Diversity Prediction Theorem

An exact decomposition of aggregate squared prediction error into average individual squared error minus prediction variance.

For numeric predictions of the same outcome, the squared error of their average equals average individual squared error minus the variance of those predictions around their average. This is an algebraic identity, often called the Diversity Prediction Theorem. Use the same averaging weights throughout; it does not require independent forecasts.

For example, predictions of 8 and 12 average to 10. If the outcome is 10, average individual squared error is 4, prediction variance is 4, and the crowd’s squared error is zero. But a group need not beat its best member. Adding a wildly inaccurate estimate can increase individual error more than it increases helpful disagreement. Numeric disagreement, relevant information, and demographic diversity are distinct quantities.

When to use it

Useful for numerical forecasting and for understanding why averages can improve on average individual performance.

How it can help

Evaluate both forecast accuracy and differences among forecasts when building an aggregate.