15.3 Bayesian inference and decision making

Bayesian inference revolutionizes how we update beliefs using data. It combines prior knowledge with new evidence, allowing us to make probability statements about hypotheses. This approach contrasts with frequentist methods, offering a more intuitive way to quantify uncertainty.

Bayes' theorem is the cornerstone of this process, helping us calculate posterior probabilities. By applying Bayesian principles, we can make informed decisions in various fields, from medical diagnoses to spam filtering, using updated beliefs based on observed data.

Bayesian Inference vs Frequentist

Fundamental Principles and Approaches

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• Bayesian inference uses prior knowledge and observed data to update probabilities of hypotheses
• Bayes' theorem calculates posterior probabilities forming the core principle of Bayesian inference
• Bayesian approach treats parameters as random variables with probability distributions
• Frequentist approach considers parameters as fixed but unknown constants
• Prior distribution in Bayesian inference represents initial beliefs about parameters before observing data
• Likelihood function quantifies the probability of observing the data given specific parameter values
• Posterior distribution combines prior knowledge with observed data providing a full probability distribution for parameters
• Bayesian inference allows for direct probability statements about parameters and hypotheses unlike frequentist methods

Key Components of Bayesian Analysis

• Prior distribution encapsulates existing knowledge or beliefs about parameters (uninformative priors, informative priors)
• Likelihood function represents the probability of observing the data given the model parameters
• Posterior distribution results from combining prior and likelihood providing updated parameter estimates
• Marginal likelihood normalizes the posterior ensuring probabilities sum to 1
• Credible intervals in Bayesian analysis differ from frequentist confidence intervals providing direct probability statements
• Bayesian model comparison uses Bayes factors to assess relative evidence for competing models
• Hierarchical Bayesian models incorporate multiple levels of uncertainty in parameter estimation

Updating Beliefs with Bayes' Theorem

Mathematical Formulation and Interpretation

• Bayes' theorem expressed as $P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$ where P(A|B) represents the posterior probability
• Prior probability P(A) represents initial beliefs about the hypothesis before observing new data
• Likelihood P(B|A) quantifies the probability of observing the data given that the hypothesis is true
• Marginal likelihood P(B) acts as a normalizing constant ensuring posterior probabilities sum to 1
• Posterior probability P(A|B) combines prior beliefs with observed data to update probability estimates
• Strength of evidence required to change prior beliefs depends on the strength and specificity of those prior beliefs
• Odds form of Bayes' theorem $\frac{P(A|B)}{P(A^c|B)} = \frac{P(B|A)}{P(B|A^c)} \times \frac{P(A)}{P(A^c)}$ illustrates how evidence updates prior odds

Practical Applications and Considerations

• Bayesian updating allows for incorporation of multiple pieces of evidence sequentially
• Iterative application uses the posterior from one analysis as the prior for the next
• Conjugate priors simplify calculations by resulting in posteriors of the same distribution family
• Bayesian inference handles small sample sizes more effectively than frequentist approaches
• Updating medical diagnoses based on test results (sensitivity, specificity)
• Spam filtering using naive Bayes classifiers to update email classification probabilities
• A/B testing in marketing updating conversion rate estimates as new data is collected

Posterior Distributions for Model Parameters

Construction and Interpretation

• Posterior distribution proportional to the product of prior distribution and likelihood function
• Conjugate priors result in posterior distributions of the same family as the prior (beta-binomial, normal-normal)
• Numerical methods like Markov Chain Monte Carlo (MCMC) approximate posteriors when analytical solutions unavailable
• Choice of prior distribution can significantly impact the posterior especially with small samples or weak likelihoods
• Posterior predictive distributions derived from posterior distribution make predictions about future observations
• Credible intervals from posterior distributions provide probability statements about parameter values
• Posterior mode, mean, and median serve as point estimates for parameters depending on loss functions

Computational Methods and Considerations

• Gibbs sampling iteratively samples from conditional distributions to approximate joint posterior
• Metropolis-Hastings algorithm uses proposal distributions to explore parameter space and approximate posteriors
• Hamiltonian Monte Carlo improves sampling efficiency for complex posterior distributions
• Variational inference provides faster approximate posteriors for high-dimensional problems
• Importance sampling reweights samples from a proposal distribution to estimate posterior expectations
• Sensitivity analysis assesses impact of prior choices on resulting posterior distributions
• Convergence diagnostics (Gelman-Rubin statistic, trace plots) ensure reliable MCMC posterior approximations

Decision Making with Posterior Probabilities

Bayesian Decision Theory Fundamentals

• Bayesian decision theory combines posterior probabilities with utility functions to determine optimal actions
• Expected utility of an action calculated by summing products of outcome utilities and posterior probabilities
• Principle of maximum expected utility states rational decision maximizes expected utility
• Bayesian risk defined as expected loss under uncertainty minimized for optimal decisions
• Posterior predictive probabilities assess likelihood of future events informing decision-making
• Value of information quantified to determine whether additional data collection worthwhile before deciding
• Loss functions (squared error, absolute error) formalize the consequences of incorrect decisions

Advanced Decision-Making Techniques

• Multi-attribute utility theory extends Bayesian decision-making to scenarios with multiple potentially conflicting objectives
• Influence diagrams graphically represent decision problems incorporating probabilistic dependencies
• Sequential decision-making uses dynamic programming to optimize decisions over time (medical treatment plans)
• Bayesian experimental design optimizes data collection to maximize information gain for decision-making
• Robust Bayesian analysis considers families of prior distributions to assess decision sensitivity
• Partial identification methods handle situations with limited prior information or model uncertainty
• Group decision-making aggregates multiple experts' posterior probabilities (linear opinion pool, logarithmic opinion pool)