian Bayesian state estimation”. IEE Proceedings-F 140.2, pp. 107–113.
3. P. Guarniero, A. M. Johansen and A. Lee (2016+). The Iterated Auxiliary Particle Filter. To appear in Journal of the American Statistical Association.