Bayesian decision theory

Bayesian decision theory chooses actions by combining probabilities about uncertain states with the consequences and preferences associated with each action.

Bayes' theorem: A rule for updating probabilities of hypotheses when new evidence arrives. It supplies the probability updates that inform decisions under uncertainty.
Decision problem: A formal description of available actions, uncertain states, outcomes, and preferences. Bayesian decision theory applies probability and preference to this structured choice.
Medical decision-making: The selection of healthcare actions by weighing evidence, outcomes, risks, and patient preferences. Diagnosis and treatment choices combine uncertain probabilities with consequences valued by patients.
Frequentist decision theory: A framework for evaluating decision rules by their long-run performance across repeated samples. It assesses procedures through repeated-sampling properties rather than posterior beliefs alone.
Thomas Bayes: An English statistician and minister whose posthumously published work established a rule for updating probabilities. His probability rule provides the framework's defining approach to uncertain belief.
Bayesian inference: A method of updating probability distributions over hypotheses using evidence and Bayes' theorem. It produces beliefs about uncertain states before actions are evaluated.
Prior probability: A probability distribution over hypotheses before incorporating the evidence under analysis. It expresses initial uncertainty that later evidence can update.
Clinical decision analysis: A structured use of decision models to compare medical strategies under uncertainty. It makes expected outcomes and trade-offs explicit in healthcare choices.
Minimax decision rule: A rule that selects the action with the smallest worst-case loss. It prioritizes protection against the most damaging state instead of posterior average performance.
Pierre-Simon Laplace: A French mathematician who developed probability theory and applied Bayesian reasoning to scientific inference. His work expanded the mathematical treatment of probability and evidence underlying Bayesian decisions.

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