
Discriminative algorithms 
Generative models 
Objective 
Discriminative models are designed to model the conditional probability of the output variable given the input data. They are used for classification and regression tasks, focusing on modeling the decision boundary between classes. 
Generative models aim to model the joint probability distribution of both input and output data. They can generate new data samples that are similar to the training data. 
Use Cases 
Used for tasks such as image classification, text classification, sentiment analysis, and any problem that involves predicting a label or value for given input data. 
Used for tasks like image generation, text generation, data synthesis, and anomaly detection, where generating new data samples is required. 
Training Data 
Typically trained on labeled data, where the input data is paired with the corresponding output labels or values. 
Trained on unlabeled data or data with minimal labels, as they aim to learn the underlying data distribution. 
Output 
Provide a direct mapping from input to output, making them suitable for tasks that involve classification or regression. 
Can generate new data samples by sampling from the learned data distribution, which is valuable for tasks like data generation or data completion. 
Complexity 
Generally simpler to train and require fewer parameters, as they only need to learn the decision boundary between classes. 
Often more complex and require a larger number of parameters, as they must model the entire data distribution. 
Interpretability 
Easier to interpret because they explicitly model the relationship between input and output variables. 
Less interpretable, as their primary focus is on capturing the data distribution, which may not have a direct interpretation. 
Data Efficiency 
Typically require a larger amount of labeled data for training, as they are specific to the task at hand. 
Can be more dataefficient, as they can learn from unlabeled or weakly labeled data by capturing the underlying structure of the data. 
Inference 
Generally faster for making predictions, as they directly compute the conditional probability of the output given the input. 
May be slower for inference, as they involve sampling from the learned distribution. 
Mathematical differences 
The primary equation associated with discriminative models is the conditional probability of the output variable (Y) given the input data (X):
P(Y  X) Or,
h_{θ}(x) = 0
1
The most common discriminative models include logistic regression, support vector machines, and neural networks for classification problems. The equation for logistic regression, for example, looks like this:
P(Y = 1  X) = 1 / (1 + e^{(w · X)})
Where:
 P(Y = 1  X) is the probability of the output being 1 given the input X.
 e represents the base of the natural logarithm.
 w is the weight vector, and X is the input data.

Generative models are more focused on modeling the joint probability distribution of both input (X) and output (Y):
P(X,Y)
Common generative models include Gaussian Mixture Models (GMM), Hidden Markov Models (HMM), Variational Autoencoders (VAEs), and Generative Adversarial Networks (GANs). The equations for these models can be more complex, but here's a simplified example for a Gaussian Mixture Model:
P(X, Y) = Σ_{i} P(Y = i) * P(X  Y = i)
Where:
 P(Y = i) represents the prior probability of class i.
 X is feature.
 Y is class.
 P(X  Y = i) represents the probability of the input X given class i.

Cost function 
Conditional likelihood:
 which is p of y given x 
 which is p of x given y 
Advantages 

Classification Tasks: Discriminative models are typically superior for classification tasks, where the goal is to assign a label or category to input data. Examples include image classification, sentiment analysis, and spam detection. Algorithms like logistic regression, support vector machines, and deep neural networks are often used for discriminative tasks.

Highly Imbalanced Data: Discriminative models are wellsuited for handling imbalanced datasets. They can focus on learning decision boundaries and class separations, making them effective for scenarios where certain classes are rare.

Supervised Learning: When you have access to labeled data, discriminative models are usually the goto choice. They learn the conditional probability distribution of the output given the input (P(YX)), which is particularly useful when you want to make predictions based on the input data.

Feature Engineering: Discriminative models allow for better feature engineering. You can design and select features that are most relevant to the specific classification task, which can lead to improved performance.


Data Generation: Generative models are superior when the goal is to generate new data that is similar to the training data. This is useful in applications like image generation, text generation, and data augmentation.

Unsupervised Learning: Generative models are often used in unsupervised learning tasks, where you don't have access to labeled data. They can learn the underlying probability distribution of the data (P(X)) and discover patterns and structure within the data.

Anomaly Detection: Generative models can be used for anomaly detection. If the model has learned the distribution of normal data, it can identify deviations as anomalies.

SemiSupervised Learning: In scenarios where you have limited labeled data and a large amount of unlabeled data, generative models can be used to improve the performance of semisupervised learning algorithms.

Data Augmentation: Generative models can be used to augment training data by generating additional samples, which can help improve the performance of discriminative models.

Assumptions 
Weaker assumptions 
Stronger assumptions (you can aprove from the weaker assumptions to the stronger assumptions) 
Examples 
Logistic Regression, Support Vector Machines (SVM), Neural Networks (in the context of classification), Random Forests. 
Gaussian Mixture Models (GMM), Hidden Markov Models (HMM), Variational Autoencoders (VAEs), and Generative Adversarial Networks (GANs). 
Comparison 
Logistic regression 
Gaussian Mixture Model 


