Please, help me understand the difference between a generative and a discriminative algorithm,

Let's say you have input data x and you want to classify the data into labels y. A generative model learns the joint probability distribution p(x,y) and a discriminative model learns the conditional probability distribution p(y|x) - which you should read as "the probability of y given x".

Here's a really simple example. Suppose you have the following data in the form (x,y):

(1,0), (1,0), (2,0), (2, 1)

p(x,y) is

y=0 y=1
-----------
x=1 | 1/2 0
x=2 | 1/4 1/4
p(y|x) is

y=0 y=1
-----------
x=1 | 1 0
x=2 | 1/2 1/2
If you take a few minutes to stare at those two matrices, you will understand the difference between the two probability distributions.

The distribution p(y|x) is the natural distribution for classifying a given example x into a class y, which is why algorithms that model this directly are called discriminative algorithms. Generative algorithms model p(x,y), which can be transformed into p(y|x) by applying Bayes rule and then used for classification. However, the distribution p(x,y) can also be used for other purposes. For example, you could use p(x,y) to generate likely (x,y) pairs.

From the description above, you might be thinking that generative models are more generally useful and therefore better, but it's not as simple as that. This paper is a very popular reference on the subject of discriminative vs. generative classifiers, but it's pretty heavy going. The overall gist is that discriminative models generally outperform generative models in classification tasks This post was edited by Viaan Prakash at September 10, 2020 4:22 PM IST

A generative algorithm models how the data was generated in order to categorize a signal. It asks the question: based on my generation assumptions, which category is most likely to generate this signal?

A discriminative algorithm does not care about how the data was generated, it simply categorizes a given signal.

Imagine your task is to classify a speech to a language.

You can do it by either:

1. learning each language, and then classifying it using the knowledge you just gained

or

2. determining the difference in the linguistic models without learning the languages, and then classifying the speech.

The first one is the generative approach and the second one is the discriminative approach.

Check this reference for more details: http://www.cedar.buffalo.edu/~srihari/CSE574/Discriminative-Generative.pdf.

Here's the most important part from the lecture notes of CS299 (by Andrew Ng) related to the topic, which really helps me understand the difference between discriminative and generative learning algorithms.

Suppose we have two classes of animals, elephant (y = 1) and dog (y = 0). And x is the feature vector of the animals.

Given a training set, an algorithm like logistic regression or the perceptron algorithm (basically) tries to find a straight line — that is, a decision boundary — that separates the elephants and dogs. Then, to classify a new animal as either an elephant or a dog, it checks on which side of the decision boundary it falls, and makes its prediction accordingly. We call these discriminative learning algorithm.

Here's a different approach. First, looking at elephants, we can build a model of what elephants look like. Then, looking at dogs, we can build a separate model of what dogs look like. Finally, to classify a new animal, we can match the new animal against the elephant model, and match it against the dog model, to see whether the new animal looks more like the elephants or more like the dogs we had seen in the training set. We call these generative learning algorithm.

This post was edited by Jainew Nanda at September 19, 2020 11:46 AM IST

Generally, there is a practice in machine learning community not to learn something that you don’t want to. For example, consider a classification problem where one's goal is to assign y labels to a given x input. If we use generative model

p(x,y)=p(y|x).p(x)
we have to model p(x) which is irrelevant for the task in hand. Practical limitations like data sparseness will force us to model p(x) with some weak independence assumptions. Therefore, we intuitively use discriminative models for classification.

The different models are summed up in the table below: