Statistical Inference

11.

Introduction to Bayesian Inference

11.1.

Bayesian vs Frequentist Philosophy

11.1.1.

Different Interpretations of Probability

11.1.2.

Subjective vs Objective Probability

11.1.3.

Parameter as Random Variable

11.1.4.

Role of Prior Information

11.2.

11.2.1.

Mathematical Statement

11.2.2.

Components and Interpretation

11.2.3.

Updating Beliefs with Data

11.2.4.

Continuous Parameter Version

11.3.

Components of Bayesian Analysis

11.3.1.

Prior Distribution

11.3.1.1.

Expressing Prior Beliefs

11.3.1.2.

Types of Priors

11.3.1.2.1.

Informative Priors

11.3.1.2.2.

Non-informative Priors

11.3.1.2.3.

Conjugate Priors

11.3.1.2.4.

Jeffreys Priors

11.3.2.

Likelihood Function

11.3.2.1.

Role in Bayesian Analysis

11.3.2.2.

Same as in Frequentist Analysis

11.3.3.

Posterior Distribution

11.3.3.1.

Combining Prior and Likelihood

11.3.3.2.

Normalization Constant

11.3.3.3.

Posterior as Compromise

11.3.4.

Marginal Likelihood (Evidence)

11.3.4.1.

Role in Normalization

11.3.4.2.

Model Comparison

11.4.

Bayesian Estimation

11.4.1.

Posterior Point Estimates

11.4.1.1.

11.4.1.2.

Posterior Median

11.4.1.3.

Posterior Mode (MAP)

11.4.1.4.

Loss Functions and Optimal Estimators

11.4.2.

Credible Intervals

11.4.2.1.

Definition and Interpretation

11.4.2.2.

Equal-Tailed Intervals

11.4.2.3.

Highest Posterior Density Intervals

11.4.2.4.

Comparison with Confidence Intervals

11.5.

Bayesian Hypothesis Testing

11.5.1.

11.5.1.1.

Definition and Calculation

11.5.1.2.

Interpretation and Evidence Scale

11.5.1.3.

Comparison with P-values

11.5.2.

Posterior Probabilities of Hypotheses

11.5.3.

Decision Theory Approach

11.6.

Computational Methods

11.6.1.

Analytical Solutions

11.6.2.

Numerical Integration

11.6.3.

Markov Chain Monte Carlo

11.6.4.

11.6.5.

Metropolis-Hastings Algorithm

Previous

10. Simple Linear Regression Inference

Go to top

Next

12. Non-parametric Methods