📊Bayesian Statistics Unit 8 – Hierarchical Bayesian models

Key Concepts and Foundations

• Hierarchical Bayesian models extend standard Bayesian models by introducing multiple levels of parameters and distributions
• Enable modeling of complex, multi-level data structures where parameters are related hierarchically
• Incorporate prior knowledge and uncertainty at different levels of the model hierarchy
• Bayesian inference combines prior information with observed data to update beliefs about parameters
• Hyperparameters define the distributions of lower-level parameters, allowing information sharing across groups or units
  • Example: In a clinical trial, patient-level parameters may be governed by group-level hyperparameters
• Partial pooling balances individual-level estimates with group-level information, improving overall parameter estimation
• Exchangeability assumes that units within a group are interchangeable given the group-level parameters
• Hierarchical models can handle nested or clustered data structures (students within schools, patients within hospitals)

Hierarchical Model Structure

• Hierarchical models consist of multiple levels of parameters and distributions, organized in a tree-like structure
• Each level of the hierarchy corresponds to a different source of variability or grouping in the data
• Lower-level parameters are typically assumed to be conditionally independent given the higher-level parameters
• The top level of the hierarchy represents the hyperparameters, which govern the distributions of lower-level parameters
• Intermediate levels capture group-specific or unit-specific effects, allowing for heterogeneity across groups
• The bottom level represents the observed data, which are modeled using likelihood functions
• Directed acyclic graphs (DAGs) visually represent the dependencies and relationships between variables in a hierarchical model
  • Nodes represent variables, and edges indicate probabilistic dependencies
• Plate notation is a compact way to represent repeated structures or replicated units in a hierarchical model

Prior and Hyperprior Distributions

• Prior distributions encode initial beliefs or knowledge about model parameters before observing data
• In hierarchical models, priors are specified at multiple levels of the hierarchy
• Hyperpriors are distributions placed on the hyperparameters, representing uncertainty about the higher-level parameters
• Conjugate priors lead to analytically tractable posterior distributions, simplifying inference
  • Example: Beta prior for a binomial likelihood, or Gamma prior for a Poisson likelihood
• Non-informative or weakly informative priors can be used when prior knowledge is limited or to let the data drive inference
• Informative priors incorporate domain expertise or previous study results into the model
• Hierarchical priors allow for borrowing strength across groups or units, improving parameter estimation
• Prior predictive checks assess the plausibility of the chosen priors by simulating data from the prior distribution
• Sensitivity analysis investigates the robustness of posterior inferences to different prior choices

Likelihood Functions in Hierarchical Models

• Likelihood functions specify the probability of observing the data given the model parameters
• In hierarchical models, likelihood functions are defined at the lowest level of the hierarchy, linking the observed data to the parameters
• The choice of likelihood function depends on the nature of the data and the assumed data-generating process
• Common likelihood functions include Gaussian (normal) for continuous data, Binomial for binary data, and Poisson for count data
• Hierarchical likelihoods allow for modeling heterogeneity across groups or units
  • Example: Each group may have its own set of likelihood parameters, governed by higher-level distributions
• Mixture models combine multiple likelihood components to model complex data distributions
• Latent variable models introduce unobserved variables to capture underlying structure or missing information
• Likelihood-based inference involves maximizing the likelihood function or sampling from the posterior distribution
• Model comparison techniques, such as Bayes factors or cross-validation, assess the relative fit of different likelihood specifications

Posterior Inference Techniques

• Posterior inference combines prior knowledge with observed data to update beliefs about model parameters
• The posterior distribution represents the updated probability distribution of the parameters given the data
• Bayes' theorem is the fundamental rule for updating beliefs, expressing the posterior as a product of the prior and the likelihood
• Markov chain Monte Carlo (MCMC) methods are commonly used for sampling from the posterior distribution
  • Metropolis-Hastings algorithm proposes new parameter values and accepts or rejects them based on a probability ratio
  • Gibbs sampling updates each parameter conditionally on the current values of the other parameters
• Variational inference approximates the posterior distribution with a simpler, tractable distribution
• Laplace approximation constructs a Gaussian approximation to the posterior distribution around the mode
• Posterior predictive checks assess the fit of the model by comparing observed data to data simulated from the posterior predictive distribution
• Credible intervals quantify the uncertainty in parameter estimates, providing a range of plausible values
• Bayesian model averaging combines inferences from multiple models, weighted by their posterior probabilities

Computational Methods and Tools

• Markov chain Monte Carlo (MCMC) algorithms are the primary computational tools for Bayesian inference in hierarchical models
• Gibbs sampling is a special case of MCMC that updates each parameter conditionally on the others
• Metropolis-Hastings algorithm proposes new parameter values and accepts or rejects them based on a probability ratio
• Hamiltonian Monte Carlo (HMC) uses gradient information to efficiently explore the posterior distribution
• Stan is a popular probabilistic programming language for specifying and fitting Bayesian models
  • Provides a high-level interface for defining models and performing inference
• JAGS (Just Another Gibbs Sampler) is another widely used tool for MCMC sampling in Bayesian models
• Variational inference algorithms approximate the posterior distribution with a simpler, tractable distribution
• Laplace approximation constructs a Gaussian approximation to the posterior distribution around the mode
• Bayesian optimization techniques can be used for model selection and hyperparameter tuning
• Parallel computing and distributed algorithms enable scaling Bayesian inference to large datasets and complex models

Applications and Case Studies

• Hierarchical Bayesian models have been applied in various domains, including social sciences, ecology, epidemiology, and marketing
• In educational research, hierarchical models can account for student-level and school-level factors affecting academic performance
• In clinical trials, hierarchical models can estimate treatment effects while accounting for patient heterogeneity and site-specific variations
• Ecological studies use hierarchical models to analyze population dynamics and species distributions across different habitats
• Marketing applications include customer segmentation, choice modeling, and brand preference analysis
• Spatial and spatio-temporal models incorporate geographical information and temporal dependencies into hierarchical structures
• Bayesian meta-analysis combines results from multiple studies while accounting for study-level heterogeneity
• Bayesian networks and graphical models represent complex dependencies and conditional relationships in hierarchical data
• Bayesian nonparametric models, such as Dirichlet process mixtures, allow for flexible modeling of unknown distributions

Advantages and Limitations

• Hierarchical Bayesian models provide a principled framework for incorporating prior knowledge and uncertainty at multiple levels
• Enable borrowing strength across groups or units, improving parameter estimation and prediction accuracy
• Can handle complex, multi-level data structures and account for heterogeneity and dependencies
• Allow for probabilistic statements and quantification of uncertainty through posterior distributions and credible intervals
• Facilitate model comparison and selection using Bayesian model averaging and Bayes factors
• Provide a coherent approach to combining information from multiple sources and propagating uncertainty
• Computational complexity can be a challenge, especially for large datasets and complex model structures
• Specifying appropriate priors and hyperpriors requires careful consideration and sensitivity analysis
• Convergence diagnostics and model checking are important to ensure the reliability of posterior inferences
• Interpreting and communicating results from hierarchical models may require additional effort and expertise
• Bayesian methods may be computationally intensive compared to frequentist approaches, although advances in computational tools and algorithms have mitigated this issue