Blending two noisy models into one estimate

Question

Two independent models estimate an asset's fair value. Model A says $50$ with a standard deviation of $3$ on its estimate; Model B says $56$ with a standard deviation of $2$ .

What single estimate should you use, and what is the standard deviation of that combined estimate?

Accepted Answer

When two independent estimates disagree, you do not average them equally, you weight each by its precision, which is one over its variance. A sharper estimate (smaller variance) deserves more say: \hat\mu = \frac{XA/\sigmaA^2 + XB/\sigmaB^2}{1/\sigmaA^2 + 1/\sigmaB^2}. With XA = 50,\ \sigmaA = 3 (variance 9, precision 1/9) and XB = 56,\ \sigmaB = 2 (variance 4, precision 1/4): \hat\mu = \frac{50/9 + 56/4}{1/9 + 1/4} = \frac{5.556 + 14}{0.1111 + 0.25} = \frac{19.556}{0.3611} \approx 54.15. The combined estimate is 54.15, pulled toward Model B because B is more precise (1/4 vs 1/9, so B carries…

Blending two noisy models into one estimate

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