Probability Theory

1. Bayesian Probability
   1. Prior and Posterior Distributions
      1. Definition of Prior Distribution
         1. Understanding subjective prior beliefs
         2. Objective versus Subjective Priors
         3. Types of Priors: Informative vs Non-informative
         4. Examples of Priors in Applications
            1. Uniform Priors
            2. Jeffreys' Priors
      2. Definition of Posterior Distribution
         1. Calculating Posterior Distributions
         2. Models for Posterior Prediction
         3. Impact of Prior Selection on Posterior
         4. Bayesian Credible Intervals
      3. Likelihood Function
         1. Role in connecting prior and posterior
         2. Construction of the likelihood in models
         3. Interpretation and Properties
      4. Normalizing Constant
         1. Ensuring posterior distribution integrates to 1
         2. Methods to approximate or calculate
   2. Bayesian Updating
      1. Formal Process of Bayesian Updating
         1. Sequential Updating: Incorporating new data
         2. Impact of new evidence on beliefs
         3. Practical computational methods:
            1. Markov Chain Monte Carlo (MCMC)
            2. Variational Inference
      2. Real-world Examples and Applications
         1. Updating risk assessments
         2. Adaptive learning in machine learning
   3. Conjugate Priors
      1. Definition and Importance of Conjugate Priors
         1. Simplifying the computation of posteriors
      2. Common Conjugate Prior Examples
         1. Beta and Binomial Model
         2. Normal Distributions in Linear Regression
      3. Derivation and Implications for Modeling
      4. Limitations and Extensions of Conjugate Priors
   4. Bayesian Inference
      1. Comparing Bayesian and Frequentist Inference
         1. Differences in interpretation of probability
         2. Advantages of Bayesian methodologies
      2. Methods of Bayesian Inference
         1. Analytical Solutions
         2. Numeric and Simulation-based Methods
      3. Applications of Bayesian Inference
         1. Hierarchical Models
         2. Bayesian Networks
      4. Challenges in Bayesian Inference
         1. Computational Cost
         2. Sensitivity to the Choice of Priors
         3. Interpretation of Results
   5. Bayesian Decision Theory
      1. Foundations of Decision Theory in Bayesian Context
      2. Decision-Making under Uncertainty
         1. Utility and Loss Functions
         2. Expected Value of Perfect Information (EVPI)
      3. Practical Applications
         1. Medical Decision Making
         2. Economics and Market Predictions
   6. Advanced Topics in Bayesian Methods
      1. Bayesian Nonparametrics
         1. Dirichlet Processes
         2. Gaussian Processes
      2. Bayesian Model Selection
         1. Bayes Factors
         2. Model Comparison Techniques
      3. Hierarchical Bayesian Models
         1. Multilevel Models
         2. Applications and Examples in Data Analysis
      4. Computational Approaches
         1. Advances in MCMC
         2. Hamiltonian Monte Carlo (HMC)
         3. Importance Sampling
      5. Bayesian Robustness
         1. Sensitivity Analysis
         2. Dealing with Model Uncertainty