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Understanding Bayesian Probability
What is Bayesian Probability?
Bayesian probability is an interpretation of probability that expresses a degree of certainty about events. It allows for the updating of probabilities as new evidence becomes available. Key concepts include:
- Prior probability - Initial belief before evidence
- Likelihood - Probability of evidence given hypothesis
- Posterior probability - Updated belief after evidence
- Bayes' theorem - Mathematical framework for updates
- Evidence incorporation - Systematic belief updating
Key Formulas
Bayes' Theorem:
P(H|E) = P(E|H) * P(H) / P(E)
Law of Total Probability:
P(E) = P(E|H) * P(H) + P(E|not H) * P(not H)
Likelihood Ratio:
LR = P(E|H) / P(E|not H)
Odds Form:
Posterior Odds = LR * Prior Odds
Properties
Fundamental Properties
- Coherence - Consistent probability assignments
- Conditionalization - Systematic updating
- Dutch book arguments - Rational betting behavior
- Calibration - Alignment with frequencies
- Exchangeability - Order independence
Advanced Concepts
- Conjugate priors
- Hierarchical models
- Model selection
- Parameter estimation
- Predictive inference
Applications
Scientific Research
- Hypothesis testing
- Parameter estimation
- Model comparison
- Experimental design
Machine Learning
- Classification
- Pattern recognition
- Neural networks
- Decision systems
Real World
- Medical diagnosis
- Risk assessment
- Quality control
- Financial forecasting