Last updated: 2025-03-05

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Knit directory: fiveMinuteStats/analysis/

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Prerequisites

Prerequisites will go here.

  • MCMC sampling (and specifically Gibbs sampling) will be used to illustrate some of the key ideas so you should be familiar with those.

Introduction

Add introduction here.

Ridge regression

We will use the ridge regression model as our running example to illustrate the use of variational inference to perform posterior inferences. Although variational inference methods aren’t really needed because the math works out very well for this model, its convenient mathematical properties will be helpful for understanding the variational approximations since we can compare the approximations to the exact calculations.

Although you may have seen the ridge regression model elsewhere, it hasn’t been introduced in another of the other fiveMinuteStats vignettes, so we will briefly (re)introduce it here.

Ridge regression can be introduced in different ways. Here we will view it as a Bayesian model (with a likelihood and a prior, and our interest is in performing posterior inferences using this model).

We start with a standard linear regression model: \[ y_i \sim N(\mathbf{x}_i^T\mathbf{b}, \sigma^2), \quad i = 1, \ldots, n. \]

Here, \(i\) is a sample, and the data for sample \(i\) are the output \(y_i\) and the \(p\) inputs \(x_{i1}, \ldots, x_{ip}\) stored as a vector \(\mathbf{x}_i\). Typically one also includes an intercept term, and we will ignore this detail here for simplicity (noting that it is easy to add without fundamentally changing the math).

Give posterior mean and posterior covariance matrix.

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