Definition

1. Noun:
   • A theorem in probability theory and statistics: Bayes' theorem describes how to update the probability of a hypothesis (or cause) based on new evidence (or an observed event). It mathematically relates the conditional and prior probabilities of events.

Usage

• Bayes' theorem is used to calculate the probability of an event based on prior knowledge of conditions that might be related to the event.
• It is a fundamental concept in Bayesian statistics, machine learning, and data science for making inferences from data.

Examples

• Noun:
  • Bayes' theorem is essential for understanding diagnostic tests in medicine.
  • The researcher applied Bayes' theorem to update the model's predictions with the new data.
  • A simple explanation of Bayes' theorem involves calculating the probability of having a disease given a positive test result.

Advanced Usage

• "Posterior probability": The updated probability of a hypothesis, calculated using Bayes' theorem, which combines prior probability and new evidence.
  • The posterior probability, computed via Bayes' theorem, indicated the hypothesis was likely true.
• "Bayesian inference": The process of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence becomes available.
  • Bayesian inference relies fundamentally on the application of Bayes' theorem.

Variants and Related Words

• Bayesian (adj): Relating to or based on the methods and philosophy of Thomas Bayes, especially the use of Bayes' theorem.
  • Bayesian analysis provides a coherent framework for updating beliefs.
• Prior probability (n): The initial probability of a hypothesis before considering new evidence, a key component in Bayes' theorem.
• Likelihood (n): The probability of observing the evidence given that a hypothesis is true, another key component in Bayes' theorem.

Synonyms

• Bayes' rule: Another name for Bayes' theorem.
• Bayesian theorem: A less common synonym for Bayes' theorem.

Related Phrases

• "Apply Bayes' theorem": To use the theorem for a specific calculation.
  • To solve this problem, you need to apply Bayes' theorem correctly.
• "According to Bayes' theorem": As stated or derived from the theorem.
  • According to Bayes' theorem, the probability must be recalculated with the new information.

Noun

1. (statistics) a theorem describing how the conditional probability of a set of possible causes for a given observed event can be computed from knowledge of the probability of each cause and the conditional probability of the outcome of each cause