Gaussian Mixture Models Quiz

Answers and explanations appear as you take the quiz.

1. What do GMM responsibilities express for one observation?
   • Posterior membership probabilities across components
   • A hard component label assigned before the model is fit
   • The prior mixing weights shared by every observation
   • Density values normalized across observations rather than components
2. Which covariance structure lets each component have a general covariance matrix of its own?
   • Spherical
   • Full
   • Tied
   • Diagonal
3. What does a diagonal covariance assumption exclude within each component?
   • Unequal mixture weights
   • Different component means
   • Off-diagonal feature covariances
   • Soft responsibilities
4. How does k-means differ from a GMM in the guide?
   • GMM assigns every point to all components equally
   • K-means estimates component covariances; GMM uses only nearest centroids
   • Both optimize the same likelihood and always return identical assignments
   • K-means minimizes squared distances; GMM estimates a probabilistic mixture density
5. Why can different GMM initializations produce different fits?
   • The likelihood is convex, so initial values cannot matter
   • The mixture likelihood is nonconvex and can have local optima
   • A covariance matrix prevents mixture components from overlapping
   • Adding components always improves held-out likelihood
6. What tradeoff comes with full covariance matrices in higher dimensions?
   • They remove the need for model selection
   • They force all components to share one mean
   • More parameters and greater overfitting risk with limited data
   • They make every responsibility equal
7. What does permuting component labels do to a fitted mixture density?
   • It changes soft assignments into known labels
   • It makes the covariance matrices diagonal
   • It guarantees better held-out fit
   • It leaves the same density when parameters and weights are permuted together
8. Why should a responsibility of 0.51 not be presented as certain cluster membership?
   • It leaves substantial probability on competing components
   • It is a guaranteed likelihood ratio of one
   • It proves the model selected too many components
   • It means the observation has zero density