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Bias vs variance
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**Bias vs Variance**
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The bias-variance tradeoff is a fundamental concept in machine learning that refers to the balance between two types of errors: **bias** and **variance**.

### Bias

*   The error caused by **bias** measures how far off the predictions of a model are from the true value. It represents the difference between the predicted mean of the model and the correct value.
*   A high bias indicates that the model is underfitting, meaning it's too simple to capture the underlying patterns in the data.

### Variance

*   The error due to **variance** measures how much the predictions of a model vary for a given point. It reflects the impact of randomness in the data on the prediction results.
*   A high variance indicates that the model is overfitting, meaning it's too complex and fits the noise in the training data.

### Diagrammatic Representation

The bias-variance tradeoff can be visualized using a **target diagram**, where the center represents a perfect model that predicts the true values.

### Mathematical Definition

The prediction error at point x can be decomposed into bias, variance, and irreducible error:

$$\text{Err}(x) = \left(E[\hat{f}(x)] - f(x)\right)^2 + E\left[(\hat{f}(x) - E[\hat{f}(x)])^2\right] + \sigma^2_\epsilon$$

### Related Concepts

*   **Optimism in the Face of Uncertainty - Upper Confidence Bound (UCB)**: Discusses balancing exploration and exploitation in a multi-armed bandit setting, related to bias-variance tradeoff.
*   **Gaussian Filtering**: Describes linear filtering using a Gaussian kernel, which can smooth out noise and reduce variance.
*   **Logistic Regression**: Introduces maximum likelihood estimation (MLE) for logistic regression, related to bias-variance tradeoff in model selection.

These concepts can be used to discuss the bias-variance tradeoff in more detail.
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