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State Bayes' theorem and what each term means.

easy

Answer

  • P(θ  ∣  D)  =  P(D  ∣  θ)  ⋅  P(θ)  /  P(D)P({\theta}\; \mid \;D)\; = \;P(D\; \mid \;{\theta})\; \cdot \;P({\theta})\; / \;P(D).
  • Posterior = Likelihood * Prior / Evidence.
  • Prior encodes belief before seeing D.
  • Likelihood tells how well θ explains D.
  • Evidence P(D) = ∫ P(D  ∣  θ)P(D\; \mid \;{\theta}) P(θ) dθ normalizes.
  • In practice, we compute posterior up to proportionality: P(θ  ∣  D)P({\theta}\; \mid \;D) ∝ P(D  ∣  θ)  ⋅  P(θ)P(D\; \mid \;{\theta})\; \cdot \;P({\theta}), and use MCMC / variational methods to sample or approximate.
Check yourself — multiple choice
  • P(D)*P(θ)
  • Posterior ∝ Likelihood × Prior; evidence normalizes; foundation of Bayesian inference
  • Only for classification
  • Not real

Bayes: posterior ∝ likelihood × prior; evidence normalizes.

#bayesian#theory

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