How I Started Viewing Lasso and Ridge Regression Through a Bayesian Lens 🤓📊

Recently, I revisited the concepts of Lasso and Ridge regression and discovered their intriguing connection to Bayesian principles. While both are frequentist regularization techniques, they serve distinct purposes:

• Ridge Regression minimizes the usual sum of squared errors but adds an L2 penalty ∑β^2 to shrink coefficients. It’s great for handling multicollinearity 🌀 and avoids overfitting by reducing the magnitude of all coefficients. 📉
• Lasso Regression does something similar but uses an L1 penalty ∑∣β∣, which not only shrinks coefficients but can also set some to exactly zero, effectively performing feature selection. ✂️

The fascinating part is how these penalties align with Bayesian priors:

• Ridge Regression corresponds to a Gaussian prior (β∼Normal(0,σ2)). It assumes coefficients are small and shrinks them toward zero without eliminating them. 🎯
• Lasso Regression aligns with a Laplace prior (β∼Laplace(0,b)), which places a sharper peak at zero and encourages sparsity by shrinking some coefficients to exactly zero. 🔍

The real "aha" moment 🤯 is realizing that these frequentist penalties correspond to Bayesian priors, and solving Ridge or Lasso is like finding the Maximum a Posteriori (MAP) estimate—the most likely coefficients given the data and our prior beliefs. This connection makes regularization more than just a mathematical tool; it’s a way to encode assumptions about your model. 🛠️

It’s always fascinating to uncover these overlaps between statistical paradigms and see the shared ideas behind the math we use every day. 🚀✨