文章目录

• 一、Classification: Probabilistic Generative Model
• • Ideal Alternatives
  • Two Boxes example
  • Two Classes
  • Gaussian Distribution
  • Probability from Class
  • Maximum Likelihood
  • Now we can do classification
  • Three Steps
• 二、Classification: Logistic Regression
• • Step 1: Function Set
  • Step 2: Goodness of a Function
  • Step 3: Find the best function
  • Logistic Regression + Square Error
  • Generative v.s. Discriminative
  • Multi-class Classification
  • Limitation of Logistic Regression
• 总结

一、Classification: Probabilistic Generative Model

Ideal Alternatives

Function (Model):

Loss function:

The number of times f get incorrect results on training data.

Find the best function:
Example: Perceptron, SVM

Classification as Regression?
Binary classification as example ：
Training: Class 1 means the target is 1; Class 2 means the target is -1
Testing: closer to 1 → class 1; closer to -1 → class 2

Penalize to the examples that are “too correct”

Multiple class: Class 1 means the target is 1; Class 2 means the target is 2; Class 3 means the target is 3 …… problematic

Two Boxes example

From one of the boxes，where does it come from?

Two Classes

Estimating the Probabilities From training data

Given an x, which class does it belong to

Gaussian Distribution

Input: vector x, output: probability of sampling x
The shape of the function determines by mean μ and covariance matrix Σ

Probability from Class

Maximum Likelihood

The Gaussian with any mean μ and covariance matrix Σ can generate these points

Likelihood of a Gaussian with mean μ and covariance matrix Σ = the probability of the Gaussian samples x^1,x2,x^3, …… ,x^79

Now we can do classification

Testing data: 47% accuracy
All: hp, att, sp att,
de, sp de, speed (6 features)

Modifying Model：

Three Steps

Function Set (Model):

Goodness of a function:
The mean μ and covariance Σ that maximizing the likelihood (the probability of generating data)
Find the best function: easy

Probability Distribution

Posterior Probability：

二、Classification: Logistic Regression

Step 1: Function Set

Step 2: Goodness of a Function

Step 3: Find the best function

Logistic Regression + Square Error

Generative v.s. Discriminative

Usually people believe discriminative model is better
Benefit of generative model
With the assumption of probability distribution
less training data is needed
more robust to the noise
Priors and class-dependent probabilities can be estimated from different sources.

Multi-class Classification

Limitation of Logistic Regression

总结