Week 5 Flashcards
(49 cards)
“What is the goal of the Metropolis-Hastings (MH) sampler?”
To produce a Markov chain of values x(1), x(2), … whose equilibrium distribution is a target distribution π.
“In the MH algorithm, how is a candidate value x’ generated at iteration i?”
It is proposed from a transition probability (proposal distribution) q(x(i-1), x’).
“What is the formula for the acceptance probability α(x(i-1), x’) in the general Metropolis-Hastings algorithm?”
α(x(i-1), x’) = min{1, [π(x’)q(x’, x(i-1))] / [π(x(i-1))q(x(i-1), x’)]}
“How is the next state x(i) determined after calculating the acceptance probability α and drawing u ~ U(0, 1)?”
x(i) = x’ if u < α(x(i-1), x’), otherwise x(i) = x(i-1).
“What is the proposal distribution q(x(i-1), x’) also called?”
The proposal distribution.
“What are the two main types of Metropolis-Hastings samplers characterized by their proposal distribution?”
Metropolis-Hastings random walk sampler and Metropolis-Hastings independence sampler.
“How is the proposed value x’ generated in an MH random walk sampler?”
x’ = x(i-1) + εi, where the distribution of εi is symmetric (e.g., Normal, t).
“What condition simplifies the acceptance probability formula in an MH random walk sampler?”
The symmetry of the proposal distribution, which means q(x(i-1), x’) = q(x’, x(i-1)).
“What is the simplified acceptance probability α(x(i-1), x’) for an MH random walk sampler?”
α(x(i-1), x’) = min{1, π(x’) / π(x(i-1))}
“What is the characteristic of the proposal distribution q(x(i-1), x’) in an MH independence sampler?”
It does not depend on the previous state x(i-1), i.e., q(x(i-1), x’) = q(x’).
“What is the acceptance probability α(x(i-1), x’) for an MH independence sampler?”
α(x(i-1), x’) = min{1, [π(x’)q(x(i-1))] / [π(x(i-1))q(x’)]}
“When does the MH independence sampler method work well?”
When the proposal distribution q(x) is a good approximation of the target distribution π(x).
“In the outlier modelling example, what distribution is used to model the scores yi|μ?”
A t-distribution with ν degrees of freedom, location μ, and scale σ: yi|μ ~ tν(μ, σ²).
“What is the formula for the probability density function fμ(yi) of the tν(μ, σ²) distribution?”
fμ(yi) = [Γ((ν+1)/2) / (√(νπσ²)Γ(ν/2))] * [1 + (yi - μ)² / (νσ²)]^(-(ν+1)/2)
“In the outlier example, what were the assumed fixed values for σ² and ν?”
σ² = 260 and ν = 3.
“In the outlier example, what was the prior distribution assumed for μ?”
μ ~ N(100, 100²).
“How is the posterior distribution π(μ|y) related to the prior and likelihood in the outlier example?”
π(μ|y) ∝ π(μ) * Π[i=1 to 21] fμ(yi).
“In the MH random walk implementation for the outlier example, what proposal distribution was used for μ’?”
μ’ ~ N(μ(i-1), τ²).
“What was the acceptance probability formula used in the MH random walk for the outlier example?”
α(μ(i-1), μ’) = min{1, π(μ’|y) / π(μ(i-1)|y)}.
“In the outlier example MH run, what was the posterior mean of μ approximately?”
Approximately 94.6.
“In the outlier example MH run, what was the 95% credible interval for μ?”
(87.3, 101.8).
“What parameter controls the variance of the proposal distribution in the MH random walk sampler?”
τ (or τ² depending on parameterization).
“What happens if the proposal variance τ² is too small in an MH random walk?”
The chain typically moves often but with very small jumps (slow exploration).
“What happens if the proposal variance τ² is too large in an MH random walk?”
The chain typically moves rarely (low acceptance rate) but with large jumps when it does move.
