Editing Bayesian Statistics (section)

== <span style="color: #FFFFFF;">Applying</span> ==
'''Calculating Posterior Probability (Medical Test):'''
<syntaxhighlight lang="python">
def bayes_update(prior, sensitivity, specificity):
    """
    prior: P(Disease)
    sensitivity: P(+ | Disease)
    specificity: P(- | No Disease)
    """
    # P(Positive Test | No Disease)
    false_positive_rate = 1 - specificity
    
    # P(Evidence) = P(E|H)P(H) + P(E|~H)P(~H)
    p_evidence = (sensitivity * prior) + (false_positive_rate * (1 - prior))
    
    # Posterior P(H|E)
    posterior = (sensitivity * prior) / p_evidence
    return posterior

# Rare disease: 0.1% of population
# Test: 95% sensitive, 95% specific
prior_prob = 0.001
result = bayes_update(prior_prob, 0.95, 0.95)

print(f"Prior Probability: {prior_prob*100:.2f}%")
print(f"Posterior Probability after Positive Test: {result*100:.2f}%")
# Even with a '95% accurate' test, you only have ~1.8% chance of being sick!
</syntaxhighlight>

; Bayesian Applications
: '''Spam Filters''' → Updating the probability that an email is "Spam" based on the presence of words like "Free" or "Viagra."
: '''Self-Driving Cars''' → Constantly updating the "Belief State" of where other cars are based on noisy sensor data (Kalman Filters).
: '''Medical Diagnosis''' → Doctors implicitly use Bayesian reasoning when they consider a patient's symptoms alongside their age and history.
: '''Scientific Discovery''' → Using "Bayesian Search Theory" to find missing ships or planes (like the search for MH370).
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