I am trying to identify the parameters of a discrete-time nonlinear state space model: \begin{equation} \begin{aligned} x_k & = f(x_{k-1},\theta)+q_{k-1}\\ y_k & = h(x_k,\theta)+r_k \end{aligned} \end{equation}

The MLE is one of the most commonly used estimators in statistics, another choice is to use joint Kalman filter where the unknown system states and parameters are concatenated in a single higher dimensional joint state vector.

Can someone give us an idea of what method should we use based on convergence and consistency?

  • 1
    $\begingroup$ I have never heard of the term "joint Kalman filter." Do you have a reference? $\endgroup$
    – Taylor
    Nov 14, 2017 at 3:14

1 Answer 1


In section 6.2.1 this paper explains the inevitable degeneracy problem you will run into if you try to augment the state vector with the parameter. I assume that is what you mean by "joint Kalman filter." To deal with that, you can add "artificial dynamics" to this parameter part of the process. They mention that this method might "perform satisfactorily in practice," but that

[i]t is difficult to quantify how much bias is introduced in the resulting estimates by the introduction of this artificial dynamics. Additionally, these methods require a significant amount of tuning, for example, choosing the variance of the artificial dynamic noise or the kernel width.

If you're not talking about using particle filtering, then every complaint except the degeneracy one will still remain. Depends on the application, I'd say. Your joint filter will be a lot faster, but who knows if it's giving you the right numbers because it's a completely different model.


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