Bayesian Linear Regression

A Bayesian linear regression is an approach to linear regression in which the statistical analysis is undertaken within the context of Bayesian inference. Wikipedia: Bayesian linear regression

Key Statements
# Inputs: x_train, y_train, x_test, y_test.

# Fit the model.
from sklearn.linear_model import BayesianRidge
model = BayesianRidge().fit(x_train, y_train)

# Get predictions.
y_predict = model.predict(x_test)

# Get the results.
from sklearn.metrics import mean_squared_error, r2_score
mean_squared_error = mean_squared_error(y_test, y_predict)
r2_score = r2_score(y_test, y_predict)
Working End-to-End Example
# Step 1: Import the libraries.
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

import numpy as np
import pandas as pd
from sklearn.linear_model import BayesianRidge
from sklearn.metrics import mean_squared_error, r2_score
from sklearn.model_selection import train_test_split

# Step 2: Set up the constants.
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

# The target feature is the price at which the house sold.

# We'll set aside 20% of the data to test the model.

# There are some columns we won't use in this model.
FEATURES_TO_REMOVE = ['id', 'date']

# We need to know which features are categorical.
CATEGORICAL_FEATURES = ['waterfront', 'condition', 'zipcode']

# Step 3: Load in the raw data.
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

# This assumes the data is in the same directory as this script.
# Here we load the data into a pandas DataFrame.
raw_data = pd.read_csv('kc_house_data.csv')

# It's helpful to take a quick look at the data.
print('Sample of loaded data:')

target_data = raw_data[TARGET_FEATURE]
for percent in [25, 50, 75]:
    label = 'Target Feature, %d%% Percentile:' % percent
    print(label, np.percentile(target_data, percent))

# Step 4: Set up the data.
# ~~~~~~~~~~~~~~~~~~~~~~~~

# Separate the X and Y values.
y_data = raw_data[TARGET_FEATURE]

# Using drop() doesn't change raw_data, only the return value.
# The axis=1 keyword tells pandas to drop a column (not a row).
x_data = raw_data.drop(TARGET_FEATURE, axis=1)

# Remove the unused features.
x_data = x_data.drop(FEATURES_TO_REMOVE, axis=1)

# To include an intercept, add a new column with a constant.
x_data['intercept'] = 1.0

# Turn categorical variables into dummy columns (0 or 1 values).
# Do this to avoid assuming a meaningful order of categories.
# Use drop_first to avoid multicollinearity among features.
x_data = pd.get_dummies(

# It's helpful to double check that the final data looks good.
print('Sample of data to use:')

# Split the data into training and test sets.
x_train, x_test, y_train, y_test = train_test_split(

# Step 5: Fit the model.
# ~~~~~~~~~~~~~~~~~~~~~~

model = BayesianRidge().fit(x_train, y_train)

# Yes, that's it!

# Step 6: Get the results.
# ~~~~~~~~~~~~~~~~~~~~~~~~

# Get predictions for the test set.
y_predict = model.predict(x_test)

mean_squared_error = mean_squared_error(y_test, y_predict)
print('Mean Squared Error: %.2f' % mean_squared_error)

r2_score = r2_score(y_test, y_predict)
print('R2 Score: %.2f' % r2_score)

# As a custom metric, we're curious to check how many of the
# predictions were within 20% of the true prediction.
percent_diff = np.abs(y_predict - y_test) / y_test
result = (percent_diff < 0.2).sum() / len(y_test)
print('Percent within 20%% of target value: %.2f' % result)


This classifier uses L2 regularization ("ridge regression") to avoid overfitting; it chooses the regularization coefficient based on the data itself.

The scikit-learn implementation uses priors with gamma distributions, which makes the model a Bayesian ridge regression.