Blend an unbiased noisy forecast with a biased sharp one

Question

You have two independent forecasts of the same quantity $\mu$ . Forecast A is unbiased but noisy, with variance $\sigma_A^2 = 8$ . Forecast B is sharp but biased: it is off by $b = 2$ on average and has variance $\sigma_B^2 = 1$ . You blend them as $w\,A + (1-w)\,B$ .

What weight $w$ on forecast A minimizes the mean squared error of the blend?

Accepted Answer

The idea. With inverse-variance weighting you would compare only the two variances, but that ignores B's bias. Bias is real error too, so B should be judged by its total error, its variance plus its squared bias, not its variance alone. Effective error of each forecast. Forecast A, being unbiased, has effective error equal to its variance, 8. Forecast B has effective error \sigmaB^2 + b^2 = 1 + 2^2 = 5. Weight inversely to effective error. Treating \beta = \sigmaB^2 + b^2 = 5 as B's error, the MSE-optimal weight on A is wA = \frac{\beta}{\sigmaA^2 + \beta} = \frac{5}{8 + 5} = \frac{5}{13} \ap…

Blend an unbiased noisy forecast with a biased sharp one

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