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Bayesian Statistics & Inference

Master the art and science of Bayesian statistical inference: from philosophical foundations to practical applications, learn to update beliefs with data and quantify uncertainty in a principled way.

Intermediate to Advanced

12 Lessons

8-12 Hours

Learning Objectives

Master comprehensive Bayesian statistical inference concepts and methods

Master the philosophical foundations of Bayesian statistics and parameter randomness
Understand the three types of information: population, sample, and prior information
Learn prior distribution construction using moments, quantiles, and expert knowledge methods
Apply conjugate prior families: Beta-Binomial, Gamma-Poisson, Normal-Normal models
Master Bayesian point estimation: posterior mean, median, and mode estimators
Construct credible intervals and understand their interpretation versus confidence intervals
Perform Bayesian prediction with posterior predictive distributions
Explore advanced methods: empirical Bayes, hierarchical models, and MCMC techniques

Course Content Navigation

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Mathematical Foundations

Core mathematical framework underlying Bayesian inference

Bayes' Theorem - The Heart of Bayesian Inference

Posterior = (Likelihood × Prior) / Evidence

\pi(\theta|\tilde{x}) = \frac{p(\tilde{x}|\theta) \pi(\theta)}{p(\tilde{x})}

Key Components:

π(θ|x̃): Posterior distribution - updated beliefs about θ
p(x̃|θ): Likelihood function - probability of data given θ
π(θ): Prior distribution - initial beliefs about θ
p(x̃): Marginal likelihood - normalizing constant

Beta-Binomial Conjugacy Example

Classic example of conjugate prior updating

\text{Beta}(a,b) + \text{Binomial}(n,x) \rightarrow \text{Beta}(a+x, b+n-x)

Key Components:

Prior: θ ~ Beta(a,b) represents beliefs about success probability
Data: x successes in n trials from Binomial(n,θ)
Posterior: θ|data ~ Beta(a+x, b+n-x)
Interpretation: add successes to a, failures to b

Bayesian Credible Interval

Direct probability statement about parameter location

P(\theta_L \leq \theta \leq \theta_U | \text{data}) = 1-\alpha

Key Components:

1-α probability that θ lies in the interval given the data
Different from confidence intervals (frequency interpretation)
Can be computed using posterior quantiles
Natural for decision-making and risk assessment

Posterior Predictive Distribution

Distribution of future observations accounting for parameter uncertainty

p(z|\tilde{x}) = \int p(z|\theta) \pi(\theta|\tilde{x}) d\theta

Key Components:

z: future observation to be predicted
Integration over all possible parameter values
Weighted by posterior probability of each θ
Includes both aleatory and epistemic uncertainty

Core Topics & Concepts

Comprehensive coverage of Bayesian statistical methods

Statistical Inference Information Types

Population Information: probability distribution types and parameter characteristics
Sample Information: observations obtained through sampling (common to classical and Bayesian)
Prior Information: historical data, expert knowledge, and previous studies

Bayesian vs Classical Philosophy

Parameter Randomness: θ as random variable with probability distribution
Information Integration: combining prior knowledge with sample data
Uncertainty Quantification: probability statements about parameters
Sequential Learning: updating beliefs as new data arrives

Prior Distribution Construction

Subjective Method: expert opinions and domain knowledge elicitation
Moments Method: matching prior mean and variance to beliefs
Quantiles Method: using percentile assessments from experts
Reference Priors: objective priors (Jeffreys, uniform) for minimal influence

Conjugate Prior Families

Beta-Binomial: Beta(a,b) prior + Binomial likelihood → Beta(a+x, b+n-x) posterior
Gamma-Poisson: Gamma(α,β) prior + Poisson data → Gamma(α+Σx, β+n) posterior
Normal-Normal: Normal prior for mean with known variance
Inverse Gamma-Normal: Inverse Gamma prior for variance with known mean

Bayesian Point Estimation

Posterior Mean: E[θ|data] - optimal under squared loss
Posterior Median: 50th percentile - optimal under absolute loss
Posterior Mode (MAP): most probable value - optimal under 0-1 loss
Weighted Average Property: balance between prior and data based on precision

Credible Intervals

Equal-tail intervals: using α/2 and (1-α/2) quantiles
Highest Posterior Density (HPD): shortest interval with given probability
One-sided bounds: for directional hypotheses and safety applications
Computational methods: direct inversion, Monte Carlo, transformations

Bayesian Prediction

Posterior Predictive Distribution: p(z|data) = ∫ p(z|θ)π(θ|data)dθ
Uncertainty Decomposition: aleatory (irreducible) + epistemic (parameter)
Prediction Intervals: quantifying uncertainty in future observations
Model-based Forecasting: applications in risk assessment and decision making

Advanced Bayesian Methods

Empirical Bayes: using data to estimate hyperparameters
Hierarchical Bayes: multi-level parameter structures with hyperpriors
MCMC Methods: Metropolis-Hastings, Gibbs sampling, Hamiltonian MC
Model Comparison: Bayes factors, DIC, cross-validation

Real-World Applications

See how Bayesian methods solve practical problems across domains

Medical Diagnosis

Updating disease probability based on test results

Example:

Prior: disease prevalence → Test result → Posterior: probability of disease

Key Advantages:

+Natural incorporation of base rates
+Sequential testing
+Uncertainty quantification

Quality Control

Manufacturing process monitoring with historical knowledge

Example:

Prior: historical defect rates → Current batch data → Updated process assessment

Key Advantages:

+Continuous improvement
+Small sample robustness
+Cost-effective decisions

Financial Risk Assessment

Portfolio risk modeling with market uncertainty

Example:

Prior: historical volatility → Recent market data → Updated risk estimates

Key Advantages:

+Dynamic risk management
+Regulatory compliance
+Stress testing

Machine Learning

Uncertainty-aware AI and model selection

Example:

Prior: regularization preferences → Training data → Posterior over models

Key Advantages:

+Robust predictions
+Active learning
+Model interpretability

Bayesian vs Classical Statistics

Understanding the fundamental philosophical and practical differences

Aspect	Classical	Bayesian	Bayesian Advantage
Parameter Nature	Fixed unknown constant	Random variable with distribution	Natural uncertainty representation
Information Used	Sample data only	Prior knowledge + sample data	Incorporates domain expertise
Interval Interpretation	95% of intervals contain parameter	95% probability parameter in interval	Direct probability statement
Small Sample Performance	May have poor coverage	Stabilized by prior information	Better finite-sample properties
Sequential Analysis	Requires stopping rules	Natural updating framework	Flexible data collection

What Makes Bayesian Statistics Unique?

Philosophical Foundations:

• Parameters as random variables: Uncertainty about unknown quantities
• Prior knowledge integration: Combine existing knowledge with new data
• Subjective probability: Degrees of belief represented as probabilities
• Natural uncertainty quantification: Full probability distributions, not just point estimates

Practical Advantages:

• Intuitive interpretation: "Probability that parameter is in this range"
• Sequential learning: Update beliefs as new data arrives
• Decision-theoretic framework: Optimal actions under uncertainty
• Hierarchical modeling: Natural framework for complex, multi-level problems

Study Tips & Best Practices

Learning Strategy:

• Start with philosophy: Understand the paradigm shift from classical statistics
• Master conjugate pairs: Learn Beta-Binomial and Gamma-Poisson thoroughly
• Practice interpretation: Focus on probability statements about parameters
• Work through examples: Apply methods to real datasets

Implementation Tips:

• Prior specification: Start with weakly informative priors
• Sensitivity analysis: Check robustness to prior choices
• Computational tools: Learn MCMC for complex problems
• Model checking: Use posterior predictive checks

Explore Further

Additional resources to deepen your understanding

Practice Problems12 comprehensive exercises Formula ReferenceComplete formula collection Bayesian CalculatorsInteractive computation tools All TopicsMathematical Statistics hub

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