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Regression

Preliminaries

  • Goal
    • Introduction to Bayesian (Linear) Regression
  • Materials
    • Mandatory
      • These lecture notes
    • Optional
      • Bishop pp. 152-158

Regression - Illustration

Given a set of (noisy) data measurements, find the 'best' relation between an input variable $x \in \mathbb{R}^D$ and input-dependent outcomes $y \in \mathbb{R}$

Regression vs Density Estimation

  • Observe $N$ IID data pairs $D=\{(x_1,y_1),\dotsc,(x_N,y_N)\}$ with $x_n \in \mathbb{R}^D$ and $y_n \in \mathbb{R}$.
  • Assume that we are interested in (a model for) the responses $y_n$ for given inputs $x_n$?, I.o.w. we are interested in building a model for the conditional distribution $p(y|x)$.
  • Note that, since $p(x,y)=p(y|x)\, p(x)$, building a model $p(y|x)$ is similar to density estimation with the assumption that $x$ is drawn from a uniform distribution.

Bayesian Linear Regression

  • Next, we discuss (1) model specification, (2) Inference and (3) a prediction application for a Bayesian linear regression problem.
1. Model Specification

  • In a regression model, we try to 'explain the data' by a purely deterministic term $f(x_n,w)$, plus a purely random term $\epsilon_n$ for 'unexplained noise',

    $$ y_n = f(x_n,w) + \epsilon_n $$

  • In a linear regression model, i.e., linear w.r.t. the parameters $w$, we assume that $$f(x_n,w)= \sum_{j=0}^{M-1} w_j \phi_j(x_n) = w^T \phi(x_n)$$ where $\phi_j(x)$ are called basis functions.
    • For notational simplicity, from now on we will assume $f(x_n,w) = w^T x_n$, with $x_n \in \mathbb{R}^M$.
  • In ordinary linear regression , the noise process $\epsilon_n$ is zero-mean Gaussian with constant variance, i.e. $$ y_n = w^T x_n + \mathcal{N}(0,\beta^{-1}) \,. $$
  • Hence, given a data set $D=\{(x_1,y_1),\dotsc,(x_N,y_N)\}$, the likelihood for an ordinary linear regression model is
$$\begin{align*} p(y\,|\,\mathbf{X},w,\beta) &= \mathcal{N}(y\,|\,\mathbf{X} w,\beta^{-1} \mathbf{I}) \\ &= \prod_n \mathcal{N}(y_n\,|\,w^T x_n,\beta^{-1}) \tag{B-3.10} \end{align*}$$

where $w = \left(\begin{matrix} w_1 \\ w_2 \\ \vdots \\ w_{M} \end{matrix} \right)$, the $(N\times M)$-dim matrix $\mathbf{X} = \left(\begin{matrix}x_1^T \\ x_2^T \\ \vdots \\ x_N^T \end{matrix} \right) = \left(\begin{matrix}x_{11},x_{12},\dots,x_{1M}\\ x_{21},x_{22},\dots,x_{2M} \\ \vdots \\ x_{N1},x_{N2},\dots,x_{NM} \end{matrix} \right) $ and $y = \left(\begin{matrix} y_1 \\ y_2 \\ \vdots \\ y_N \end{matrix} \right)$.

  • For full Bayesian learning we should also choose a prior $p(w)$:
$$\begin{equation} p(w\,|\,\alpha) = \mathcal{N}(w\,|\,0,\alpha^{-1}\mathbf{I}) \tag{B-3.52} \end{equation}$$
  • For simplicity, we will assume that $\alpha$ and $\beta$ are known.
2. Inference
  • We'll do Bayesian inference for the parameters $w$.
$$\begin{align*} p(w|D) &\propto p(D|w)\cdot p(w) \\ &= \mathcal{N}(y\,|\,\mathbf{X} w,\beta^{-1} \mathbf{I}) \cdot \mathcal{N}(w\,|\,0,\alpha^{-1}\mathbf{I}) \\ &\propto \exp \left( \frac{\beta}{2} \left( {y - \mathbf{X}w } \right)^T \left( {y - \mathbf{X}w } \right) + \frac{\alpha}{2}w^T w \right)\tag{B-3.55} \\ &= \exp\left( \frac{1}{2} w^T\left(\beta \mathbf{X}^T\mathbf{X} + \alpha \mathbf{I}\right)w - (\beta \mathbf{X}^Ty)^T w + \frac{\beta}{2}y^T y \right) \\ &\propto \mathcal{N}_c\left(w\,|\,\eta_N,\Lambda_N \right) \end{align*}$$

with natural parameters (see the natural parameterization of Gaussian):

$$\begin{align*} \eta_N &= \beta\mathbf{X}^Ty \\ \Lambda_N &= \beta \mathbf{X}^T\mathbf{X} + \alpha \mathbf{I} \end{align*}$$

  • Or equivalently (in the moment parameterization of the Gaussian): $$\begin{align*} p(w|D) &= \mathcal{N}\left(w\,|\,m_N,S_N \right) \tag{B-3.49} \\ m_N &= \beta S_N \mathbf{X}^T y \tag{B-3.53}\\ S_N &= \left(\alpha \mathbf{I} + \beta \mathbf{X}^T \mathbf{X}\right)^{-1} \tag{B-3.54} \end{align*}$$

  • Note that B-3.53 and B-3.54 combine to $$ m_N = \left(\frac{\alpha}{\beta}\mathbf{I} + \mathbf{X}^T \mathbf{X} \right)^{-1} \mathbf{X}^T y\,. $$

  • (Bishop Fig.3.7) Illustration of sequential Bayesian learning for a simple linear model of the form $y(x, w) = w_0 + w_1 x$. (Bishop Fig.3.7, detailed description at Bishop, pg.154.)

3. Application: predictive distribution
  • Assume we are interested in the distribution $p(y_\bullet \,|\, x_\bullet, D)$ for a new input $x_\bullet$. This can be worked out as (exercise B-3.10)
$$\begin{align*} p(y_\bullet \,|\, x_\bullet, D) &= \int p(y_\bullet \,|\, x_\bullet, w) p(w\,|\,D)\,\mathrm{d}w \\ &= \int \mathcal{N}(y_\bullet \,|\, w^T x_\bullet, \beta^{-1}) \mathcal{N}(w\,|\,m_N,S_N)\,\mathrm{d}w \\ &= \int \mathcal{N}(y_\bullet \,|\, z, \beta^{-1}) \mathcal{N}(z\,|\,x_\bullet^T m_N,x_\bullet^T S_N x_\bullet) \,\mathrm{d}z \quad \text{(sub. }z=x_\bullet^T w \text{)} \\ &= \int \underbrace{\mathcal{N}(z \,|\, y_\bullet, \beta^{-1}) \mathcal{N}(z\,|\,x_\bullet^T m_N,x_\bullet^T S_N x_\bullet)}_{\text{Use Gaussian product formula SRG-6}}\,\mathrm{d}z \\ &= \int \mathcal{N}\left(y_\bullet\,|\, m_N^T x_\bullet, \sigma_N^2(x_\bullet) \right) \underbrace{\mathcal{N}\left(z\,|\, m_z, S_z\right)}_{\text{integrate this out}} \mathrm{d}z\\ &= \mathcal{N}\left(y_\bullet\,|\, m_N^T x_\bullet, \sigma_N^2(x_\bullet) \right) \end{align*}$$

with $$\begin{align*} \sigma_N^2(x_\bullet) = \beta^{-1} + x^T_\bullet S_N x_\bullet \tag{B-3.59} \end{align*}$$

  • So, the uncertainty $\sigma_N^2(x_\bullet)$ about the output contains comprises both uncertainty about the process ($\beta^{-1}$) and about the model parameters ($x^T_\bullet S_N x_\bullet$).

Example Predictive Distribution

  • As an example, let's do Bayesian Linear Regression for a synthetic sinusoidal data set and a model with 9 Gaussian basis functions $$\begin{align*} y &=\sum_{m=1}^9 w_m \phi_m(x_n) + \mathcal{N}(0,\beta^{-1}) \\ \phi_m(x_n) &= \exp\left( \frac{x_n-\mu_m}{\sigma^2}\right) \end{align*}$$

  • The predictive distributions for $y$ are shown in the following plots (Bishop, Fig.3.8)

  • And some plots of draws of posteriors for the functions $w^T \phi(x)$ (Bishop, Fig.3.9)

Maximum Likelihood Estimation for Linear Regression Model

  • Recall the posterior mean for the weight vector $$ m_N = \left(\frac{\alpha}{\beta}\mathbf{I} + \mathbf{X}^T \mathbf{X} \right)^{-1} \mathbf{X}^T y $$ where $\alpha$ is the prior precision for the weights.
  • The Maximum Likelihood solution for $w$ is obtained by letting $\alpha \rightarrow 0$, which leads to
$$\begin{equation*} \hat w_{\text{ML}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T y \end{equation*}$$
  • The matrix $\mathbf{X}^\dagger \equiv (\mathbf{X}^T \mathbf{X})^{-1}\mathbf{X}^T$ is also known as the Moore-Penrose pseudo-inverse (which is sort-of-an-inverse for non-square matrices).
  • Note that if we have fewer training samples than input dimensions, i.e., if $N<M$, then $\mathbf{X}^T \mathbf{X}$ will not be invertible and maximum likelihood blows up. The Bayesian solution does not suffer from this problem.

Least-Squares Regression

  • (You may say that) we don't need to work with probabilistic models. E.g., there's also the deterministic least-squares solution: minimize sum of squared errors,
$$\begin{align*} \hat w_{\text{LS}} &= \arg\min_{w} \sum_n {\left( {y_n - w ^T x_n } \right)} ^2 = \arg\min_{w} \left( {y - \mathbf{X}w } \right)^T \left( {y - \mathbf{X} w } \right) \end{align*}$$
  • Setting the gradient $ \frac{\partial \left( {y - \mathbf{X}w } \right)^T \left( {y - \mathbf{X}w } \right)}{\partial w} = -2 \mathbf{X}^T \left(y - \mathbf{X} w \right) $ to zero yields the so-called normal equations $\mathbf{X}^T\mathbf{X} \hat w_{\text{LS}} = \mathbf{X}^T y$ and consequently
$$ \hat w_{\text{LS}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T y $$

which is the same answer as we got for the maximum likelihood weights $\hat w_{\text{ML}}$.

  • $\Rightarrow$ Least-squares regression ($\hat w_{\text{LS}}$) corresponds to the (probabilistic) maximum likelihood solution ($\hat w_{\text{ML}}$) if the probabilistic model includes the following assumptions:
    1. The observations are independently and identically distributed (IID) (this determines how errors are combined), and
    2. The noise signal $\epsilon_n \sim \mathcal{N}(0,\,\beta^{-1})$ is Gaussian distributed (determines the error metric)
  • If you use the Least-Squares method, you cannot see (nor modify) these assumptions. The probabilistic method forces you to state all assumptions explicitly!

Not Identically Distributed Data

  • Let's do an example regarding changing our assumptions. What if we assume that the variance of the measurement error varies with the sampling index, $\epsilon_n \sim \mathcal{N}(0,\beta_n^{-1})$?
  • The likelihood is now (using $\Lambda \triangleq \mathrm{diag}(\beta_n)$ ) $$ p(y\,|\,\mathbf{X},w,\Lambda) = \mathcal{N}(y\,|\,\mathbf{X} w,\Lambda^{-1} ) \,. $$
  • Combining this likelihood with the prior $p(w) = \mathcal{N}(w\,|\,0,\alpha^{-1}\mathbf{I})$ leads to a posterior
$$\begin{align*} p(w|D) &\propto p(D|w)\cdot p(w) \\ &= \mathcal{N}(y\,|\,\mathbf{X} w,\Lambda^{-1} \mathbf{I}) \cdot \mathcal{N}(w\,|\,0,\alpha^{-1}\mathbf{I}) \\ &\propto \exp \left\{ \frac{1}{2} \left( {y - \mathbf{X}w } \right)^T \Lambda \left( {y - \mathbf{X}w } \right) + \frac{\alpha}{2}w^T w \right\} \\ &\propto \mathcal{N}\left(w\,|\,m_N,S_N \right) \end{align*}$$

with $$\begin{align*} m_N &= S_N \mathbf{X}^T \Lambda y \\ S_N &= \left(\alpha \mathbf{I} + \mathbf{X}^T \Lambda \mathbf{X}\right)^{-1} \end{align*}$$

  • And maximum likelihood solution $$ \hat{w}_{\text{ML}} = \left. m_N\right|_{\alpha \rightarrow 0} = \left(\mathbf{X}^T \Lambda \mathbf{X}\right)^{-1} \mathbf{X}^T \Lambda y $$
  • This maximum likelihood solution is also called the Weighted Least Squares (WLS) solution. (Note that we just stumbled upon it, the crucial aspect is appropriate model specification!)
  • Note also that the dimension of $\Lambda$ grows with the number of data points. In general, models for which the number of parameters grow as the number of observations increase are called non-parametric models.

CODE EXAMPLE

We'll compare the Least Squares and Weighted Least Squares solutions for a simple linear regression model with input-dependent noise:

$$\begin{align*} x &\sim \text{Unif}[0,1]\\ y|x &\sim \mathcal{N}(f(x), v(x))\\ f(x) &= 5x - 2\\ v(x) &= 10e^{2x^2}-9.5\\ \mathcal{D} &= \{(x_1,y_1),\ldots,(x_N,y_N)\} \end{align*}$$
In [1]:
using Pkg;Pkg.activate("probprog/workspace/");Pkg.instantiate();
IJulia.clear_output();
In [2]:
using PyPlot, LinearAlgebra

# Model specification: y|x ~ 𝒩(f(x), v(x))
f(x) = 5*x .- 2 
v(x) = 10*exp.(2*x.^2) .- 9.5 # input dependent noise variance
x_test = [0.0, 1.0]
plot(x_test, f(x_test), "k--") # plot f(x)

# Generate N samples (x,y), where x ~ Unif[0,1]
N = 50
x = rand(N)
y = f(x) + sqrt.(v(x)) .* randn(N)
plot(x, y, "kx"); xlabel("x"); ylabel("y") # Plot samples

# Add constant to input so we can estimate both the offset and the slope
_x = [x ones(N)]
_x_test = hcat(x_test, ones(2))

# LS regression
w_ls = pinv(_x) * y
plot(x_test, _x_test*w_ls, "b-") # plot LS solution

# Weighted LS regression
W = Diagonal(1 ./ v(x)) # weight matrix
w_wls = inv(_x'*W*_x) * _x' * W * y
plot(x_test, _x_test*w_wls, "r-") # plot WLS solution
ylim([-5,8]); legend(["f(x)", "D", "LS linear regr.", "WLS linear regr."],loc=2);
In [3]:
open("../../styles/aipstyle.html") do f
    display("text/html", read(f, String))
end
In [ ]: