Conditions for Equivalence of MAP and Ridge Regression Estimates
1. Introduction
We'll explore the conditions under which Maximum A Posteriori (MAP) estimates in Bayesian inference are equivalent to Ridge Regression estimates in frequentist statistics.
2. Ridge Regression
Ridge Regression minimizes the following objective function:
\[
L(\beta) = \|y - X\beta\|^2 + \lambda\|\beta\|^2
\]
The Ridge estimate is given by:
\[
\hat{\beta}_{Ridge} = (X'X + \lambda I)^{-1}X'y
\]
where λ > 0 is the regularization parameter.
3. Bayesian Framework
In the Bayesian framework, we seek to maximize the posterior probability:
\[
P(\beta|y) \propto P(y|\beta)P(\beta)
\]
Where P(β|y) is the posterior, P(y|β) is the likelihood, and P(β) is the prior.
4. Conditions for Equivalence
MAP estimates are equal to Ridge Regression estimates under the following conditions:
4.1 Likelihood
The likelihood is Gaussian:
\[
P(y|\beta) \propto \exp\left(-\frac{1}{2\sigma^2}\|y - X\beta\|^2\right)
\]
This assumes that the errors are normally distributed with variance σ².
4.2 Prior
The prior on β is also Gaussian:
\[
P(\beta) \propto \exp\left(-\frac{\lambda}{2\sigma^2}\|\beta\|^2\right)
\]
This is equivalent to assuming β ~ N(0, σ²/λ · I).
4.3 Relationship between λ and prior variance
The regularization parameter λ in Ridge Regression corresponds to the ratio of error variance to prior variance in the Bayesian setting:
\[
\lambda = \frac{\sigma^2}{\tau^2}
\]
where τ² is the variance of the prior distribution on β.
5. Proof of Equivalence
Under these conditions, the posterior distribution is:
\[
P(\beta|y) \propto \exp\left(-\frac{1}{2\sigma^2}(\|y - X\beta\|^2 + \lambda\|\beta\|^2)\right)
\]
The MAP estimate maximizes this posterior, which is equivalent to minimizing:
\[
\|y - X\beta\|^2 + \lambda\|\beta\|^2
\]
This is exactly the Ridge Regression objective function.
6. Implications and Interpretations
- Ridge Regression can be interpreted as a Bayesian point estimate with a specific Gaussian prior on the coefficients.
- The regularization in Ridge Regression is equivalent to imposing a prior belief that the coefficients are centered around zero.
- The strength of regularization (λ) in Ridge Regression corresponds to the precision of the prior in the Bayesian setting.
- This equivalence provides a Bayesian justification for Ridge Regression and allows for Bayesian interpretation of its results.
7. Limitations
- The equivalence holds only for Gaussian likelihoods and priors.
- In practice, the error variance σ² is often unknown and needs to be estimated, which can complicate the exact equivalence.
- The assumption of a zero-centered Gaussian prior may not always be appropriate for all problems.