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Following this not-very-formal-discussion here, a question raised in my head: is Kalman filter originally a frequentist or a bayesian tool?

I know that many statistical tools can be interpreted from both a frequentist and bayesian standpoint and Kalman filter is one of them, but since I have mostly seen it applied in Bayesian context (maybe because a recursive approach is more immediate in bayesian, by update of the prior as new info comes along), I was wondering if it has been thought by a bayesian or if it has just been "imported" from classical statistics.

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    $\begingroup$ I always saw it as a derivative version of the Weiner filter or Wiener-Kolmogorov filter. $\endgroup$ Commented Dec 1, 2016 at 17:50
  • $\begingroup$ I wouldn't say it is inherently, or "originally" either Bayesian or Frequentist. How you interpret probability has no bearing on whether the Kalman filtering is the right tool for a given problem. The Kalman filter can be thought of as tracking a latent (unobserved) trajectory based on noisy data, and there is no reason that a Frequentist cannot model the unobserved trajectory as a random entity. It would essentially be treating the trajectory as a random effect; conceptually, a Frequentist could talk about a population of random trajectories that they model as a Gaussian process. $\endgroup$
    – guy
    Commented Aug 28, 2018 at 18:28
  • $\begingroup$ (continued...) To me, considering the Kalman filter as being more naturally Bayesian or Frequentist falls in the same line of misconceptions as stating that every method that uses Bayes theorem is Bayesian. Bayes vs Frequentist methods are centered on how we interpret probability; the Kalman filter is a valid tool for computing conditional probabilities, irrespective of your philosophy. $\endgroup$
    – guy
    Commented Aug 28, 2018 at 18:31

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Kalman filter is the analytical implementation of Bayesian filtering recursions for linear Gaussian state space models. For this model class the filtering density can be tracked in terms of finite-dimensional sufficient statistics which do not grow in time$^*$. So I would say that it is pretty Bayesian and as you stated it is considered in Bayesian context in general.

However, the origins of Kalman filtering can be traced up to Gauss. He invented recursive least squares for prediction of orbits (Gauss, C. F. Theory of the Combination of Observations Least Subject to Errors (translated by G. W. Stewart). Philadelphia: SIAM Publishers, 1995.) which I assume can be considered frequentist or classical in some sense.


$^*$(btw other exact finite-dimensional nonlinear filters exist like Benes, Daum filters but there is no Fisher-Koopman-Darmois-Pitman theorem for filtering). For general models your best bet is sequential Monte Carlo.

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    $\begingroup$ Summary: the origins of Kalman filtering can be traced up to Gauss <...> [and] can be considered frequentist or classical in some sense. $\endgroup$ Commented Aug 28, 2018 at 18:47

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