The Bayesian Information Criterion, often abbreviated BIC, is a metric that is used to compare the goodness of fit of different regression models.

In practice, we fit several regression models to the same dataset and choose the model with the lowest BIC value as the model that best fits the data.

We use the following formula to calculate BIC:

BIC: (RSS+log(n)dσ̂²) / n

where:

• d: The number of predictors
• n: Total observations
• σ̂: Estimate of the variance of the error associate with each response measurement in a regression model
• RSS: Residual sum of squares of the regression model
• TSS: Total sum of squares of the regression model

To calculate the BIC of several regression models in Python, we can use the statsmodels.regression.linear_model.OLS() function, which has a property called bic that tells us the BIC value for a given model.

The following example shows how to use this function to calculate and interpret the BIC for various regression models in Python.

Example: Calculate BIC of Regression Models in Python

Suppose we would like to fit two different using variables from the mtcars dataset.

First, we’ll load this dataset:

from sklearn.linear_modelimport LinearRegression
import statsmodels.apias sm
import pandas as pd

#define URL where dataset is located
url = "https://raw.githubusercontent.com/Statology/Python-Guides/main/mtcars.csv"

#read in data
data = pd.read_csv(url)

#view head of data
data.head()

        model	          mpg	  cyl	disp  hp    drat wt   qsec  vs am gear carb
0	Mazda RX4	  21.0  6     160.0 110  3.90 2.620 16.46   0  1  4	4
1	Mazda RX4 Wag	  21.0  6     160.0 110  3.90 2.875 17.02   0  1  4	4
2	Datsun 710	  22.8  4     108.0 93   3.85 2.320 18.61   1  1  4	1
3	Hornet 4 Drive	  21.4  6     258.0 110  3.08 3.215 19.44   1  0  3	1
4	Hornet Sportabout 18.7	8     360.0 175  3.15 3.440 17.02   0  0  3	2

Next, we’ll fit the following two regression models:

• Model 1: mpg = β₀ + β₁(disp)+ β₂(qsec)
• Model 2: mpg = β₀ + β₁(disp)+ β₂(wt)

The following code shows how to fit the first model and calculate the BIC:

#define response variable
y = data['mpg']

#define predictor variables
x = data[['disp', 'qsec']]

#add constant to predictor variables
x = sm.add_constant(x)

#fit regression model
model = sm.OLS(y, x).fit()

#view BIC of model
print(model.bic)

174.23905634994506

The BIC of this model turns out to be 174.239.

#define response variable
y = data['mpg']

#define predictor variables
x = data[['disp', 'wt']]

#add constant to predictor variables
x = sm.add_constant(x)

#fit regression model
model = sm.OLS(y, x).fit()

#view BIC of model
print(model.bic)

166.56499196301334

The BIC of this model turns out to be 166.565.

Since the second model has a lower BIC value, it is the better fitting model. 

Once we’ve identified this model as the best, we can proceed to fit the model and analyze the results including the R-squared value and the beta coefficients to determine the exact relationship between the set of predictor variables and the .

Two other metrics that are commonly used to compare the fit of regression models are AIC and adjusted R-squared.

The following tutorials explain how to calculate each of these metrics for regression models in Python: