Which of the following increases monotonically as flexibility increases in statistical learning?

In statistical learning, as flexibility of a model increases, the model's ability to capture the training data's variability also increases. This often leads to a corresponding increase in variance. High flexibility allows the model to fit the noise and idiosyncrasies in the training data, which can lead to varying predictions for different datasets drawn from the same distribution.

Variance reflects how much the model's predictions would change if it were trained on a different dataset. Therefore, a more flexible model, capable of fitting a wider variety of patterns, will tend to show higher variance as it adapts closely to the specific training data.

In contrast, as flexibility increases, training mean squared error (MSE) typically decreases because a more flexible model can better minimize errors on the training set. Test MSE, however, does not necessarily follow a consistent pattern; it may initially decrease and then increase due to overfitting as flexibility increases. Squared bias, which describes the error due to assumptions made by the model, tends to decrease as flexibility increases.

Thus, the relationship between model flexibility and variance is well established, establishing that variance increases monotonically as flexibility increases.