BAYESIAN STATISTICAL METHODS Comprehensive Final
Examination — 300 Questions Based on the 2nd Edition by
Brian J. Reich and Sujit K. Ghosh ISBN: 9781032486321 |
Chapters 1–26 | Detailed Answers with Rationales
1. In the context of Bayesian hierarchical modeling, the term "borrowing
strength" refers to:
- A) Using the same prior distribution for all parameters in the model
- B) Sharing information across groups to improve estimates for groups with
sparse data
- C) Assuming all group-level parameters are exactly equal to each other
- D) Using an empirical Bayes approach to estimate hyperparameters from the
data
Answer: B
Rationale: Borrowing strength is the key advantage of hierarchical models,
where information is shared across groups, improving estimates for groups
with limited data (partial pooling). Option A describes using a common prior
but not necessarily borrowing strength; Option C describes complete pooling;
Option D describes empirical Bayes, which is a specific approach to estimating
hyperparameters.
,2. In a hierarchical model with group-specific parameters θⱼ and
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.
3. 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 and is measured by criteria
such as p_D in DIC or p_waic in WAIC.
4. 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 for all parameters
- 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 because the normalizing
constant is not well-defined. Equal prior model probabilities are not required
(B), nor are identical priors (A) or equal parameter counts (D).
5. 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 that performs variable
selection. Option C describes the spike; Option D is not a standard slab
specification.
6. In Bayesian nonparametric regression using Gaussian processes, the
covariance function encodes assumptions about:
- A) The mean structure of the response variable
- 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.
7. 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
Examination — 300 Questions Based on the 2nd Edition by
Brian J. Reich and Sujit K. Ghosh ISBN: 9781032486321 |
Chapters 1–26 | Detailed Answers with Rationales
1. In the context of Bayesian hierarchical modeling, the term "borrowing
strength" refers to:
- A) Using the same prior distribution for all parameters in the model
- B) Sharing information across groups to improve estimates for groups with
sparse data
- C) Assuming all group-level parameters are exactly equal to each other
- D) Using an empirical Bayes approach to estimate hyperparameters from the
data
Answer: B
Rationale: Borrowing strength is the key advantage of hierarchical models,
where information is shared across groups, improving estimates for groups
with limited data (partial pooling). Option A describes using a common prior
but not necessarily borrowing strength; Option C describes complete pooling;
Option D describes empirical Bayes, which is a specific approach to estimating
hyperparameters.
,2. In a hierarchical model with group-specific parameters θⱼ and
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.
3. 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 and is measured by criteria
such as p_D in DIC or p_waic in WAIC.
4. 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 for all parameters
- 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 because the normalizing
constant is not well-defined. Equal prior model probabilities are not required
(B), nor are identical priors (A) or equal parameter counts (D).
5. 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 that performs variable
selection. Option C describes the spike; Option D is not a standard slab
specification.
6. In Bayesian nonparametric regression using Gaussian processes, the
covariance function encodes assumptions about:
- A) The mean structure of the response variable
- 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.
7. 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
