9.2 Prior and posterior distributions

Prior and posterior distributions are fundamental concepts in Bayesian statistics. They allow us to incorporate existing knowledge into our analyses and update our beliefs as new data becomes available. This process of combining prior information with observed data forms the core of Bayesian inference.

Bayesian methods offer a flexible framework for statistical reasoning. By using prior distributions, we can account for uncertainty in our initial beliefs, while posterior distributions provide a complete picture of our updated knowledge after observing data. This approach enables more nuanced decision-making and uncertainty quantification.

Concept of prior distributions

• Prior distributions form the foundation of Bayesian inference in Theoretical Statistics
• Encapsulate existing knowledge or beliefs about parameters before observing data
• Allow incorporation of expert knowledge or historical information into statistical analysis

Types of prior distributions

• Continuous priors include normal, gamma, and beta distributions
• Discrete priors encompass Poisson, binomial, and negative binomial distributions
• Improper priors have infinite mass but can still lead to proper posteriors
• Jeffreys priors derived from Fisher information matrix
• Empirical priors estimated from data rather than specified a priori

Informative vs non-informative priors

• Informative priors contain substantial information about the parameter
• Non-informative priors aim to have minimal impact on posterior inference
• Uniform priors assign equal probability to all possible parameter values
• Reference priors maximize expected Kullback-Leibler divergence between prior and posterior
• Weakly informative priors provide some constraint while allowing data to dominate

Conjugate prior distributions

• Conjugate priors result in posteriors from the same distribution family
• Simplify calculations by providing closed-form posterior expressions
• Beta-binomial conjugacy used for proportion estimation
• Normal-normal conjugacy applied in mean estimation with known variance
• Gamma-Poisson conjugacy employed for rate parameter inference

Elicitation of prior information

• Structured interviews with domain experts to quantify beliefs
• Probability encoding techniques translate verbal descriptions into numerical priors
• Historical data analysis informs prior parameter choices
• Meta-analysis of previous studies synthesizes prior knowledge
• Sensitivity analysis assesses robustness to different prior specifications

Likelihood function

• Likelihood function quantifies the plausibility of observed data given parameter values
• Plays a crucial role in connecting prior beliefs with empirical evidence
• Forms the bridge between frequentist and Bayesian approaches in Theoretical Statistics

Role in Bayesian inference

• Represents the information contained in the observed data about the parameters
• Modifies prior beliefs to form posterior distribution
• Likelihood principle states all relevant information is contained in the likelihood function
• Serves as a weighting function for prior distribution in Bayes' theorem
• Determines the relative influence of prior and data on posterior inference

Relationship to prior distribution

• Prior and likelihood combined through multiplication in Bayes' theorem
• Likelihood dominates posterior when sample size is large or prior is weak
• Prior dominates posterior when sample size is small or prior is strong
• Conjugate priors chosen to simplify likelihood-prior interaction
• Non-conjugate priors require numerical integration or sampling methods

Maximum likelihood estimation

• Finds parameter values that maximize the likelihood function
• Serves as a point estimate in both frequentist and Bayesian frameworks
• Asymptotically efficient under certain regularity conditions
• Can be used to construct confidence intervals in frequentist inference
• Often serves as a starting point for more complex Bayesian analyses

Posterior distributions

• Posterior distributions represent updated beliefs after observing data
• Combine prior knowledge with likelihood information
• Central to Bayesian inference and decision-making in Theoretical Statistics

Bayes' theorem application

• Posterior probability proportional to prior probability times likelihood
• Normalizing constant ensures posterior integrates to one
• Conjugate priors simplify posterior calculations
• Numerical methods required for complex models or non-conjugate priors
• Sequential updating allows incorporation of new data over time

Interpretation of posterior probabilities

• Represent degree of belief in parameter values after observing data
• Allow for probabilistic statements about parameters (credible intervals)
• Provide full uncertainty quantification beyond point estimates
• Enable direct probability statements about hypotheses
• Facilitate decision-making under uncertainty

Point estimates from posteriors

• Posterior mean minimizes squared error loss
• Posterior median minimizes absolute error loss
• Maximum a posteriori (MAP) estimate maximizes posterior density
• Posterior mode coincides with MAP for unimodal distributions
• Choice of point estimate depends on loss function and decision problem

Updating prior beliefs

• Bayesian updating allows for sequential incorporation of new information
• Reflects the dynamic nature of knowledge acquisition in scientific inquiry
• Fundamental to adaptive learning systems in Theoretical Statistics

Sequential Bayesian updating

• Posterior from one analysis becomes prior for the next
• Allows for real-time updating as new data arrives
• Maintains computational efficiency by avoiding reprocessing of old data
• Particularly useful in online learning and streaming data contexts
• Enables adaptive experimental design and sequential decision-making

Posterior as new prior

• Encapsulates all available information up to current time point
• Simplifies storage and computation by summarizing historical data
• Facilitates transfer learning across related problems or domains
• Allows for incorporation of multiple data sources or expert opinions
• Enables hierarchical modeling and meta-analysis frameworks

Computational methods

• Advanced computational techniques enable Bayesian analysis of complex models
• Overcome limitations of analytical solutions in high-dimensional problems
• Essential tools for modern Bayesian inference in Theoretical Statistics

Markov Chain Monte Carlo

• Generates samples from posterior distribution through random walks
• Metropolis-Hastings algorithm provides a general MCMC framework
• Gibbs sampling simplifies MCMC for conditionally conjugate models
• Hamiltonian Monte Carlo improves efficiency in high dimensions
• Diagnostics (Gelman-Rubin, effective sample size) assess convergence and mixing

Gibbs sampling

• Iteratively samples from full conditional distributions of each parameter
• Particularly efficient for hierarchical and conditionally conjugate models
• Easily parallelizable for high-dimensional problems
• Automatic tuning methods available (adaptive Gibbs)
• Useful for missing data imputation and latent variable models

Metropolis-Hastings algorithm

• Proposes new parameter values and accepts/rejects based on probability ratio
• Allows sampling from arbitrary target distributions
• Tuning of proposal distribution crucial for efficiency
• Adaptive methods automatically adjust proposal during sampling
• Forms the basis for more advanced MCMC techniques (tempering, slice sampling)

Sensitivity analysis

• Assesses the robustness of Bayesian inferences to modeling assumptions
• Critical for understanding the reliability and generalizability of results
• Essential component of rigorous Bayesian analysis in Theoretical Statistics

Impact of prior choice

• Compares results across different prior specifications
• Assesses influence of prior on posterior inferences
• Identifies potential prior-data conflict
• Helps determine appropriate level of prior informativeness
• Guides selection of default priors for routine analyses

Robustness of posterior inferences

• Examines stability of conclusions across different model specifications
• Assesses sensitivity to outliers and influential observations
• Evaluates impact of different likelihood functions
• Compares results from Bayesian and frequentist approaches
• Guides reporting of uncertainty in final inferences and decisions

Applications in decision theory

• Bayesian decision theory provides a framework for optimal decision-making
• Integrates probabilistic inference with utility-based decision rules
• Fundamental to many areas of applied statistics and machine learning

Loss functions

• Quantify consequences of decisions under uncertainty
• Squared error loss leads to posterior mean as optimal estimator
• Absolute error loss results in posterior median as optimal estimator
• 0-1 loss function for classification problems
• Custom loss functions tailored to specific application domains

Bayesian decision rules

• Minimize expected posterior loss
• Account for full posterior uncertainty in decision-making
• Allow for asymmetric costs of different types of errors
• Incorporate prior probabilities of different states of nature
• Enable optimal experimental design and sample size determination

Hierarchical Bayesian models

• Hierarchical models capture complex dependencies in multi-level data
• Allow for partial pooling of information across groups or individuals
• Powerful tool for analyzing clustered or longitudinal data in Theoretical Statistics

Multilevel priors

• Specify priors at different levels of data hierarchy
• Group-level priors inform individual-level parameters
• Enable borrowing of strength across groups or individuals
• Naturally handle unbalanced designs and missing data
• Facilitate modeling of random effects and variance components

Hyperparameters

• Parameters of prior distributions in hierarchical models
• Control degree of shrinkage or pooling across groups
• Often assigned weakly informative priors
• Can be estimated from data (empirical Bayes) or given informative priors
• Sensitivity analysis assesses impact of hyperprior choices

Empirical Bayes methods

• Empirical Bayes combines Bayesian and frequentist approaches
• Estimates prior parameters from the data itself
• Bridges gap between fully Bayesian and classical methods in Theoretical Statistics

Estimation of prior parameters

• Maximum likelihood estimation of hyperparameters
• Method of moments for simple conjugate models
• EM algorithm for more complex hierarchical models
• Cross-validation techniques for tuning hyperparameters
• Parametric and nonparametric approaches to prior estimation

Advantages and limitations

• Provides data-driven prior specification
• Computationally efficient compared to full Bayesian analysis
• Can lead to improved estimation in high-dimensional problems
• May underestimate uncertainty by treating estimated priors as known
• Potential for overfitting if sample size is small relative to model complexity

Bayesian vs frequentist approaches

• Comparison of two fundamental paradigms in statistical inference
• Ongoing debate in statistical theory and practice
• Important for understanding the foundations of Theoretical Statistics

Philosophical differences

• Bayesian approach treats parameters as random variables
• Frequentist approach considers parameters as fixed but unknown
• Bayesian inference based on posterior probabilities
• Frequentist inference relies on sampling distributions and p-values
• Bayesian methods naturally incorporate prior information

Practical implications

• Bayesian methods provide direct probability statements about parameters
• Frequentist methods focus on long-run properties of estimators
• Bayesian approach handles small samples and complex models more naturally
• Frequentist methods often computationally simpler for standard problems
• Choice between approaches often depends on specific application and available resources