Solution Manual: Bayesian Statistical Methods, 1st
Edition by Brian J. Reich - Chapters 1-20,
9781032093185 | Rationals Included
This comprehensive final examination covers the fundamental concepts,
computational methods, and applied techniques presented in *Bayesian
Statistical Methods* by Brian J. Reich and Sujit K. Ghosh. Select the single best
answer for each question. Each question is followed by a detailed rationale
explaining the correct answer and the underlying Bayesian principle.
SECTION I: BASICS OF BAYESIAN INFERENCE (Chapters 1–2)
1. In Bayesian inference, the posterior distribution is proportional to:
- A) The likelihood function only
- B) The prior distribution only
- C) The likelihood function multiplied by the prior distribution
- D) The marginal likelihood divided by the prior
Answer: C — The likelihood function multiplied by the prior distribution**
Rationale: Bayes' theorem states that the posterior distribution is
proportional to the likelihood times the prior: π(θ|y) ∝ f(y|θ) × π(θ). The
marginal likelihood (or evidence) serves as a normalizing constant. Option A
ignores the prior; Option B ignores the data; Option D inverts the relationship.
,2. In a Beta-Binomial conjugate model, if the prior is Beta(α, β) and the data
consist of y successes in n trials, the posterior distribution is:**
- A) Beta(α + y, β + n − y)
- B) Beta(α − y, β − n + y)
- C) Binomial(n, y)
- D) Beta(α + n, β + y)
Answer: A — Beta(α + y, β + n − y)
Rationale: For the Beta-Binomial conjugate pair, the posterior updates the
prior parameters by adding the number of successes (y) to α and the number
of failures (n − y) to β. This demonstrates the interpretability of conjugate
priors as adding "pseudo-observations" to the data.
3. Jeffreys' prior is an example of which type of prior?
- A) Conjugate prior
- B) Objective prior
- C) Informative prior
- D) Empirical Bayes prior
Answer: B — Objective prior
Rationale: Jeffreys' prior is a non-informative (objective) prior derived from
the Fisher information matrix. It is designed to be invariant under
reparameterization, making it a popular choice when little prior information is
,available. Option A is incorrect because Jeffreys' prior is not necessarily
conjugate; Option C is incorrect because it is non-informative; Option D is
incorrect because it does not use data to estimate hyperparameters.
4. In the Normal-Normal conjugate model for estimating a mean μ with
known variance σ², if the prior is N(μ₀, τ₀²) and the data have sample mean ȳ
based on n observations, the posterior mean is a weighted average of:
- A) The prior mean and the sample mean
- B) The prior mean and the prior variance
- C) The sample mean and the sample variance
- D) The prior variance and the sample variance
Answer: A — The prior mean and the sample mean
Rationale: In the Normal-Normal model, the posterior mean is a precision-
weighted average of the prior mean and the sample mean. The weights are
proportional to the inverse variances (precision) of the prior and the sampling
distribution. This shrinkage property is fundamental to Bayesian inference.
5. An improper prior is characterized by:
- A) Integrating to a finite positive constant
- B) Integrating to 1
- C) Not integrating to a finite value
- D) Being a conjugate prior
Answer: C — Not integrating to a finite value
, Rationale: Improper priors, such as π(θ) ∝ 1, do not integrate to a finite value.
They are often used as objective priors but require that the resulting posterior
be proper (integrable). Option A describes a proper prior; Option B describes a
probability distribution; Option D is unrelated.
6. The posterior predictive distribution is used to:
- A) Estimate the posterior mean
- B) Make predictions for new observations
- C) Compute the marginal likelihood
- D) Select the prior distribution
Answer: B — Make predictions for new observations
Rationale: The posterior predictive distribution p(ỹ|y) = ∫ p(ỹ|θ) π(θ|y) dθ is
used to make predictions for new data by averaging the sampling distribution
over the posterior uncertainty. Option A is a point estimation task; Option C is
model comparison; Option D is prior selection.
7. In the Poisson-Gamma conjugate model for estimating a rate λ, if the prior
is Gamma(α, β) and the data consist of y total events observed over t units of
exposure, the posterior distribution is:
- A) Gamma(α + y, β + t)
- B) Gamma(α − y, β − t)
- C) Poisson(λ)
- D) Gamma(α + t, β + y)
Edition by Brian J. Reich - Chapters 1-20,
9781032093185 | Rationals Included
This comprehensive final examination covers the fundamental concepts,
computational methods, and applied techniques presented in *Bayesian
Statistical Methods* by Brian J. Reich and Sujit K. Ghosh. Select the single best
answer for each question. Each question is followed by a detailed rationale
explaining the correct answer and the underlying Bayesian principle.
SECTION I: BASICS OF BAYESIAN INFERENCE (Chapters 1–2)
1. In Bayesian inference, the posterior distribution is proportional to:
- A) The likelihood function only
- B) The prior distribution only
- C) The likelihood function multiplied by the prior distribution
- D) The marginal likelihood divided by the prior
Answer: C — The likelihood function multiplied by the prior distribution**
Rationale: Bayes' theorem states that the posterior distribution is
proportional to the likelihood times the prior: π(θ|y) ∝ f(y|θ) × π(θ). The
marginal likelihood (or evidence) serves as a normalizing constant. Option A
ignores the prior; Option B ignores the data; Option D inverts the relationship.
,2. In a Beta-Binomial conjugate model, if the prior is Beta(α, β) and the data
consist of y successes in n trials, the posterior distribution is:**
- A) Beta(α + y, β + n − y)
- B) Beta(α − y, β − n + y)
- C) Binomial(n, y)
- D) Beta(α + n, β + y)
Answer: A — Beta(α + y, β + n − y)
Rationale: For the Beta-Binomial conjugate pair, the posterior updates the
prior parameters by adding the number of successes (y) to α and the number
of failures (n − y) to β. This demonstrates the interpretability of conjugate
priors as adding "pseudo-observations" to the data.
3. Jeffreys' prior is an example of which type of prior?
- A) Conjugate prior
- B) Objective prior
- C) Informative prior
- D) Empirical Bayes prior
Answer: B — Objective prior
Rationale: Jeffreys' prior is a non-informative (objective) prior derived from
the Fisher information matrix. It is designed to be invariant under
reparameterization, making it a popular choice when little prior information is
,available. Option A is incorrect because Jeffreys' prior is not necessarily
conjugate; Option C is incorrect because it is non-informative; Option D is
incorrect because it does not use data to estimate hyperparameters.
4. In the Normal-Normal conjugate model for estimating a mean μ with
known variance σ², if the prior is N(μ₀, τ₀²) and the data have sample mean ȳ
based on n observations, the posterior mean is a weighted average of:
- A) The prior mean and the sample mean
- B) The prior mean and the prior variance
- C) The sample mean and the sample variance
- D) The prior variance and the sample variance
Answer: A — The prior mean and the sample mean
Rationale: In the Normal-Normal model, the posterior mean is a precision-
weighted average of the prior mean and the sample mean. The weights are
proportional to the inverse variances (precision) of the prior and the sampling
distribution. This shrinkage property is fundamental to Bayesian inference.
5. An improper prior is characterized by:
- A) Integrating to a finite positive constant
- B) Integrating to 1
- C) Not integrating to a finite value
- D) Being a conjugate prior
Answer: C — Not integrating to a finite value
, Rationale: Improper priors, such as π(θ) ∝ 1, do not integrate to a finite value.
They are often used as objective priors but require that the resulting posterior
be proper (integrable). Option A describes a proper prior; Option B describes a
probability distribution; Option D is unrelated.
6. The posterior predictive distribution is used to:
- A) Estimate the posterior mean
- B) Make predictions for new observations
- C) Compute the marginal likelihood
- D) Select the prior distribution
Answer: B — Make predictions for new observations
Rationale: The posterior predictive distribution p(ỹ|y) = ∫ p(ỹ|θ) π(θ|y) dθ is
used to make predictions for new data by averaging the sampling distribution
over the posterior uncertainty. Option A is a point estimation task; Option C is
model comparison; Option D is prior selection.
7. In the Poisson-Gamma conjugate model for estimating a rate λ, if the prior
is Gamma(α, β) and the data consist of y total events observed over t units of
exposure, the posterior distribution is:
- A) Gamma(α + y, β + t)
- B) Gamma(α − y, β − t)
- C) Poisson(λ)
- D) Gamma(α + t, β + y)
