Hassan Ijaz
Ai, Web & Design
Bayes' theorem and Bayesian inference
Medical test simulator where users adjust disease prevalence and test accuracy sliders to see how posterior probabilities change, with visual prior/posterior distributions
Concept Overview
Bayes' theorem is a fundamental result in probability theory that describes how to update beliefs in light of new evidence. It forms the foundation of Bayesian inference and has applications from medical diagnosis to machine learning.
The Formula
P(A|B) = (P(B|A) × P(A))/ P(B)
P(A|B): Posterior probability - What we want to find
P(B|A): Likelihood - How likely is the evidence given our hypothesis
P(A): Prior probability - Our initial belief before seeing evidence
P(B): Marginal likelihood - Total probability of the evidence
Alternative Forms
P(B) = P(B|A) × P(A) + P(B|A') × P(A')
Law of Total Probability - useful when P(B) is hard to calculate directly
P(A|B) ∝ P(B|A) × P(A)
Proportional form - often used when comparing hypotheses
Medical Testing Example
Consider a disease test with:
- Disease prevalence: 1% (prior)
- Test sensitivity: 99% (true positive rate)
- Test specificity: 95% (true negative rate)
If someone tests positive, what's the probability they have the disease?
P(Disease|Positive) = 0.99 × 0.01 / (0.99 × 0.01 + 0.05 × 0.99) ≈ 16.7%
Surprisingly low! This is because the disease is rare, so most positives are false positives.
Key Insights
- Base Rate Matters:Rare conditions remain unlikely even with positive tests
- Evidence Strength Varies:Some evidence is more informative than others
- Sequential Updates:Can apply Bayes' theorem multiple times as new evidence arrives
- Prior Selection Matters:Different priors can lead to different conclusions
Common Applications
Medical Diagnosis
Updating disease probability based on test results and symptoms
Spam Filtering
Classifying emails based on word frequencies
Machine Learning
Naive Bayes classifiers, Bayesian neural networks
A/B Testing
Bayesian approaches to experiment analysis
Remember: Bayes' theorem is about updating beliefs rationally. The posterior combines what we knew before (prior) with what the data tells us (likelihood).
Use the medical test simulator below to see how changing disease prevalence and test accuracy affects the posterior probability. Watch how the visual prior and posterior distributions change as you adjust the parameters.
Interactive Visualization
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Medical test simulator where users adjust disease prevalence and test accuracy sliders to see how posterior probabilities change, with visual prior/posterior distributions