Detection theory

Detection theory is the mathematical study of deciding which signal, event, or hypothesis best explains noisy observations. It quantifies decision rules and the errors they produce.

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Likelihood ratio test: A decision rule that compares the likelihoods of observed data under competing hypotheses. It implements detection by comparing how well each hypothesis explains the same observation.
Statistical hypothesis testing: A framework for evaluating competing claims about data using probability models and decision rules. Detection theory specializes this framework to deciding which signal or event produced observations.
Radar: A system that detects objects and estimates their range or motion by transmitting and receiving radio waves. Radar uses detection rules to decide whether echoes indicate targets amid clutter and noise.
Bayesian decision theory: A framework for choosing actions by minimizing expected loss under uncertainty and prior probabilities. It supplies a broader decision framework when detection errors have unequal consequences.
Receiver operating characteristic: A curve showing the trade-off between true-positive and false-positive rates across decision thresholds. It displays how changing a detector’s threshold changes its two main error rates.
Likelihood function: A function measuring how probable observed data are under different parameter values or hypotheses. Likelihoods rank competing explanations of the same noisy data.
Digital communication: The transmission of information as discrete symbols represented by signals. A receiver detects which transmitted symbol best explains a noisy waveform.
Minimax decision rule: A decision rule that minimizes the worst-case risk across possible states or models. It contrasts with Bayesian rules by guarding against the worst case rather than averaging over priors.
Bayes risk: The expected loss of a decision rule, averaged over hypotheses and observations. It lets detection rules account for prior probabilities and unequal costs of mistakes.
Bayes' theorem: A rule for updating probabilities of hypotheses using observed evidence. It converts likelihood evidence and prior beliefs into posterior probabilities for decisions.