📊 1. Discriminative vs Generative Learning

• Joint Probability:

  p(x, y) = p(y | x) p(x) or p(x | y) p(y)

• Discriminative Learning:

  • Focuses on directly modeling p(y | x) — the probability of the output y given input x.
  • Example: “Who won?” in classification.
  • Typical models: Logistic Regression, SVMs.

• Generative Learning:

  • Focuses on modeling p(x | y) and p(y).
  • Learns how data is generated for each class, then uses Bayes’ theorem.
  • Example: Naive Bayes, Gaussian Mixture Models.

✅ Key difference:

Discriminative: direct decision boundary.

Generative: models full data distribution for each class.

📈 2. Likelihood (MLE) vs Posterior (MAP)

• Likelihood:

  P(D | θ) (How likely is the data given my model parameters)

  • This is the Maximum Likelihood Estimation (MLE).
  • Goal: find parameter θ that maximizes the likelihood.

• Posterior:

  P(θ | D) (How probable are the parameters given my data)

  • This is the Maximum A Posteriori Estimation (MAP).
  • Combines prior beliefs and observed data.

✅ Key idea:

MLE: purely data-driven.

MAP: combines data and prior.

🎲 3. Putting It Together: Bayesian Intuition

p(heads | θ, D) * p(D | θ) dθ

• To know the probability of an event (e.g., coin comes up heads):
  • Multiply the model’s prediction (p(heads | θ, D))
  • By the probability of seeing that model given your data (p(D | θ))
  • Integrate over all possible θ.

✅ This is the Bayesian way: combine your model’s predictions with how well the model fits your data.