The likelihood principle is a cornerstone of statistical inference, guiding how we draw conclusions from data. It states that all relevant information for inference about a parameter is contained in the likelihood function , influencing how we update our beliefs based on observations.
This principle aligns closely with Bayesian methods, forming the basis for updating prior beliefs to posterior distributions. It challenges traditional frequentist approaches, encouraging a focus on parameter estimation and model comparison rather than binary hypothesis testing decisions.
Definition of likelihood principle
Fundamental concept in statistical inference guides how to draw conclusions from observed data
Asserts that all relevant information for inference about a parameter is contained in the likelihood function
Plays a crucial role in Bayesian statistics by influencing how we update our beliefs based on observed data
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All information about the parameter θ contained in a sample x is given by the likelihood function L(θ|x)
Two likelihood functions containing the same information lead to identical inferences about θ
Mathematically expressed as: If L1(θ|x1) ∝ L2(θ|x2), then inferences about θ should be the same for both samples
Intuitive explanation
Compares the plausibility of different parameter values given observed data
Focuses on the relative support for different parameter values rather than their absolute probabilities
Emphasizes the importance of how likely the observed data is under different parameter values
Historical context
Introduced by George Barnard and separately by Leonard Savage in the early 1960s
Developed as a response to limitations of traditional frequentist approaches
Gained prominence with the rise of Bayesian statistics and computational methods
Foundations of likelihood principle
Sufficiency principle
States that a sufficient statistic contains all relevant information about a parameter
Implies that inference should depend only on the sufficient statistic, not the full dataset
Examples of sufficient statistics include sample mean for normal distribution with known variance
Conditionality principle
Asserts that inference should be based only on the experiment actually performed
Eliminates consideration of hypothetical experiments that could have been conducted
Helps focus analysis on relevant data and avoid misleading conclusions
Relationship to Bayesian inference
Likelihood principle naturally aligns with Bayesian approach to statistical inference
Forms the basis for updating prior beliefs to posterior distributions in Bayesian analysis
Allows incorporation of prior knowledge while still respecting the information in the data
Implications for statistical inference
Frequentist vs Bayesian approaches
Likelihood principle more closely aligned with Bayesian methods than frequentist approaches
Frequentist methods often violate the likelihood principle (p-values, confidence intervals)
Bayesian methods naturally adhere to the likelihood principle through use of Bayes' theorem
Impact on hypothesis testing
Challenges traditional null hypothesis significance testing based on p-values
Encourages focus on parameter estimation and model comparison rather than binary decisions
Promotes use of likelihood ratios or Bayes factors for comparing hypotheses
Influence on parameter estimation
Supports use of maximum likelihood estimation and Bayesian posterior estimation
Discourages use of estimators that depend on sampling distribution (unbiased estimators)
Emphasizes importance of considering full likelihood function, not just point estimates
Applications of likelihood principle
Maximum likelihood estimation
Method for finding parameter values that maximize the likelihood of observed data
Widely used in various fields (economics, psychology, biology)
Provides basis for many statistical techniques (logistic regression, generalized linear models)
Likelihood ratio tests
Compare relative support for two nested models or hypotheses
Calculate ratio of likelihoods under different parameter constraints
Used in various contexts (model selection, hypothesis testing)
Model selection criteria
Akaike Information Criterion (AIC) based on likelihood and model complexity
Bayesian Information Criterion (BIC) incorporates likelihood and sample size
Deviance Information Criterion (DIC) extends model selection to Bayesian hierarchical models
Criticisms and limitations
Violation in some scenarios
Can lead to paradoxical results in certain situations (Basu's example)
May not always align with intuitive notions of evidence
Potential issues with improper prior distributions in Bayesian analysis
Challenges in implementation
Computational difficulties in calculating likelihoods for complex models
Sensitivity to model misspecification or outliers
Requires careful consideration of model assumptions and parameterization
Alternative principles
Frequentist principles (repeated sampling, control of long-run error rates)
Fiducial inference proposed by R.A. Fisher
Decision-theoretic approaches focusing on loss functions and utilities
Likelihood principle in practice
Examples in data analysis
Estimating population parameters from sample data (mean, variance)
Fitting regression models to predict outcomes based on predictors
Analyzing survival data to estimate hazard rates and treatment effects
Software implementations
R packages (stats, bbmle) for maximum likelihood estimation and model fitting
Python libraries (scipy.stats, statsmodels) for likelihood-based inference
Stan and JAGS for Bayesian modeling incorporating likelihood principle
Best practices for application
Carefully specify model assumptions and parameterization
Conduct sensitivity analyses to assess robustness of results
Use multiple methods (maximum likelihood, Bayesian) to cross-validate findings
Profile likelihood
Technique for dealing with nuisance parameters in likelihood-based inference
Involves maximizing likelihood over nuisance parameters for each value of parameter of interest
Useful for constructing confidence intervals and hypothesis tests
Marginal likelihood
Integral of likelihood function over parameter space
Key component in Bayesian model selection and averaging
Challenging to compute for complex models, often requiring approximation methods
Integrated likelihood
Similar to marginal likelihood but integrates over subset of parameters
Used in Bayesian hierarchical models and mixed-effects models
Allows for more efficient inference in presence of nuisance parameters
Likelihood principle in Bayesian statistics
Role in posterior distribution
Likelihood function combines with prior distribution to form posterior distribution
Determines how much the prior beliefs are updated based on observed data
Crucial in balancing prior knowledge with empirical evidence
Influence on prior selection
Likelihood principle suggests using proper priors to avoid improper posterior distributions
Encourages use of weakly informative priors when little prior knowledge is available
Helps in identifying potential conflicts between prior beliefs and observed data
Connection to Bayes factors
Bayes factors compare marginal likelihoods of competing models
Provide a Bayesian alternative to frequentist hypothesis testing
Allow for quantifying evidence in favor of one model over another