How many models to hit a variance target?

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Each model prediction has variance \sigma^2 = 100 and pairwise correlation \rho = 0.3. You want the averaged prediction's variance to be at most 40. What is the smallest number of models B that achieves this?

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The idea. Averaging drives the variance down toward its floor ho\sigma^2 = 30, which is below your target of 40, so the target is reachable. Solve for how many models it takes. Set up the inequality. ho\sigma^2 + \frac{1 - ho}{B}\sigma^2 \le 40 \implies 30 + \frac{0.7 	imes 100}{B} \le 40 \implies \frac{70}{B} \le 10. Solve. B \ge 7. So you need at least 7 models. Check: at B = 7 the variance is 30 + 70/7 = 30 + 10 = 40, exactly the target; at B = 6 it is 30 + 70/6 \approx 41.7, just over. Notice the floor is 30: if your target had been below 30, no finite number of these correlated mo…

How many models to hit a variance target?

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