9.2 Prior and posterior distributions
7 min read•august 21, 2024
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 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
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- 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
- 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
- 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
- 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 () 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
Key Terms to Review (18)
Bayes' theorem: Bayes' theorem is a mathematical formula used to update the probability of a hypothesis based on new evidence. This theorem illustrates how conditional probabilities are interrelated, allowing one to revise predictions or beliefs when presented with additional data. It forms the foundation for concepts like prior and posterior distributions, playing a crucial role in decision-making under uncertainty.
Bayesian Inference: Bayesian inference is a statistical method that applies Bayes' theorem to update the probability of a hypothesis as more evidence or information becomes available. This approach combines prior beliefs with new data to produce posterior probabilities, allowing for continuous learning and refinement of predictions. It plays a crucial role in understanding relationships through conditional probability, sufficiency, and the formulation of distributions, particularly in complex settings like multivariate normal distributions and hypothesis testing.
Bayesian Posterior: The Bayesian posterior is the updated probability distribution of a parameter after observing new evidence or data, calculated using Bayes' theorem. This concept highlights how prior beliefs about a parameter (the prior distribution) are revised in light of new information, resulting in the posterior distribution, which serves as the foundation for making statistical inferences.
Conjugate Prior: A conjugate prior is a type of prior distribution that, when combined with a likelihood function from a given statistical model, results in a posterior distribution that is in the same family as the prior. This property simplifies the process of Bayesian updating since it allows for analytical solutions, making computations more manageable. Conjugate priors help in maintaining consistency in the modeling process and provide a systematic way to incorporate prior beliefs into the analysis.
Empirical Bayes: Empirical Bayes is a statistical approach that combines prior information with observed data to improve the estimation of parameters in a Bayesian framework. It uses data to estimate the prior distribution, allowing for a more informed posterior distribution without requiring subjective priors. This method bridges the gap between Bayesian and frequentist statistics by providing a practical way to apply Bayesian principles in real-world problems.
Full Bayes: Full Bayes refers to the complete Bayesian approach to statistical inference where prior beliefs are updated with new evidence to obtain a posterior distribution. This method emphasizes the importance of incorporating all available information through prior distributions, which represent initial beliefs about parameters, and combining them with likelihood functions derived from observed data. The result is a comprehensive framework for making probabilistic statements about unknown parameters.
Importance Sampling: Importance sampling is a statistical technique used to estimate properties of a particular distribution while sampling from a different distribution. This method is especially useful when dealing with high-dimensional integrals or rare events, allowing for more efficient simulations. It focuses on sampling from regions of interest in the probability space, which improves the accuracy of estimates for probabilities or expectations without requiring a proportional amount of computational resources.
Informative prior: An informative prior is a type of prior distribution in Bayesian statistics that reflects specific knowledge or beliefs about a parameter before observing any data. This concept plays a crucial role in updating beliefs based on new evidence, as it allows for incorporating existing information into the analysis, leading to a more nuanced posterior distribution. Informative priors can significantly influence the results of Bayesian inference, especially when data is sparse or limited.
Likelihood function: The likelihood function is a fundamental concept in statistics that measures the probability of observing the given data under different parameter values in a statistical model. It connects closely to estimation techniques, allowing us to determine the most likely parameters that could have generated the observed data. The likelihood function is crucial in various statistical methodologies, including parameter estimation and hypothesis testing, serving as a bridge between frequentist and Bayesian approaches.
Marginal likelihood: Marginal likelihood is the probability of observing the data given a model, integrated over all possible parameter values of that model. It plays a crucial role in Bayesian inference as it allows for the comparison of different models by evaluating how well each model explains the observed data. Understanding marginal likelihood helps in determining the posterior distribution and making informed decisions about which model is most appropriate.
Markov Chain Monte Carlo: Markov Chain Monte Carlo (MCMC) is a class of algorithms used to sample from probability distributions based on constructing a Markov chain. The key idea is that through this chain, we can approximate complex distributions that might be difficult to sample from directly, making it especially useful in Bayesian inference and estimation. MCMC allows us to derive posterior distributions, apply Bayes' theorem effectively, and estimate parameters by drawing samples that converge to the desired distribution over time.
Model Comparison: Model comparison is a statistical approach used to evaluate and contrast different models in order to determine which one best explains the data at hand. It involves analyzing prior and posterior distributions to assess how well each model fits the observed data, guiding researchers in selecting the most appropriate model based on criteria such as predictive accuracy and complexity. This process is crucial for understanding uncertainty and making informed decisions based on the models' performance.
Normalizing constant: A normalizing constant is a factor used to ensure that a probability distribution sums or integrates to one. In the context of prior and posterior distributions, it is crucial for making sure that the total probability is valid. This constant helps in adjusting the likelihood of outcomes so they fit within the bounds of a proper probability measure, facilitating accurate Bayesian inference.
Pierre-Simon Laplace: Pierre-Simon Laplace was a prominent French mathematician and astronomer known for his foundational contributions to statistical theory and probability. His work laid the groundwork for modern inferential statistics, including the development of Bayes' theorem, which relates conditional probabilities and is essential for understanding the roles of prior and posterior distributions as well as the concept of conjugate priors.
Posterior updating: Posterior updating is the process of revising beliefs or estimates about a parameter after observing new evidence, utilizing Bayes' theorem to combine prior distributions with likelihoods. This concept is essential in Bayesian statistics, as it allows statisticians to adjust their understanding based on incoming data, thereby refining predictions and inferences about unknown quantities.
Prior belief adjustment: Prior belief adjustment refers to the process of modifying initial beliefs or assumptions about a parameter based on new evidence or data. This concept is central to Bayesian statistics, where prior beliefs are quantified into prior distributions that are updated to form posterior distributions as new information is incorporated. The adjustments made reflect a more accurate understanding of the uncertainty surrounding the parameter in light of the observed data.
Thomas Bayes: Thomas Bayes was an 18th-century statistician and theologian best known for formulating Bayes' theorem, a fundamental principle in probability theory that describes how to update the probability of a hypothesis based on new evidence. His work laid the groundwork for Bayesian inference, allowing for the use of prior knowledge to refine estimates and improve decision-making processes across various fields.
Uninformative Prior: An uninformative prior is a type of prior distribution that provides minimal or no specific information about the parameters of interest in a Bayesian analysis. This kind of prior is often used to reflect a state of ignorance regarding the parameter values, allowing the data to play a more significant role in shaping the posterior distribution. The aim is to avoid introducing bias into the analysis while still enabling the incorporation of prior beliefs when necessary.