What is the relationship between model flexibility and test mean squared error (MSE)?

The relationship between model flexibility and test mean squared error (MSE) is nuanced. As model flexibility increases, it typically allows the model to better fit the training data, capturing its underlying structure more precisely. However, flexibility also runs the risk of overfitting, where the model learns not just the true signal but also the noise present in the training data.

When a model is too flexible, it generally results in lower training MSE as it can respond to nearly every nuance of the training data. However, when evaluating the model’s performance on unseen data (the test set), this overly flexible model may perform poorly because it has adapted too closely to the training data, leading to an increase in the test MSE. In contrast, a well-fitted model that balances flexibility and complexity tends to have a lower test MSE, as it captures essential patterns without being overly responsive to noise.

Ultimately, a moderate level of flexibility tends to minimize test MSE, which is why it is understood that model flexibility and test MSE are intricately related, but the relationship implies that test MSE tends to decrease to a point with increased flexibility before potentially increasing again as the risk of overfitting becomes significant. Thus, while flexibility may initially lower MSE