Bayesian Inference: Complete Problems

# Write your bayes_theorem function here
def bayes_theorem(prior, likelihood, evidence_probability):
    """Calculate posterior probability using Bayes' theorem

    Args:
        prior: P(H) - prior probability of hypothesis
        likelihood: P(E|H) - likelihood of evidence given hypothesis
        evidence_probability: P(E) - total probability of evidence

    Returns:
        posterior: P(H|E) - posterior probability
    """
    # Your code here
    pass

# Example usage:
# prior = 0.01  # P(Disease)
# likelihood = 0.95  # P(Positive|Disease)
# evidence_prob = 0.06  # P(Positive)
# posterior = bayes_theorem(prior, likelihood, evidence_prob)
# print(f"Posterior probability: {posterior:.3f}")

# Write your BeliefUpdater class here
class BeliefUpdater:
    """Class for sequential Bayesian belief updates"""

    def __init__(self, initial_belief):
        """Initialize with prior belief P(H)"""
        self.current_belief = initial_belief

    def update_belief(self, likelihood, evidence_probability):
        """Update belief with new evidence

        Args:
            likelihood: P(E|H) - likelihood of new evidence
            evidence_probability: P(E) - probability of this evidence

        Returns:
            new_belief: Updated P(H|E)
        """
        # Your code here
        pass

    def get_current_belief(self):
        """Return current belief probability"""
        # Your code here
        pass

# Example usage:
# updater = BeliefUpdater(0.01)  # Start with 1% belief
# updater.update_belief(0.95, 0.06)  # First test
# print(f"After first test: {updater.get_current_belief():.3f}")
# updater.update_belief(0.90, 0.15)  # Second test
# print(f"After second test: {updater.get_current_belief():.3f}")

# Write code to test your functions with the cancer screening example from Problem 1
# Use your bayes_theorem function and BeliefUpdater class
# Verify you get: 3.6% → 20.9% → 96.2%

# Test the basic function:
# prior = 0.036  # 3.6%
# likelihood = 0.92  # First test sensitivity
# evidence_prob = 0.07664  # P(+) for first test
# posterior1 = bayes_theorem(prior, likelihood, evidence_prob)

# Test the sequential class:
# updater = BeliefUpdater(0.036)  # Start with 3.6%
# updater.update_belief(0.92, 0.07664)  # First test
# updater.update_belief(0.95, ?)  # Second test (you need to calculate P(++))
# print(f"Final belief: {updater.get_current_belief():.3f}")