Bias and variance are two important concepts in machine learning that describe the sources of error in a model”s predictions. Bias refers to the error that results from oversimplifying the underlying relationship between the input features and the output variable, while variance refers to the error that results from being too sensitive to fluctuations in the training data.

In machine learning, we strive to minimize both bias and variance in order to build a model that can accurately predict on unseen data. A model with high bias may be too simplistic and underfit the training data, while a model with high variance may overfit the training data and fail to generalize to new data.

Example

Below is an implementation example in Python that illustrates how bias and variance can be analyzed using the Boston Housing dataset −

import numpy as np
import pandas as pd
from sklearn.datasets import load_boston

boston = load_boston()
X = boston.data
y = boston.target
from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2,
random_state=42)
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error

lr = LinearRegression()
lr.fit(X_train, y_train)

train_preds = lr.predict(X_train)
train_mse = mean_squared_error(y_train, train_preds)
print("Training MSE:", train_mse)

test_preds = lr.predict(X_test)
test_mse = mean_squared_error(y_test, test_preds)
print("Testing MSE:", test_mse)

Output

The output shows the training and testing mean squared errors (MSE) of the linear regression model. The training MSE is 21.64 and the testing MSE is 24.29, indicating that the model has a moderate level of bias and variance.

Training MSE: 21.641412753226312
Testing MSE: 24.291119474973456

Reducing Bias and Variance

To reduce bias, we can use more complex models that can capture non-linear relationships in the data.

Example

Let”s try a polynomial regression model −

from sklearn.preprocessing import PolynomialFeatures

poly = PolynomialFeatures(degree=2)
X_train_poly = poly.fit_transform(X_train)
X_test_poly = poly.transform(X_test)

pr = LinearRegression()
pr.fit(X_train_poly, y_train)

train_preds = pr.predict(X_train_poly)
train_mse = mean_squared_error(y_train, train_preds)
print("Training MSE:", train_mse)

test_preds = pr.predict(X_test_poly)
test_mse = mean_squared_error(y_test, test_preds)
print("Testing MSE:", test_mse)

Output

The output shows the training and testing MSE of the polynomial regression model with degree=2. The training MSE is 5.31 and the testing MSE is 14.18, indicating that the model has a lower bias but higher variance compared to the linear regression model.

Training MSE: 5.31446956670908
Testing MSE: 14.183558207567042

Example

To reduce variance, we can use regularization techniques such as ridge regression or lasso regression. In the following example, we will be using ridge regression −

from sklearn.linear_model import Ridge

ridge = Ridge(alpha=1)
ridge.fit(X_train_poly, y_train)

train_preds = ridge.predict(X_train_poly)
train_mse = mean_squared_error(y_train, train_preds)
print("Training MSE:", train_mse)

test_preds = ridge.predict(X_test_poly)
test_mse = mean_squared_error(y_test, test_preds)
print("Testing MSE:", test_mse)

Output

The output shows the training and testing MSE of the ridge regression model with alpha=1. The training MSE is 9.03 and the testing MSE is 13.88 compared to the polynomial regression model, indicating that the model has a lower variance but slightly higher bias.

Training MSE: 9.03220937860839
Testing MSE: 13.882093755326755

Example

We can further tune the hyperparameter alpha to find the optimal balance between bias and variance. Let”s see an example −

from sklearn.model_selection import GridSearchCV

param_grid = {''alpha'': np.logspace(-3, 3, 7)}
ridge_cv = GridSearchCV(Ridge(), param_grid, cv=5)
ridge_cv.fit(X_train_poly, y_train)

train_preds = ridge_cv.predict(X_train_poly)
train_mse = mean_squared_error(y_train, train_preds)
print("Training MSE:", train_mse)

test_preds = ridge_cv.predict(X_test_poly)
test_mse = mean_squared_error(y_test, test_preds)
print("Testing MSE:", test_mse)

Output

The output shows the training and testing MSE of the ridge regression model with the optimal alpha value.

Training MSE: 8.326082686584716
Testing MSE: 12.873907256619141

The training MSE is 8.32 and the testing MSE is 12.87, indicating that the model has a good balance between bias and variance.