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Suppose my model is reversible in time (e.g. GPS + accelerometers for a vehicle), so that I can run Kalman filter forwards or backwards. Kalman filter, of course, cannot be symmetric, because it is not casual. But will Kalman smoother give the same results when run forwards and backwards (up to numeric errors)? If no, is there a filter that does guarantee this?

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I think that any filter that is symmetric in time needs to be non-causal. –  thang Jan 25 '13 at 10:10
    
@thang: OK, so? Kalman smoother is not casual. –  ybungalobill Jan 25 '13 at 11:01
    
I think that you can build a Kalman filter to be symmetric, but in that case it is almost useless. For example, just use a degenerate case that does nothing. –  thang Jan 25 '13 at 11:02
    
@thang: I'm not talking about Kalman filter, but about Kalman smoother‌​. –  ybungalobill Jan 25 '13 at 11:04
    
I think the answer would still be that in general it is not symmetric, but because it is a designable filter class there are degenerate cases where it is. So that's the first part of your question. The second part is what filter guarantees symmetry. I think that any LTI filter with an impulse response that is symmetric around $t=0$ is symmetric in time. This can be shown by working out the convolution expression with x(-t) and assumption that h(t)=h(-t). –  thang Jan 25 '13 at 15:59
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Running Kalman smoother is not same thing with running Kalman filter backwards in time. They are different. In particular, RTS smoother does not take care of observations, it uses filtered estimates. Kalman filter is an online estimation algorithm and estimates the filtering distribution $p(x_t | y_{1:t})$, i.e., the posterior distribution over the hidden states given the observations up to time $t$. However, smoothing recursions give the posterior of the states given the whole observations: $p(x_t | y_{1:T})$.

Even if your observations are reversible in time (if that is what you imply by saying 'symmetric'), in that case whenever you process the filtering algorithm, the smoother will work backwards and since filtered estimates are incomplete and does not know future observations, they will typically be different.

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