How does the performance of bagging depend on instability? This question relates to Leo Breiman's paper: Bagging Predictors from 1996.
Assuming that $\mathcal{L}$ denotes the training set and $\phi$ the predictor which depends on the training set and the instance vector $\phi(\mathbf{x}, \mathcal{L})$, the author concludes after simple algebra that
$$\left( E_{\mathcal{L}}\phi(\mathbf{x}, \mathcal{L})\right)^2 \leq E_{\mathcal{L}}\phi^2(\mathbf{x}, \mathcal{L})$$
The author states that if $\phi(\mathbf{x}, \mathcal{L})$ does not change much with $\mathcal{L}$, both sides of the equation will be nearly equal and aggregation will not help.
Can someone explain why bagging improves with increased variability of the predictor $\phi$ on the training set based on the equation above?

In case someone is interested, we have to remember that the stated inequality becomes equality iff $\phi(\mathbf{x}, \mathcal{L})$ is a constant. This is only the case if the predictor does not change with the change of the underlying training set $\mathcal{L}$ during bootstrapping.
Hence the more variability in the prediction, the more improvement is to be expected with bagging.
