Linear models in general HTF Ch. 2.8.3

  • By linear models, we mean that the hypothesis function \(h_{\bf w}({\bf x})\) is a linear function of the parameters \({\bf w}\).

  • Predictions are a linear combination of feature values

  • \[h_{\bf w}({\mathbf{x}}) = \sum_{k=0}^{p} w_k \phi_k({\mathbf{x}}) = {{\boldsymbol{\phi}}}({\mathbf{x}})^{\mathsf{T}}{{\mathbf{w}}}\] where \(\phi_k\) are called basis functions As usual, we let \(\phi_0({\mathbf{x}})=1, \forall {\mathbf{x}}\), to create a bias.

  • To recover degree-\(d\) polynomial regression in one variable, set \[\phi_0(x) = 1, \phi_1(x) = x, \phi_2(x) = x^2, ..., \phi_d(x) = x^d\]

  • Basis functions are fixed for training

Linear Methods for Classification

  • Classification tasks

  • Error functions for classification

  • Logistic Regression

  • Support Vector Machines

Example: Given nucleus radius, predict cancer recurrence

ggplot(bc,aes(Radius.Mean,fill=Outcome,color=Outcome)) + geom_density(alpha=I(1/2))

Example: Solution by linear regression

  • Univariate real input: nucleus size
  • Output coding: non-recurrence = 0, recurrence = 1
  • Sum squared error minimized by the blue line

Linear regression for classification

  • The predictor shows an increasing trend towards recurrence with larger nucleus size, as expected.

  • Output cannot be directly interpreted as a class prediction.

  • Thresholding output (e.g., at 0.5) could be used to predict 0 or 1.
    (In this case, prediction would be 0 except for extremely large nucleus size.)

  • Interpret as probability? Not bounded to \([0,1]\), not consistent even for well-separated data

Probabilistic view

  • Suppose we have two possible classes: \(y\in \{0,1\}\).

  • The symbols “\(0\)” and “\(1\)” are unimportant. Could have been \(\{a,b\}\), \(\{\mathit{up},\mathit{down}\}\), whatever.

  • Rather than try to predict the class label directly, ask:
    What is the probability that a given input \({\mathbf{x}}\) to has class \(y=1\)?

  • Conditional Probability:

\[P(y=1|{\mathbf{X}}= {\mathbf{x}}) = \frac{P({\mathbf{X}}= {\mathbf{x}}, y=1)}{P({\mathbf{X}}= {\mathbf{x}})} \] - Bayes' Rule

\[ = \frac{P({\mathbf{X}}= {\mathbf{x}}| y=1)P(y=1)}{P({\mathbf{X}}= {\mathbf{x}}|y=1)P(y=1)+P({\mathbf{X}}= {\mathbf{x}}|y=0)P(y=0)} \]

Probabilistic models for binary classification

  • Can also write: \[P(y=1|{\mathbf{X}}= {\mathbf{x}})=\sigma\left(\log\frac{P(y=1|{\mathbf{X}}= {\mathbf{x}})}{P(y=0|{\mathbf{X}}= {\mathbf{x}})}\right) = \sigma\left(\log\frac{P({\mathbf{X}}= {\mathbf{x}}|y=1)P(y=1)}{P({\mathbf{X}}= {\mathbf{x}}|y=0)P(y=0)}\right)\] where \(\sigma(a) = \frac{1}{1+\exp(-a)}\), the sigmoid or logistic function.

  • Discriminative Learning:
    • Model \(\log\frac{P(y=1|{\mathbf{X}}= {\mathbf{x}})}{P(y=0|{\mathbf{X}}= {\mathbf{x}})}\) (log odds) as a function of \(\mathbf{x}\)

    • Only models how to discriminate between examples of the two classes. Does not model distribution of \(\mathbf{x}\).

  • Generative Learning:
    • Model \(P(y=1), P(y=0), P({\mathbf{X}}= {\mathbf{x}}|y=1), P({\mathbf{X}}= {\mathbf{x}}|y=0)\), then use rightmost formula above

    • Models the full joint; can actually use the model to generate (i.e. fantasize) data

Logistic regression HTF (Ch. 4.4)

  • Represent the hypothesis as a logistic function of a linear combination of inputs: \[h({\mathbf{x}}) = \sigma({\mathbf{x}}^{\mathsf{T}}{\mathbf{w}})\]

  • Interpret \(h({\mathbf{x}})\) as \(P(y=1|{\mathbf{X}}= {\mathbf{x}})\), interpret \({\mathbf{x}}^{\mathsf{T}}{\mathbf{w}}\) as the log-odds

  • How do we choose \({\bf w}\)?

  • In the probabilistic framework, observing \(\langle {\mathbf{x}}_i , 1 \rangle\) does not mean \(h({\mathbf{x}}_i)\) should be as close to \(1\) as possible.

  • Maximize probability the model assigns to the \(y_i\), given the \({\mathbf{x}}_i\).

Max Conditional Likelihood

  • Maximize probability the model assigns to the \(y_i\), given the \({\mathbf{x}}_i\).

  • Assumption 1: Examples are i.i.d. Probability of observing all \(y\)s is product \[\begin{gathered} P(Y_1=y_1, Y_2=y_2, ..., Y_n = y_n|X_1 = {\mathbf{x}}_1, X_2 = {\mathbf{x}}_2, ..., X_n = {\mathbf{x}}_n) \\ = \prod_{i=1}^n P(Y_i = y_i | X_i = {\mathbf{x}}_i)\end{gathered}\]

  • Assumption 2: \[\begin{aligned} P(y = 1|{\mathbf{X}}= {\mathbf{x}}) & = h_{\mathbf{w}}({\mathbf{x}}) = 1 / (1 + \exp(-{\mathbf{x}}^{\mathsf{T}}{\mathbf{w}}))\\ P(y = 0|{\mathbf{X}}= {\mathbf{x}}) & = (1 - h_{\mathbf{w}}({\mathbf{x}}))\\\end{aligned}\]

Max Conditional Likelihood

  • Maximize probability the model assigns to the \(y_i\), given the \({\mathbf{x}}_i\).

  • More stable to maximize log probability. Note

\[\begin{aligned} \log P(Y_i = y_i | X_i = {\mathbf{x}}_i) & = \left\{ \begin{array}{ll} \log h_{\mathbf{w}}({\mathbf{x}}_i) & \mbox{if}~y_i=1 \\ \log(1-h_{\mathbf{w}}({\mathbf{x}}_i)) & \mbox{if}~y_i=0 \end{array} \right. \end{aligned} \]

  • Therefore,

\[\log \prod_{i=1}^n P(Y_i = y_i | X_i = {\mathbf{x}}_i) = \sum_{i = 1}^n \left[y_i \log( h_{\mathbf{w}}({\mathbf{x}}_i)) + (1 - y_i) \log (1 - h_{\mathbf{w}}({\mathbf{x}}_i))\right] \]

  • Suggests an error \[\begin{aligned} \hspace{-2em} J(h_{{\mathbf{w}}}) = - \sum_{i = 1}^n \left[y_i \log( h_{\mathbf{w}}({\mathbf{x}}_i)) + (1 - y_i) \log (1 - h_{\mathbf{w}}({\mathbf{x}}_i))\right]\end{aligned}\]

  • This is the cross entropy. Number of bits to transmit \(y_i\)
    if both parties know \(h_{\mathbf{w}}\) and \({\mathbf{x}}_i\).

Back to the breast cancer problem

Logistic Regression:

## (Intercept) Radius.Mean 
##  -3.4671348   0.1296493

Least Squares:

## (Intercept) Radius.Mean 
## -0.17166939  0.02349159

Supervised Learning Methods: “Objective-driven”

Mthd. Form Objective
OLS \(h_w({\mathbf{x}}) = {\mathbf{x}}^{\mathsf{T}}{\mathbf{w}}\) \(\sum_{i=1}^n (h_{\mathbf{w}}({\mathbf{x}}_i) - y_i)^2\)
\(\approx E[Y=y|\mathbf{X}={\mathbf{x}}]\)… …using a linear function
LR \(h_w({\mathbf{x}}) = \frac{1}{1 + \mathrm{e}^{-{\mathbf{x}}^{\mathsf{T}}{\mathbf{w}}}}\) \(-\sum_{i=1}^n y_i \log h_{\mathbf{w}}({\mathbf{x}}_i) + (1-y_i) \log (1-h_{\mathbf{w}}({\mathbf{x}}_i))\)
\(\approx P(Y=y|\mathbf{X}={\mathbf{x}})\)… …using sigmoid of a linear function
  • Both model the conditional mean of \(y\) using a (transformed) linear function
  • Both use maximum conditional likelihood to estimate

Decision boundary HTF Ch. 2.3.1,2.3.2

  • How complicated is a classifier?

  • One way to think about it is in terms of its decision boundary, i.e. the line it defines for separating examples

  • Linear classifiers draw a hyperplane between examples of the different classes. Non-linear classifiers draw more complicated surfaces between the different classes.

  • For a probabilistic classifier with a cutoff of 0.5,
    the decision boundary is the curve on which: \[\frac{P(y=1|{\mathbf{X}}= {\mathbf{x}})}{P(y=0|{\mathbf{X}}= {\mathbf{x}})} = 1, \mbox{i.e., where } \log\frac{P(y=1|{\mathbf{X}}= {\mathbf{x}})}{P(y=0|{\mathbf{X}}= {\mathbf{x}})} = 0\]

Decision boundary

Class = R if \({\mathrm{Pr}}(Y=1|X=x) > 0.5\)

Decision boundary

Class = R if \({\mathrm{Pr}}(Y=1|X=x) > 0.25\)