Question

In this exercise we want to understand a little better the formula $$frac{1}{sum_{i=0}^n frac{1}{sigma_i^2}}$$ for the mean squared error by considering two alternative scenarios. In the first scenario, ? ~ N (0, 1) and we observe X = ? + W, where W ~ N (0, 1) is independent of ?. In the second scenario, the prior information on ? is extremely inaccurate: ? ~ N (0, ?_0^2), where ?_0^2 is so large that it can be treated as infinite. But in this second scenario we obtain two observations of the form X_i = ? + W_i, where the W_i are standard normals, independent of each other and of ?. The mean squared error is smaller in the first scenario. smaller in the second scenario. the same in both scenarios.

          In this exercise we want to understand a little better the formula
$$frac{1}{sum_{i=0}^n frac{1}{sigma_i^2}}$$
for the mean squared error by considering two alternative scenarios.
In the first scenario, ? ~ N (0, 1) and we observe X = ? + W, where W ~ N (0, 1) is independent of ?.
In the second scenario, the prior information on ? is extremely inaccurate: ? ~ N (0, ?_0^2), where ?_0^2 is so
large that it can be treated as infinite. But in this second scenario we obtain two observations of the form
X_i = ? + W_i, where the W_i are standard normals, independent of each other and of ?.
The mean squared error is
smaller in the first scenario.
smaller in the second scenario.
the same in both scenarios.

$In this exercise we want to understand a little better the formula frac1sumi=0^n frac1sigmai^2 for the mean squared error by considering two alternative scenarios. In the first scenario, ? N (0, 1) and we observe X = ? + W, where W N (0, 1) is independent of ?. In the second scenario, the prior information on ? is extremely inaccurate: ? N (0, ?0^2), where ?0^2 is so large that it can be treated as infinite. But in this second scenario we obtain two observations of the form Xi = ? + Wi, where the Wi are standard normals, independent of each other and of ?. The mean squared error is smaller in the first scenario. smaller in the second scenario. the same in both scenarios.$

Added by Antonia B.

Question

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