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We can solve the Riccati equation and get the steady state priors for a Kalman filter.

So why are people still allowed to have any prior they want? They can solve Riccati equation ex-ante and use the steady state prior as the initial value, and reach steady state at time 0 rather than time infinity. And this is the only 'right' prior any rational individual would use.

I understand this is a philosophical question, any comment is welcome.

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Solving the Riccati equation gives us the steady-state covariance matrix, which is only part of the prior: you have the mean of the state vector as well.

What the prior should reflect is our initial information about the value of the state vector, which in general will be different than our information after reaching the steady state.

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  • $\begingroup$ Let me rephrase my question, why not use steady state covariance matrix as initial prior for any mean of estimate? Is there anything wrong with that, I am allowed to have any prior right, and I get the benefit of not changing the covariance matrix from time to time since it's already steady state. $\endgroup$ – Zwingli Tannenbaum Nov 12 '12 at 13:35
  • $\begingroup$ You can do that but the question is: Will that reflect your a priori knowledge? Likely not, and this makes it hard to justify from a Bayesian viewpoint. $\endgroup$ – F. Tusell Nov 12 '12 at 19:18
  • $\begingroup$ can you please tell me what is the bayesian viewpoint? $\endgroup$ – Zwingli Tannenbaum Nov 13 '12 at 18:18
  • $\begingroup$ By the Bayesian viewpoint I refer to the fact that, under normality, the Kalman filter can be seen as a recipe to update the distribution of the state; you start with a prior distribution, reflecting your knowledge (or a difuse prior, if you know nothing) and update that distribution each time you process a new observation. $\endgroup$ – F. Tusell Nov 19 '12 at 21:08
  • $\begingroup$ sorry let me rephrase again. I think it's wrong, for the variance component, to use anything other than steady state value, because you can calculate the steady state variance at time 0, and that is the only reasonable prior anyone should have on how much the estimated latent state variable spreads(variance) $\endgroup$ – Zwingli Tannenbaum Dec 6 '12 at 4:35

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