Learning the concepts of Minimum Description Length (MDL) is valuable for several reasons, especially for those involved in statistics, machine learning, data science, and related fields. One of the fundamental problems in statistics and data analysis is choosing the best model from a set of potential models. The challenge is to find a model that captures the essential features of the data without overfitting. This is where methods such as MDL, AIC, BIC, etc. comes to rescue. MDL offers a principled way to balance model complexity against the goodness of fit. This is crucial in many areas, such as machine learning and statistical modeling, where overfitting is a common problem. Understanding MDL provides several benefits related to model selection, generalization to new data, strengthening theoretical foundation linked to the core principles of information theory, etc. In this blog, we will learn about concepts of MDL, how it could be applied to solve real-world problems with the help of examples.
The Minimum Description Length (MDL) principle is a method in information theory and statistics for inductive inference, particularly for model selection. It’s based on the idea that the best explanation, or model, for a set of data is the one that minimizes the total length of the description of the model and the data given the model. This principle can be seen as a formalization of Occam’s Razor in the context of model selection.
The following are the core concepts related to MDL:
Let’s understand the concept of description length.
To mathematically explain the concept of Description Length in the Minimum Description Length (MDL) principle, we’ll break it down into its two main components and discuss how each can be quantified.
This component represents the cost of specifying the model. In mathematical terms, it’s often related to the number of parameters in the model or the complexity of the model structure. There are several ways to quantify this:
Mathematically, the model complexity (Cm) might be represented as:
Cm=f(model parameters or structure)
where f is a function that quantifies complexity.
This component is about how well the model compresses or explains the data. It’s often quantified using the likelihood of the data given the model, which can be transformed into a length measure using information-theoretic concepts.
The total description length (DL) in the MDL framework is the sum of these two components:
DL = Cm+Cd
This total length represents the overall cost of both describing the model and encoding the data using that model. The goal in MDL is to minimize this total description length, striking a balance between a simple model (lower Cm) and a good fit to the data (lower Cd).
In this section, we will learn about how to use the Minimum Description Length (MDL) principle for model selection with a Python example. We’ll use a simple dataset and compare two models: a linear regression and a polynomial regression. Our goal will be to select the model that best balances complexity and data fit according to the MDL principle.
We’ll generate a synthetic dataset that follows a quadratic pattern with some noise. Then, we’ll fit two models to this data:
The following steps will be taken for model selection using MDL:
import numpy as np import matplotlib.pyplot as plt from sklearn.linear_model import LinearRegression from sklearn.preprocessing import PolynomialFeatures from sklearn.metrics import mean_squared_error from sklearn.model_selection import train_test_split # Step 1: Generate Synthetic Data np.random.seed(0) X = np.random.rand(100, 1) * 10 # Random data points y = 3 * X**2 + 2 * X + 1 + np.random.randn(100, 1) * 10 # Quadratic equation with noise # Splitting the data into training and testing sets X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0) # Step 2: Fit Models # Linear Regression Model lin_reg = LinearRegression() lin_reg.fit(X_train, y_train) y_pred_lin = lin_reg.predict(X_test) # Polynomial Regression Model poly_features = PolynomialFeatures(degree=2, include_bias=False) X_poly_train = poly_features.fit_transform(X_train) X_poly_test = poly_features.transform(X_test) poly_reg = LinearRegression() poly_reg.fit(X_poly_train, y_train) y_pred_poly = poly_reg.predict(X_poly_test) # Step 3: Calculate MDL for Each Model # Using Mean Squared Error as a proxy for negative log-likelihood # Model A: Linear Regression mdl_lin = 2 + mean_squared_error(y_test, y_pred_lin) # 2 parameters (slope, intercept) # Model B: Polynomial Regression mdl_poly = 3 + mean_squared_error(y_test, y_pred_poly) # 3 parameters (quadratic, linear, intercept) (mdl_lin, mdl_poly)
The following is output by executing the code.
(578.3917388008515, 105.85467801527344)
The above output represents the Minimum Description Length (MDL) values for the two models we considered: the linear regression model (Model A) and the polynomial regression model (Model B). The values are:
The model prediction output can be plotted for better understanding:
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