  We introduce a framework for inference in general state-space hidden Markov
models (HMMs) under likelihood misspecification. In particular, we leverage the
loss-theoretic perspective of Generalized Bayesian Inference (GBI) to define
generalised filtering recursions in HMMs, that can tackle the problem of
inference under model misspecification. In doing so, we arrive at principled
procedures for robust inference against observation contamination by utilising
the $\beta$-divergence. Operationalising the proposed framework is made
possible via sequential Monte Carlo methods (SMC), where most standard particle
methods, and their associated convergence results, are readily adapted to the
new setting. We apply our approach to object tracking and Gaussian process
regression problems, and observe improved performance over both standard
filtering algorithms and other robust filters.
