Gaussian linear dynamical systems ( LDS ) provide very efficient learning and inference algorithms , but they can typically only be applied when the observations are themselves linear with Gaussian noise . While it is possible to apply a Gaussian LDS to count vectors ( Belanger and Kakade , 2015 ) , the resulting model is misspecified in the sense that , as a continuous density , the model assigns zero probability to training and test data . However , Belanger and Kakade ( 2015 ) show that this model can still be used for several machine learning tasks with compelling performance , and that the efficient algorithms afforded by the misspecified Gaussian assumptions confer a significant computational advantage . Indeed , the authors have observed that such a Gaussian model is " worth exploring , since multinomial models with softmax link functions prevent closed - form M step updates and require expensive " computations ( Belanger and Kakade , 2014 ) ; this paper aims to help bridge precisely this gap and enable efficient Gaussian LDS computational methods to be applied while maintaining multinomial emissions and an asymptotically unbiased representation of the posteiror . While there are other approximation schemes that effectively extend some of the benefits of LDSs to nonlinear , non - Gaussian settings , such as the extended Kalman filter ( EKF ) and unscented Kalman filter ( UKF ) ( Wan and Van Der Merwe , 2000;Thrun et al . , 2005 ) , these methods do not allow for asymptotically unbiased Bayesian inference and can have complex behavior . Alternatively , particle MCMC ( pMCMC ) ( Andrieu et al . , 2010 ) with ancestor resampling ( Lindsten et al . , 2012 ) is a very powerful algorithm that provides unbiased Bayesian inference for very general state space models , but it does not enjoy the efficient block updates or conjugacy of LDSs or HMMs .