Bias-Variance Decomposition

Estimate the bias and variance of a model from the predictions of several models (for example trained on different bootstrap samples) on the same observations. predictions has shape (n_models, n_observations) and y holds the true targets.

Implement solve(predictions, y).

Returns. Return a tuple (squared_bias, variance). Squared bias is the mean over observations of $(\bar{f}(x)-y{)}^2(\backslash bar\; f(x)-y)^2$, where $\bar{f}\backslash bar\; f$ is the average prediction across models. Variance is the mean over observations of the (population, ddof=0) variance of the predictions across models.

Examples

Example 1

Input:

solve([[2, 4], [4, 6]], [3, 5])

Output:

(0.0, 1.0)

Example 2

Input:

solve([[1.0, 2.0], [1.0, 2.0], [1.0, 2.0]], [0.0, 0.0])

Output:

(2.5, 0.0)