Markov Chain Monte Carlo Quiz

Answers and explanations appear as you take the quiz.

1. Why are MCMC draws usually not equivalent to the same number of independent samples?
   • Successive states are dependent and often autocorrelated
   • The target distribution has no probabilities
   • Every proposal is always rejected
   • Only one parameter can be sampled
2. For an asymmetric Metropolis-Hastings proposal, what additional ratio term is needed?
   • The number of parameters squared
   • The reverse proposal probability divided by the forward proposal probability
   • The sample mean of the target
   • The posterior interval width
3. What does Gibbs sampling draw at each coordinate update?
   • A proposal that is always rejected
   • An independent draw from the joint posterior by definition
   • A value from that variable's full conditional given current other variables
   • A point with maximum likelihood
4. Why run chains from dispersed starting values?
   • To estimate the data likelihood without a model
   • To make all posterior values equal
   • To guarantee each chain has independent draws
   • To reveal whether separate runs explore compatible target regions
5. What does effective sample size estimate?
   • The total number of rows saved regardless of correlation
   • The independent-draw equivalent for a quantity after accounting for dependence
   • The number of parameters in the posterior
   • The probability that the model is correct
6. What does R-hat compare across chains?
   • Prediction accuracy with training error
   • The prior density with the likelihood only
   • Within-chain and between-chain behavior
   • The number of parameters with observations
7. Why does discarding warmup not guarantee convergence?
   • All chains become independent afterward
   • Warmup contains the exact posterior by definition
   • The target distribution changes after warmup
   • No fixed finite discard proves all relevant quantities have been explored
8. A chain has many saved draws but low ESS for a tail probability. What follows?
   • The tail estimate may still have substantial Monte Carlo uncertainty
   • The tail probability is exactly zero
   • The posterior mean must be biased
   • The model is necessarily misspecified