naive - What is the difference between a generative and a discriminative algorithm?

on discriminative vs. generative classifiers: a comparison of logistic regression and naive bayes (7)

Please, help me understand the difference between a generative and a discriminative algorithm, keeping in mind that I am just a beginner.

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.

A generative algorithm model will learn completely from the training data and will predict the response.

A discriminative algorithm job is just to classify or differentiate between the 2 outcomes.

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.

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.

In practice, the models are used as follows.

In discriminative models, to predict the label y from the training example x, you must evaluate:

which merely chooses what is the most likely class y considering x. It's like we were trying to model the decision boundary between the classes. This behavior is very clear in neural networks, where the computed weights can be seen as a complexly shaped curve isolating the elements of a class in the space.

Now, using Bayes' rule, let's replace the in the equation by . Since you are just interested in the arg max, you can wipe out the denominator, that will be the same for every y. So, you are left with

which is the equation you use in generative models.

While in the first case you had the conditional probability distribution p(y|x), which modeled the boundary between classes, in the second you had the joint probability distribution p(x, y), since p(x, y) = p(x | y) p(y), which explicitly models the actual distribution of each class.

With the joint probability distribution function, given a y, you can calculate ("generate") its respective x. For this reason, they are called "generative" models.

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.

The different models are summed up in the table below: 