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According to the book Probabilistic Robotics, "the Markov assumption postulates that past and future data are independent if one knows the current state $x_t$." This is a central assumption in many localization methods, where it's invoked to allow the current state of the robot to be accepted as a complete summary of the past. It is generally accepted in this context that the assumption is only an approximation, which doesn't really hold in practice — but the robustness of Bayes filters can be counted upon to accommodate for that.

My question is, does that robustness extend beyond localization relative to a map, and into odometry estimation itself? In other words, is it "alright enough" to assume that, given an odometry estimate $x_t$, the previous estimate $x_{t-1}$ and the next estimate $x_{t+1}$ are conditionally independent from each other? I would love to find any academic references arguing this point either way.

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