MMiDS 7.6: Self-Assessment Quiz

In the context of Markov Chain Monte Carlo (MCMC), what is the primary goal?

• a) To find the maximum likelihood estimate of a parameter.
• b) To generate samples from a complex target distribution.
• c) To optimize a loss function using gradient descent.
• d) To cluster data points based on similarity.

Which of the following is NOT a requirement for the Metropolis-Hastings algorithm?

• a) A target distribution \( \pi \) to sample from.
• b) A proposal distribution \( Q \) that is easy to simulate.
• c) A symmetric proposal distribution \( Q \), i.e., \( Q(x,y) = Q(y,x) \) for all \( x,y \in S \).
• d) An initial state \( x_0 \in S \).

In the Metropolis-Hastings algorithm, what is the role of the proposal chain \( Q \)?

• a) It determines the stationary distribution of the resulting Markov chain.
• b) It is used to compute the acceptance probability for the proposed moves.
• c) It generates the candidate states for the next move in the Markov chain.
• d) It ensures that the resulting Markov chain is irreducible and aperiodic.

What is the purpose of the Hastings correction in the Metropolis-Hastings algorithm?

• a) To ensure that the proposal chain is symmetric.
• b) To make the resulting Markov chain irreducible and aperiodic.
• c) To ensure that the resulting Markov chain has the desired stationary distribution.
• d) To improve the mixing time of the resulting Markov chain.

In the proof of the correctness of the Metropolis-Hastings algorithm, what is the key property that ensures the resulting Markov chain has the desired stationary distribution?

• a) Irreducibility
• b) Aperiodicity
• c) Reversibility
• d) Ergodicity

What is the main advantage of Gibbs sampling compared to the general Metropolis-Hastings algorithm?

• a) It always accepts the proposed moves.
• b) It does not require a proposal distribution.
• c) It exploits the conditional independence structure of the target distribution.
• d) It guarantees a faster convergence to the stationary distribution.

What is the role of the energy function \( E(v,h) \) in a Restricted Boltzmann Machine (RBM)?

• a) It determines the acceptance probability in the Metropolis-Hastings algorithm.
• b) It defines the joint probability distribution of the visible and hidden units.
• c) It represents the cost function to be minimized during training.
• d) It controls the learning rate of the RBM.

Which function plays a crucial role in computing the conditional probabilities in an RBM?

• a) The exponential function.
• b) The sigmoid function.
• c) The logarithmic function.
• d) The hyperbolic tangent function.

After training an RBM on handwritten digit images, what is the output of the Gibbs sampling process?

• a) A set of parameters for the RBM.
• b) A probability distribution over all possible images.
• c) A sequence of images that resemble handwritten digits.
• d) A single image representing the average of all training images.

What is the partition function \( Z \) used for in the RBM's joint probability distribution \( \pi(v, h) \)?

• a) It normalizes the energy function.
• b) It scales the weights matrix \( W \).
• c) It ensures that the probability distribution sums to one.
• d) It adjusts the biases \( b \) and \( c \).