Compute posterior for biased coin after 3H1T
Problem Description
You are presented with a coin known to be biased. One face lands with a probability of 2/3, and the other with a probability of 1/3. However, you do not know whether Heads corresponds to the 2/3 probability or Tails does.
Assume an initial prior probability: P(Heads-heavy) = P(Tails-heavy) = 1/2.
You flip the coin 4 times and observe 3 Heads and 1 Tail. The order of these outcomes is unspecified.
Address the following questions:
- a) Posterior Probability: Compute the posterior probability P(Heads-heavy | 3H, 1T).
- b) Bayes Factor: Compute the Bayes factor BF = P(data | Heads-heavy) / P(data | Tails-heavy).
- c) Posterior Predictive: Compute the posterior predictive probability P(next toss is Head | 3H, 1T).
- d) Generalization: Generalize the problem to
ntosses withkHeads.- Derive a closed-form expression for the posterior probability.
- Determine the decision rule (under 0-1 loss) for choosing between Heads-heavy and Tails-heavy.
- Show that the Bayes factor can be expressed as a simple power of 2.
- e) Order of Outcomes: Does the order of outcomes matter under this model? Explain your reasoning precisely.
- f) Asymmetric Prior: How would the posterior and predictive probabilities change if the prior were asymmetric, specifically if π = P(Heads-heavy) ≠ 1/2?
AI interview prep
Core dilemma
Inferring the true underlying state of a system (coin bias) with limited, discrete observations, while explicitly accounting for prior beliefs and quantifying uncertainty.
3 tradeoffs and 3 stress tests locked
Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Compute posterior for biased coin after 3H1T
Problem Description
You are presented with a coin known to be biased. One face lands with a probability of 2/3, and the other with a probability of 1/3. However, you do not know whether Heads corresponds to the 2/3 probability or Tails does.
Assume an initial prior probability: P(Heads-heavy) = P(Tails-heavy) = 1/2.
You flip the coin 4 times and observe 3 Heads and 1 Tail. The order of these outcomes is unspecified.
Address the following questions:
- a) Posterior Probability: Compute the posterior probability P(Heads-heavy | 3H, 1T).
- b) Bayes Factor: Compute the Bayes factor BF = P(data | Heads-heavy) / P(data | Tails-heavy).
- c) Posterior Predictive: Compute the posterior predictive probability P(next toss is Head | 3H, 1T).
- d) Generalization: Generalize the problem to
ntosses withkHeads.- Derive a closed-form expression for the posterior probability.
- Determine the decision rule (under 0-1 loss) for choosing between Heads-heavy and Tails-heavy.
- Show that the Bayes factor can be expressed as a simple power of 2.
- e) Order of Outcomes: Does the order of outcomes matter under this model? Explain your reasoning precisely.
- f) Asymmetric Prior: How would the posterior and predictive probabilities change if the prior were asymmetric, specifically if π = P(Heads-heavy) ≠ 1/2?