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Bayes' Rule

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Bayes' rule turns a test result into a belief:

  • — the prior: how likely the hypothesis was before you saw anything.
  • — the likelihood: how likely this evidence is if the hypothesis holds.
  • — the evidence: how likely this evidence is at all, counting both worlds.
  • — the posterior: your updated belief.

    • The classic result: a disease affects 1 in 100 people, and a test catches 99% of cases with a 5% false positive rate. You test positive. Your chance of having it is not 99%:

    P(E) = 0.99*0.01 + 0.05*0.99 = 0.0594
P(H|E) = 0.99*0.01 / 0.0594 = 0.1667

    About 17%. The prior dominates because the disease is rare, and no amount of intuition gets you there without doing the arithmetic. This is also exactly why accuracy is a bad metric on imbalanced data.

    Your task:

    Implement posterior(prior, likelihood, false_positive_rate) returning as a float.

    false_positive_rate is .

    Example Tests

    The rare-disease case: a positive test still leaves you probably fine

    Input: {"prior":0.01,"likelihood":0.99,"false_positive_rate":0.05}

    Expected: 0.16667

    A perfect test with no false positives confirms the hypothesis outright

    Input: {"prior":0.3,"likelihood":0.9,"false_positive_rate":0}

    Expected: 1

    Evidence that is equally likely either way leaves the prior untouched

    Input: {"prior":0.4,"likelihood":0.5,"false_positive_rate":0.5}

    Expected: 0.4

    Python
    def posterior(prior, likelihood, false_positive_rate):
    """
    Bayes' rule for a binary hypothesis.

    Args:
        prior:               P(H)
        likelihood:          P(E | H)
        false_positive_rate: P(E | not H)

    Returns:
        float: P(H | E)
    """
    # YOUR CODE HERE
    pass
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