quiz.json

{
  "questions": [
    {
      "stage": "pre",
      "question": "In Bayes' theorem, what is the 'prior'?",
      "options": ["The probability of the evidence", "Your belief about a hypothesis before observing any evidence", "The probability of the evidence given the hypothesis", "The final updated probability after seeing evidence"],
      "correct": 1,
      "explanation": "The prior P(A) represents your initial belief about hypothesis A before seeing any data. It gets updated to the posterior P(A|B) after observing evidence B."
    },
    {
      "stage": "pre",
      "question": "A rare disease affects 1 in 10,000 people. A 99% accurate test returns positive. What is the approximate probability you have the disease?",
      "options": ["99%", "About 1%", "50%", "10%"],
      "correct": 1,
      "explanation": "Despite 99% test accuracy, the disease is so rare (0.01%) that most positive results come from healthy people (false positives). Bayes' theorem gives P(sick|positive) = ~0.98%, not 99%."
    },
    {
      "stage": "post",
      "question": "What is Laplace smoothing in a Naive Bayes classifier and why is it necessary?",
      "options": ["It normalizes feature values to have zero mean and unit variance", "It adds a small count to every feature to prevent zero probabilities from unseen words, which would zero out the entire product", "It smooths the decision boundary between classes", "It reduces the dimensionality of the feature space"],
      "correct": 1,
      "explanation": "Without smoothing, a word never seen in spam training data gets P(word|spam)=0, making the entire product zero regardless of other strong spam indicators. Adding 1 to each count prevents this."
    },
    {
      "stage": "post",
      "question": "How does MAP (Maximum A Posteriori) estimation differ from MLE (Maximum Likelihood Estimation)?",
      "options": ["MAP uses a larger dataset than MLE", "MAP incorporates a prior distribution over parameters, equivalent to adding regularization, while MLE only uses the likelihood", "MAP is faster to compute than MLE", "MLE always produces better results than MAP"],
      "correct": 1,
      "explanation": "MAP maximizes P(data|params) * P(params) while MLE maximizes P(data|params) alone. The prior P(params) acts as regularization -- a Gaussian prior equals L2 regularization, a Laplace prior equals L1."
    },
    {
      "stage": "post",
      "question": "In sequential Bayesian updating, you start with Beta(1,1) and observe 7 heads and 3 tails. What is the posterior?",
      "options": ["Beta(7, 3)", "Beta(8, 4)", "Beta(1.7, 1.3)", "Normal(0.7, 0.1)"],
      "correct": 1,
      "explanation": "The Beta-Binomial conjugate update adds successes to the first parameter and failures to the second: Beta(1+7, 1+3) = Beta(8, 4). The posterior mean is 8/12 = 0.667."
    }
  ]
}