The variance floor of an infinite ensemble

Question

You average more and more model predictions, each with variance $\sigma^2 = 80$ and pairwise correlation $\rho = 0.25$ .

As the number of models grows without bound, what variance does the average approach?

Accepted Answer

The idea. Averaging shrinks only the private, uncorrelated part of the variance. The shared part, driven by correlation, stays no matter how many models you add. So there is a hard floor. Take the limit. Starting from 	ext{Var}(	ext{avg}) = ho\sigma^2 + \frac{1 - ho}{B}\sigma^2, let B 	o \infty. The second term vanishes, leaving 	ext{Var}(	ext{avg}) 	o ho\sigma^2 = 0.25 	imes 80 = 20. No matter how many correlated models you stack, the ensemble variance cannot drop below 20. This is a crucial and often-missed limit: buying more models has sharply diminishing returns once the priv…

The variance floor of an infinite ensemble

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