BAYESIAN STATISTICAL METHODS Comprehensive
Final Examination — 300 Questions Based on the 1st
Edition by Brian J. Reich and Sujit K. Ghosh ISBN:
9781032093185 | Chapters 1–26 | Detailed Answers
with Rationales Latest Update 2026-2027
1. In a hierarchical Bayesian model where group-specific parameters θⱼ are
assigned a common prior distribution with hyperparameters φ, the joint
posterior distribution can be expressed as:**
- A) π(θ, φ | y) ∝ f(y | θ) × π(θ | φ) × π(φ)
- B) π(θ, φ | y) ∝ f(y | θ, φ) × π(θ) × π(φ)
- C) π(θ, φ | y) ∝ f(y | θ) × π(θ) × π(φ | θ)
- D) π(θ, φ | y) ∝ f(y | θ, φ) × π(θ | φ)
Answer: A
Rationale: In a hierarchical model, the joint posterior is proportional to the
likelihood f(y|θ) times the prior for θ given φ, times the hyperprior for φ.
Option B incorrectly conditions the likelihood on φ; Option C reverses the
conditional relationship; Option D omits the hyperprior.
2. The effective number of parameters in a Bayesian hierarchical model is
typically:**
- A) Equal to the number of group-specific parameters
,- B) Less than the number of group-specific parameters due to shrinkage
- C) Greater than the number of group-specific parameters
- D) Equal to the number of hyperparameters
Answer: B
Rationale: In hierarchical models, the effective number of parameters is less
than the total number of group-specific parameters because shrinkage pulls
group estimates toward the population mean, reducing the effective degrees
of freedom. This is a key feature of partial pooling.
3. When comparing two non-nested Bayesian models using Bayes factors,
which of the following is a critical requirement for valid interpretation?
- A) Both models must use the same prior distributions
- B) The prior probabilities of the models must be equal
- C) The marginal likelihoods must be computed using proper priors
- D) The models must have the same number of parameters
Answer: C
Rationale: Bayes factors require proper priors for both models; improper
priors lead to undefined marginal likelihoods. Equal prior model probabilities
are not required (B), nor are identical priors (A) or equal parameter counts
(D).
,4. In the context of variable selection using spike-and-slab priors, the "slab"
component is typically assigned a:
- A) Point mass at zero
- B) Diffuse distribution with large variance
- C) Distribution with variance approaching zero
- D) Uniform distribution over the entire real line
Answer: B
Rationale: The slab component is a diffuse distribution (e.g., normal with large
variance) that allows large coefficients to be estimated with minimal
shrinkage. The spike (A) is the point mass at zero. Option C describes the
spike; Option D is not a standard slab specification.
5. In Bayesian nonparametric regression using Gaussian processes, the
covariance function encodes assumptions about:
- A) The mean structure of the response
- B) The smoothness and correlation structure of the underlying function
- C) The error variance of the observations
- D) The prior distribution of the regression coefficients
Answer: B
Rationale: The covariance (kernel) function in a Gaussian process determines
the smoothness, stationarity, and correlation properties of the latent
function. Option A is modeled by the mean function; Option C is the
observation error; Option D is relevant for parametric models.
, 6. The horseshoe prior is a continuous shrinkage prior that:
- A) Forces all coefficients to be exactly zero
- B) Has a Cauchy-like tail that allows large coefficients to remain unshrunk
- C) Is equivalent to a Laplace prior
- D) Requires the number of predictors to be less than the sample size
Answer: B
Rationale: The horseshoe prior has a Cauchy-like tail, which provides minimal
shrinkage for large coefficients while strongly shrinking small coefficients
toward zero. Unlike the Lasso (A), it does not force exact zeros. It is not
equivalent to Laplace (C), which is the Bayesian Lasso. It can handle p > n (D).
7. The marginal likelihood in Bayesian model comparison serves as:
- A) The prior predictive distribution integrated over parameters
- B) The posterior predictive distribution evaluated at new data
- C) The likelihood function evaluated at the posterior mode
- D) The prior distribution evaluated at the posterior mean
Answer: A
Rationale: The marginal likelihood is ∫ f(y|θ) π(θ) dθ, which is the prior
predictive distribution averaged over the prior. Option B is the posterior
predictive; Option C is a Laplace approximation; Option D is not a standard
quantity.
Final Examination — 300 Questions Based on the 1st
Edition by Brian J. Reich and Sujit K. Ghosh ISBN:
9781032093185 | Chapters 1–26 | Detailed Answers
with Rationales Latest Update 2026-2027
1. In a hierarchical Bayesian model where group-specific parameters θⱼ are
assigned a common prior distribution with hyperparameters φ, the joint
posterior distribution can be expressed as:**
- A) π(θ, φ | y) ∝ f(y | θ) × π(θ | φ) × π(φ)
- B) π(θ, φ | y) ∝ f(y | θ, φ) × π(θ) × π(φ)
- C) π(θ, φ | y) ∝ f(y | θ) × π(θ) × π(φ | θ)
- D) π(θ, φ | y) ∝ f(y | θ, φ) × π(θ | φ)
Answer: A
Rationale: In a hierarchical model, the joint posterior is proportional to the
likelihood f(y|θ) times the prior for θ given φ, times the hyperprior for φ.
Option B incorrectly conditions the likelihood on φ; Option C reverses the
conditional relationship; Option D omits the hyperprior.
2. The effective number of parameters in a Bayesian hierarchical model is
typically:**
- A) Equal to the number of group-specific parameters
,- B) Less than the number of group-specific parameters due to shrinkage
- C) Greater than the number of group-specific parameters
- D) Equal to the number of hyperparameters
Answer: B
Rationale: In hierarchical models, the effective number of parameters is less
than the total number of group-specific parameters because shrinkage pulls
group estimates toward the population mean, reducing the effective degrees
of freedom. This is a key feature of partial pooling.
3. When comparing two non-nested Bayesian models using Bayes factors,
which of the following is a critical requirement for valid interpretation?
- A) Both models must use the same prior distributions
- B) The prior probabilities of the models must be equal
- C) The marginal likelihoods must be computed using proper priors
- D) The models must have the same number of parameters
Answer: C
Rationale: Bayes factors require proper priors for both models; improper
priors lead to undefined marginal likelihoods. Equal prior model probabilities
are not required (B), nor are identical priors (A) or equal parameter counts
(D).
,4. In the context of variable selection using spike-and-slab priors, the "slab"
component is typically assigned a:
- A) Point mass at zero
- B) Diffuse distribution with large variance
- C) Distribution with variance approaching zero
- D) Uniform distribution over the entire real line
Answer: B
Rationale: The slab component is a diffuse distribution (e.g., normal with large
variance) that allows large coefficients to be estimated with minimal
shrinkage. The spike (A) is the point mass at zero. Option C describes the
spike; Option D is not a standard slab specification.
5. In Bayesian nonparametric regression using Gaussian processes, the
covariance function encodes assumptions about:
- A) The mean structure of the response
- B) The smoothness and correlation structure of the underlying function
- C) The error variance of the observations
- D) The prior distribution of the regression coefficients
Answer: B
Rationale: The covariance (kernel) function in a Gaussian process determines
the smoothness, stationarity, and correlation properties of the latent
function. Option A is modeled by the mean function; Option C is the
observation error; Option D is relevant for parametric models.
, 6. The horseshoe prior is a continuous shrinkage prior that:
- A) Forces all coefficients to be exactly zero
- B) Has a Cauchy-like tail that allows large coefficients to remain unshrunk
- C) Is equivalent to a Laplace prior
- D) Requires the number of predictors to be less than the sample size
Answer: B
Rationale: The horseshoe prior has a Cauchy-like tail, which provides minimal
shrinkage for large coefficients while strongly shrinking small coefficients
toward zero. Unlike the Lasso (A), it does not force exact zeros. It is not
equivalent to Laplace (C), which is the Bayesian Lasso. It can handle p > n (D).
7. The marginal likelihood in Bayesian model comparison serves as:
- A) The prior predictive distribution integrated over parameters
- B) The posterior predictive distribution evaluated at new data
- C) The likelihood function evaluated at the posterior mode
- D) The prior distribution evaluated at the posterior mean
Answer: A
Rationale: The marginal likelihood is ∫ f(y|θ) π(θ) dθ, which is the prior
predictive distribution averaged over the prior. Option B is the posterior
predictive; Option C is a Laplace approximation; Option D is not a standard
quantity.
