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A signal that decays

Expected valueMedium

A research signal has a hidden strength that halves each day: $x_{t} = 0.5 x_{t - 1} + w_{t}$ with $Var (w_{t}) = 3$ . Today's reading is $y_{t} = x_{t} + v_{t}$ with $Var (v_{t}) = 4$ . Yesterday's estimate was $10$ with variance $P = 4$ , and today's reading is $9$ . What is the expected strength today, given the reading? Give a number.

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Hint

Both the mean and the variance must be rolled through the decay before the reading comes in.

Solution

The predicted mean is $0.5 \times 10 = 5$ and the predicted variance is $0.5 \times 0.5 \times 4 + 3 = 4$ .
The gain is $\frac{4}{4 + 4} = \frac{1}{2}$ .
The estimate is $5 + \frac{1}{2} \times (9 - 5) = 7$ .

Answer: $7$

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