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In a Kalman Filter, assume we have the state equation as:

  • $x_t=F_{t}x_{t-1}+e_t$, $e_t \sim N(0,V_t))$

  • $y_t=H_t x_t+w_t$, $w_t \sim N(0,W_t))$.

The measurement equation tells us how the observations at time t are related to the underlying state at time t. Assume that we are at time t: are the observations at time t-1 considered constants or are they random variables? I imagine that they may be random variables given that even though we have already observed them, we can never be sure how far they were from the true underlying state given that we never truly observe the underlying state. Is this correct?

I think I answered my own question: I think the above position is incorrect because at each time t the state model does take a value which is determined by our model. Therefore there is no variation in the time t-1 measurement vs state because both take on values determined by the model.

Am I right?

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They're considered constants.

Every step of the Kalman filter has two sub-steps: predict, then update.

If you've predicted, but haven't updated on the most recent data point $y_t$, then you have the mean and variance of $X_t$, given all of your avaialable information $y_1,\ldots,y_{t-1}$. All of this available can be considered as a constant, because you are conditioning on them. You have $E[X_t \vert y_1,\ldots,y_{t-1}]$ and $\operatorname{Var}[X_t \vert y_1,\ldots,y_{t-1}]$.

If you've predicted *and* updated, you have more available information, and you have the same random variable of interest. So you would have $E[X_t \vert y_1,\ldots,y_{t-1},y_t]$ and $\operatorname{Var}[X_t \vert y_1,\ldots,y_{t-1},y_t]$.

we can never be sure how far they were from the true underlying state given that we never truly observe the underlying state.

Yes, that's true, but you do have certain probability distributions, along with their parameters, that define the model you are assuming is true to run a Kalman filter.

I think the above position is incorrect because at each time t the state model does take a value which is determined by our model. Therefore there is no variation in the time t-1 measurement vs state because both take on values determined by the model.

The model you are assuming is true only assumes the distributions that describe how states move through time, and how they lead to observations. However, this model assumes that you have *not* observed any state variables, ever. I'm not sure I understand that last sentence of yours that I quoted you on. But hopefully this helps.

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