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You should use validation when fitting predictive models for the following reasons:

1. To reduce overfitting and model bias: Validation techniques, such as cross-validation or hold-out validation, help assess how well your model generalizes to unseen data. By splitting your data into training and validation sets, you can detect if your model is overfitting the training data, which occurs when the model learns noise or specific patterns in the training data that don't generalize well to new data.

2. To make sure your model generalizes well to new data: Validation allows you to estimate how well your model is likely to perform on unseen data, which is essential for making accurate predictions in real-world scenarios. If your model performs well on the validation data, it suggests that it's more likely to perform well on new, unseen data.

3. To help identify the important variables: During the validation process, you can assess the importance of different variables or features in your model. Techniques like feature importance analysis or model evaluation metrics can help you identify which variables have the most significant impact on the model's predictive performance.

Using validation does not directly relate to the following statements:

1. To make sure you have fit the correct model: Validation helps evaluate the performance of your model but doesn't guarantee that you have chosen the "correct" model. Model selection often involves trying different algorithms, hyperparameters, and feature sets, and validation helps you choose the best-performing model among the options you've considered.

2. To make sure your manager can interpret your results: While validation is crucial for ensuring the quality and generalization ability of your model, its primary purpose is not to make the results more interpretable to your manager. Model interpretability may involve other techniques, such as feature explanations or visualization methods, to help stakeholders understand how the model makes predictions.

Therefore, validation is essential for assessing and improving the performance of predictive models by addressing issues like overfitting, generalization, and feature importance. However, it doesn't directly relate to ensuring you've selected the "correct" model or making the results more interpretable to others.

To find the right balance between underfitting and overfitting, you typically use techniques like cross-validation and validation datasets to assess model performance. These techniques help you select a model that generalizes well to unseen data and doesn't underfit or overfit.

In addition to this simplified mathematical description, you can also use more complex metrics like learning curves, bias-variance trade-off analysis, or measures like the mean squared error (MSE) to assess the level of underfitting or overfitting in your models.

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