Baye’s Theorem

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Bayes’ Theorem

Definition

Bayes’ Theorem is a formula for updating the probability of a hypothesis (event A) given new evidence (event B). It converts a prior probability into a posterior probability by incorporating the likelihood of the observed evidence.

Why it matters

Bayes’ Theorem lets you revise beliefs rationally when new information arrives. It’s widely used in medical testing, finance, machine learning, spam filtering, and any setting where conditional probabilities matter.

The formula

P(A|B) = [P(A) · P(B|A)] / P(B)

Where:
* P(A) = prior probability of A (before seeing B)
* P(B) = probability of observing B
* P(B|A) = likelihood: probability of B given A
* P(A|B) = posterior probability of A after observing B

Derivation (brief): P(A|B) = P(A ∩ B) / P(B). Since P(A ∩ B) = P(A) · P(B|A), substituting yields Bayes’ formula above.

Key concepts

  • Prior probability: your initial belief about A before new data.
  • Likelihood: how probable the new data is under A.
  • Posterior probability: updated belief about A after seeing the data.

Simple examples

Special considerations

  • Base-rate (prior) sensitivity: Posterior results can be dominated by the prior when the event is rare. Ignoring base rates leads to the base-rate fallacy.
  • Test characteristics: Sensitivity and specificity directly affect P(B|A) and P(B|¬A).
  • Independence and model assumptions: Bayes’ calculations assume the probabilities provided are correct and events are modeled appropriately.
  • Continuous problems: In Bayesian statistics, priors and posteriors are often continuous distributions; Bayes’ rule applies to densities as well.

When to use Bayes’ Theorem

Use Bayes’ Theorem when you need the probability of an event given related evidence—for example:
* Interpreting medical or diagnostic tests
* Updating risk assessments in finance
* Inference tasks in machine learning (classification, spam detection)
* Any decision-making where beliefs are updated with new data

Key takeaways

  • Bayes’ Theorem provides a principled way to update probabilities in light of new evidence.
  • The prior matters: low-prevalence scenarios can produce counterintuitive posteriors even with accurate tests.
  • It is foundational to Bayesian statistics and widely applicable across disciplines.

Bottom line

Bayes’ Theorem links prior belief and new evidence to produce a revised probability. Proper use requires careful specification of priors and likelihoods; when applied correctly, it yields clearer, more rational assessments of uncertain events.