Probability Theory & Bayesian Inference for Machine Learning
Module: AI/ML Premium | Difficulty: PhD Level | Prerequisites: Measure Theory, Linear Algebra
1. Measure-Theoretic Probability
1.1 Probability Spaces
1.2 Random Variables and Distributions
2. Expectation and Variance
2.1 Expected Value
2.2 Variance and Covariance
2.3 Important Inequalities
3. Bayesian Inference
3.1 Bayes' Theorem
3.2 Conjugate Priors
4. Information Theory
4.1 Entropy and KL Divergence
4.2 Mutual Information
5. Exponential Families
5.1 Definition
5.2 Properties
6. Variational Inference
6.1 Evidence Lower Bound (ELBO)
7. Monte Carlo Methods
7.1 Importance Sampling
7.2 Markov Chain Monte Carlo (MCMC)
8. Bayesian Deep Learning
8.1 Uncertainty Quantification
8.2 Types of Uncertainty
| Type | Source | Method |
|---|---|---|
| Aleatoric | Inherent noise | Heteroscedastic models |
| Epistemic | Limited data | MC Dropout, SWAG |
| Model | Wrong architecture | Bayesian model comparison |
Summary
| Concept | Formula | Application |
|---|---|---|
| Bayes' Theorem | Parameter estimation | |
| KL Divergence | Variational inference | |
| ELBO | Approximate inference | |
| Entropy | Information theory |
Research Frontier: Neural processes combine neural networks with Gaussian processes, enabling amortized Bayesian inference with learned priors.