Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

  • Tian-cheng Li
  • Jin-ya Su
  • Wei Liu
  • Juan M. Corchado


Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity.

Key words

Kalman filter Gaussian filter Time series estimation Bayesian filtering Nonlinear filtering Constrained filtering Gaussian mixture Maneuver Unknown inputs 

CLC number



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T. Li would like to acknowledge Prof. Yu-chi (Larry) Ho with Harvard University for his high patience and generous encouragement shown in repeated discussion and comments on the topics involved in Sections 3.2 and 7.2 of this paper in the last several years since 2013.


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© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of SciencesUniversity of SalamancaSalamancaSpain
  2. 2.Department of Aeronautical and Automotive EngineeringLoughborough UniversityLoughboroughUK
  3. 3.Department of Electronic and Electrical EngineeringUniversity of SheffieldSheffieldUK

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