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Probability for Machine Learning
/Bayesian Inference

Bayesian Inference

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Bayesian Inference

Bayesian inference provides a systematic framework for updating beliefs based on new evidence, combining prior knowledge with observed data. In everyday terms, it's like how we naturally revise our opinions when we encounter new information.

Core Components:

  • Prior Distribution P(H): Your initial beliefs before seeing data - like assuming a new restaurant is average quality before trying it. In ML, this might be initial guesses about model parameters.
  • Likelihood P(D|H): How well your hypothesis explains the observed data - like how likely you would see these customer reviews if the restaurant was truly excellent.
  • Posterior Distribution P(H|D): Your updated beliefs after seeing evidence, calculated using Bayes' theorem: P(H|D) = (P(D|H) ⋅ P(H))⁄P(D). This is your revised opinion about the restaurant after reading reviews.
  • Evidence P(D): The overall probability of observing your data regardless of hypothesis.

Practical ML Applications:

  • Bayesian Neural Networks: Add uncertainty estimates to predictions, critical for medical diagnosis or autonomous driving where knowing confidence matters.
  • A/B Testing: Make decisions with smaller sample sizes by incorporating prior knowledge about user behavior.
  • Natural Language Processing: Naive Bayes classifiers for spam detection and document classification.
  • Recommendation Systems: Personalize recommendations while accounting for limited user data.
  • Computer Vision: Object detection with uncertainty bounds.

Real-World Example: Consider a medical diagnosis system. A purely frequentist approach might flag a rare condition based solely on test results, creating false alarms. A Bayesian system incorporates the prior knowledge that the condition is rare, reducing false positives while properly updating beliefs when multiple indicators suggest the condition is present. This balance of prior knowledge with new evidence mirrors how experienced doctors think, making Bayesian systems particularly valuable for high-stakes decisions where both accuracy and appropriate caution matter.