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Where the filter settles

StatisticsHard

A Kalman filter tracks $x_{t} = x_{t - 1} + w_{t}$ from readings $y_{t} = x_{t} + v_{t}$ , with $Var (w_{t}) = 1$ and $Var (v_{t}) = 6$ . Once the variances stop changing, what is the posterior variance $P$ after each reading? Give a number.

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Hint

At the fixed point $P^{-} = P + Q$ and $P = (1 - K) P^{-}$ both hold, which gives $Q = K P^{-}$ .

Solution

With $\frac{R}{Q} = \frac{1 - K}{K ^{2}}$ and $\frac{R}{Q} = 6$ , we get $6 K^{2} + K - 1 = 0$ , whose positive root is $K = \frac{1}{3}$ .
Then $P^{-} = \frac{Q}{K} = \frac{1}{\frac{1}{3}} = 3$ .
Finally $P = (1 - \frac{1}{3}) \times 3 = 2$ .

Answer: $2$

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