12.2 Bayesian decision theory

Bayesian decision theory combines probability and decision-making principles to guide rational choices under uncertainty. It updates beliefs based on new evidence, maximizing expected utility in various scenarios like investments or product launches.

Key components include states of nature, actions, utilities, and probabilities. The theory uses Bayes' theorem to update beliefs, calculates expected utility, and employs decision trees and sensitivity analysis to evaluate choices and their robustness.

Foundations and Components of Bayesian Decision Theory

Foundations of Bayesian decision theory

Top images from around the web for Foundations of Bayesian decision theory

• 

  Frontiers | Indices of Effect Existence and Significance in the Bayesian Framework View original

  Is this image relevant?

• 

  Bayes' theorem - Wikipedia View original

  Is this image relevant?

• 

  Bayesian Probability Illustration Diagram | TikZ example View original

  Is this image relevant?

• 

  Frontiers | Indices of Effect Existence and Significance in the Bayesian Framework View original

  Is this image relevant?

• 

  Bayes' theorem - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Foundations of Bayesian decision theory

• 

  Frontiers | Indices of Effect Existence and Significance in the Bayesian Framework View original

  Is this image relevant?

• 

  Bayes' theorem - Wikipedia View original

  Is this image relevant?

• 

  Bayesian Probability Illustration Diagram | TikZ example View original

  Is this image relevant?

• 

  Frontiers | Indices of Effect Existence and Significance in the Bayesian Framework View original

  Is this image relevant?

• 

  Bayes' theorem - Wikipedia View original

  Is this image relevant?

1 of 3

• Bayesian probability theory interprets probability subjectively updates beliefs based on new evidence (scientific experiments)
• Decision theory principles guide rational decision-making under uncertainty maximize expected utility (investment strategies)
• Historical context traces contributions of Thomas Bayes and Pierre-Simon Laplace shaped modern statistical inference
• Relationship to classical statistical methods differs from frequentist approaches focuses on subjective probabilities

Components of Bayesian decision problems

• States of nature represent possible outcomes or scenarios form mutually exclusive and exhaustive set (weather conditions)
• Actions or decisions encompass available choices for the decision-maker constitute set of possible alternatives (product launch strategies)
• Utilities provide numerical representation of preferences defined by utility functions and their properties (risk tolerance levels)
• Prior probabilities reflect initial beliefs about likelihood of states based on existing knowledge (market trends)
• Posterior probabilities update beliefs after observing new evidence incorporate latest information (customer feedback)
• Likelihood function calculates probability of observing data given a state links observed data to underlying states

Application and Analysis of Bayesian Decision Theory

Optimal decisions under uncertainty

• Expected utility calculation uses formula $E[U(a)] = \sum_{s} P(s) \cdot U(a,s)$ sums over all possible states
• Bayes' theorem application updates probabilities with $P(s|D) = \frac{P(D|s) \cdot P(s)}{P(D)}$ incorporates new data
• Decision rules aim to maximize expected utility or minimize expected loss guide rational choice
• Value of information analysis calculates expected value of perfect information helps determine worth of additional data
• Decision trees provide graphical representation of decision problems solved through backward induction

Sensitivity of Bayesian decisions

• Sensitivity analysis techniques include one-way, two-way, and probabilistic methods assess decision robustness
• Impact of prior probability changes evaluates decision sensitivity identifies critical probability thresholds
• Utility function sensitivity examines effects of risk aversion on decisions utilizes utility elicitation methods
• Bayesian model averaging accounts for model uncertainty combines multiple models
• Robust decision-making develops strategies for decisions under deep uncertainty considers multiple scenarios
• Value of information revisited determines when to gather additional information balances cost and benefit of new data