9.5 Bayesian hypothesis testing

Bayesian hypothesis testing offers a powerful alternative to traditional statistical methods. It allows researchers to directly compare competing hypotheses and quantify evidence strength using prior knowledge and observed data.

This approach provides a flexible framework for updating beliefs, interpreting results as probabilities, and making decisions under uncertainty. Understanding its principles and applications is crucial for modern statistical analysis across various fields.

Fundamentals of Bayesian inference

• Bayesian inference forms a cornerstone of Theoretical Statistics by providing a framework for updating beliefs based on observed data
• This approach contrasts with traditional frequentist methods by incorporating prior knowledge and quantifying uncertainty in parameter estimates

Prior and posterior distributions

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• Prior distribution represents initial beliefs about parameters before observing data
• Posterior distribution updates prior beliefs after incorporating observed data
• Shapes of distributions reflect uncertainty in parameter estimates
• Conjugate priors (Beta-Binomial, Normal-Normal) simplify posterior calculations

Likelihood function

• Quantifies probability of observing data given specific parameter values
• Plays crucial role in connecting prior beliefs with observed evidence
• Often represented as $L(\theta|x) = P(x|\theta)$ where θ represents parameters and x represents data
• Likelihood principle states all relevant information in data is contained in likelihood function

Bayes' theorem

• Fundamental equation in Bayesian inference: $P(\theta|x) = \frac{P(x|\theta)P(\theta)}{P(x)}$
• Posterior probability $P(\theta|x)$ calculated by combining prior probability $P(\theta)$ with likelihood $P(x|\theta)$
• Denominator $P(x)$ acts as normalizing constant, ensuring posterior integrates to 1
• Allows for sequential updating of beliefs as new data becomes available

Bayesian vs frequentist approaches

• Bayesian and frequentist approaches represent two major paradigms in Theoretical Statistics
• Understanding differences between these approaches crucial for interpreting statistical results and choosing appropriate methods

Philosophical differences

• Bayesian approach treats parameters as random variables with probability distributions
• Frequentist approach considers parameters as fixed, unknown constants
• Bayesians incorporate prior knowledge, frequentists rely solely on observed data
• Interpretation of probability differs: Bayesians use subjective probabilities, frequentists use long-run frequencies
• Bayesian inference focuses on entire posterior distribution, frequentist inference often uses point estimates and confidence intervals

Practical implications

• Bayesian methods allow for incorporating prior knowledge in analysis
• Frequentist methods often easier to compute and have well-established procedures
• Bayesian results directly interpretable as probabilities of hypotheses
• Frequentist p-values often misinterpreted as posterior probabilities
• Small sample inference often more reliable with Bayesian methods
• Bayesian approach provides natural framework for sequential updating of beliefs

Bayesian hypothesis testing framework

• Bayesian hypothesis testing provides alternative to traditional null hypothesis significance testing
• Allows for direct comparison of competing hypotheses and quantification of evidence strength

Null and alternative hypotheses

• Null hypothesis (H0) typically represents status quo or no effect
• Alternative hypothesis (H1) represents deviation from null or presence of effect
• Both hypotheses assigned prior probabilities reflecting initial beliefs
• Hypotheses can be simple (point) or composite (range of parameter values)
• Formulation of hypotheses crucial for meaningful interpretation of results

Bayes factor

• Quantifies relative evidence in favor of one hypothesis over another
• Defined as ratio of marginal likelihoods: $BF_{10} = \frac{P(data|H1)}{P(data|H0)}$
• Values greater than 1 indicate support for alternative hypothesis
• Can be interpreted as updating factor for prior odds to posterior odds
• Allows for continuous assessment of evidence strength, unlike p-values

Posterior odds

• Ratio of posterior probabilities of competing hypotheses
• Calculated by multiplying prior odds by Bayes factor
• Posterior odds = (Prior odds) × (Bayes factor)
• Directly interpretable as relative plausibility of hypotheses given data
• Can be converted to posterior probabilities for individual hypotheses

Prior selection

• Choice of prior distribution crucial in Bayesian analysis
• Reflects initial beliefs or state of knowledge before observing data
• Can significantly impact posterior inference, especially with small sample sizes

Informative vs non-informative priors

• Informative priors incorporate specific prior knowledge about parameters
  • Can be based on previous studies, expert opinion, or theoretical considerations
  • Tend to have narrower distributions, reflecting stronger beliefs
• Non-informative priors aim to have minimal impact on posterior inference
  • Often have flat or very wide distributions
  • Examples include uniform priors or Jeffreys priors
• Trade-off between incorporating prior knowledge and letting data dominate inference

Conjugate priors

• Prior distributions that result in posterior distributions of same family
• Simplify calculations by allowing closed-form solutions for posterior
• Common conjugate pairs include:
  • Beta prior for binomial likelihood
  • Normal prior for normal likelihood with known variance
  • Gamma prior for Poisson likelihood
• Choice of conjugate prior often motivated by computational convenience

Sensitivity analysis

• Assesses robustness of conclusions to choice of prior distribution
• Involves repeating analysis with different prior specifications
• Helps identify when prior choice significantly impacts results
• Important for establishing credibility of Bayesian analyses
• Can involve varying hyperparameters or entire prior family

Computation of Bayes factors

• Calculating Bayes factors often involves complex integrals
• Various methods available depending on model complexity and computational resources

Analytical solutions

• Closed-form solutions available for some simple models with conjugate priors
• Exact Bayes factors can be derived for nested linear models with certain priors
• Analytical solutions provide fast and precise results when applicable
• Limited to relatively simple models and specific prior choices

Numerical integration methods

• Used when analytical solutions not available but integrals still tractable
• Methods include trapezoidal rule, Simpson's rule, and Gaussian quadrature
• Accuracy depends on integration method and number of evaluation points
• Suitable for low to moderate dimensional problems

Monte Carlo techniques

• Employ random sampling to approximate complex integrals
• Methods include importance sampling, bridge sampling, and reversible jump MCMC
• Allow for computation of Bayes factors in high-dimensional models
• Accuracy improves with increased number of samples
• Computationally intensive but widely applicable to complex models

Interpretation of Bayes factors

• Bayes factors quantify relative evidence for competing hypotheses
• Interpretation guidelines help translate numerical values into meaningful conclusions

Evidence categories

• Commonly used scale proposed by Harold Jeffreys:
  • BF 1-3: Barely worth mentioning
  • BF 3-10: Substantial evidence
  • BF 10-30: Strong evidence
  • BF 30-100: Very strong evidence
  • BF >100: Decisive evidence
• Alternative scales exist, emphasizing different evidence thresholds
• Interpretation should consider context and consequences of decisions

Decision making

• Bayes factors inform but do not determine decisions
• Consider prior probabilities and loss functions for formal decision theory
• Can be used in conjunction with other criteria (effect size, practical significance)
• Allows for continuous updating of evidence as new data becomes available
• Facilitates comparison of non-nested models or multiple hypotheses

Bayesian model comparison

• Extends hypothesis testing to compare multiple competing models
• Provides framework for selecting among complex statistical models

Model selection criteria

• Bayes factors can be used to compare pairs of models
• Posterior model probabilities allow comparison of multiple models simultaneously
• Information criteria (BIC, DIC) approximate Bayesian model comparison
• Cross-validation methods assess predictive performance of models
• Trade-off between model fit and complexity often considered

Posterior model probabilities

• Represent probability of each model being true given observed data
• Calculated by normalizing marginal likelihoods across all considered models
• Allow for model averaging to account for model uncertainty
• Can be used to compute inclusion probabilities for individual predictors
• Provide natural way to quantify model selection uncertainty

Bayesian point estimation

• Focuses on obtaining single-value estimates of parameters from posterior distribution
• Complements interval estimation and hypothesis testing in Bayesian inference

Maximum a posteriori estimation

• Finds parameter value that maximizes posterior probability density
• Analogous to maximum likelihood estimation in frequentist framework
• Often used as point estimate when full posterior is complex
• Can be computed using optimization algorithms
• May not be representative of entire posterior distribution

Posterior mean and median

• Posterior mean minimizes expected squared error loss
• Posterior median minimizes expected absolute error loss
• Both account for entire posterior distribution, unlike MAP estimate
• Easily computed from MCMC samples of posterior
• Choice between mean and median depends on posterior shape and loss function

Credible intervals

• Bayesian alternative to frequentist confidence intervals
• Directly interpretable as probability statements about parameters

Highest posterior density intervals

• Interval containing specified probability mass (95%) with highest posterior density
• May consist of multiple disjoint intervals for multimodal posteriors
• Provides shortest interval for given probability content
• Computationally intensive for complex posteriors
• Invariant under parameter transformations

Equal-tailed intervals

• Interval with equal posterior probability in both tails
• Easier to compute than HPD intervals, especially from MCMC samples
• Often similar to HPD for approximately symmetric posteriors
• Not invariant under parameter transformations
• May be longer than HPD for skewed posteriors

Bayesian power analysis

• Assesses ability of study design to provide evidence for hypotheses
• Incorporates prior information and uncertainty in design phase

Sample size determination

• Determines required sample size to achieve desired level of evidence
• Often based on achieving target Bayes factor with specified probability
• Accounts for prior distribution and expected effect size
• Can consider multiple hypotheses or models simultaneously
• Allows for sequential designs with interim analyses

Prior predictive distributions

• Represent predictions about future data based on prior beliefs
• Used to simulate potential study outcomes for power analysis
• Incorporate uncertainty in both parameters and data generation process
• Allow for assessment of design robustness to different prior specifications
• Facilitate computation of expected Bayes factors or posterior probabilities

Limitations and criticisms

• Understanding limitations crucial for appropriate application of Bayesian methods
• Addressing criticisms important for advancing statistical practice

Subjectivity of priors

• Choice of prior distribution can significantly impact results, especially with small samples
• Critics argue this introduces unwarranted subjectivity into analysis
• Defenders emphasize importance of incorporating all available information
• Sensitivity analyses and robust priors can mitigate concerns
• Transparency in prior specification and justification essential

Computational challenges

• Complex models often require sophisticated MCMC techniques
• Convergence issues and long computation times can be problematic
• Approximation methods may introduce additional uncertainty
• Requires specialized software and computational expertise
• Ongoing research aims to develop more efficient algorithms and software tools

Applications in research

• Bayesian methods increasingly used across various scientific disciplines
• Provides flexible framework for complex research questions

Clinical trials

• Adaptive designs allow for interim analyses and sample size re-estimation
• Prior information from previous studies can be formally incorporated
• Facilitates decision making about treatment efficacy and safety
• Allows for probabilistic statements about treatment effects
• Useful for rare diseases or ethical constraints limiting sample sizes

Social sciences

• Hierarchical models account for nested data structures
• Latent variable models for measuring unobserved constructs
• Informative priors can incorporate expert knowledge or previous findings
• Allows for direct probability statements about hypotheses of interest
• Facilitates meta-analysis and cumulative knowledge building

Machine learning

• Bayesian neural networks quantify uncertainty in predictions
• Probabilistic topic models for text analysis and document classification
• Gaussian processes for flexible non-parametric regression
• Bayesian optimization for hyperparameter tuning
• Provides natural framework for online learning and transfer learning