week 6

Question 1

“What additional specification is needed to implement a fully hierarchical model beyond the prior π(θ|φ)?”

Answer 1

A hyperprior distribution π(φ) for the hyperparameter(s) φ.

Question 2

“In a fully hierarchical model (Y|θ ~ fθ, θ|φ ~ π(θ|φ), φ ~ π(φ)), how is the joint posterior π(θ, φ | y) expressed proportionally?”

Answer 2

π(θ, φ | y) ∝ L(y|θ) * π(θ|φ) * π(φ), where L(y|θ) is the likelihood Π fθ(yi).

Question 3

“What is the main challenge mentioned regarding the posterior distribution of hierarchical models?”

Answer 3

In many cases, it cannot be evaluated analytically.

Question 4

“What sampling approach is suggested as suitable for handling hierarchical models?”

Answer 4

Gibbs sampling.

Question 5

“In the school data example, what does yij represent?”

Answer 5

The score of the j-th student in the i-th school.

Question 6

“In the school data example, what does ni represent?”

Answer 6

The number of pupils who took the test in school i.

Question 7

“Specify the hierarchical model structure used for the school data example.”

Answer 7

yij|μi, σ² ~ N(μi, σ²); μi|μ0, σ0² ~ N(μ0, σ0²); σ⁻² ~ Ga(a, b); μ0 ~ N(μ00, σ00²); σ0⁻² ~ Ga(a0, b0).

Question 8

“How is the joint posterior distribution π(μ1,…,μ38, σ⁻², μ0, σ0⁻² | y) expressed proportionally in the school data example?”

Answer 8

∝ [Π(i=1 to 38) Π(j=1 to ni) f(yij|μi, σ²)] * [Π(i=1 to 38) π(μi|μ0, σ0²)] * π(σ⁻²) * π(μ0) * π(σ0⁻²).

Question 9

“What is the full conditional distribution for the school-specific mean μi in the example?”

Answer 9

N( (Σ(j=1 to ni) yij / σ² + μ0 / σ0²) / (ni / σ² + 1 / σ0²), 1 / (ni / σ² + 1 / σ0²) ).

Question 10

“What is the full conditional distribution for the inverse variance σ⁻² in the school data example?”

Answer 10

Ga( a + (1/2)Σ(i=1 to 38) ni, b + (1/2)Σ(i=1 to 38) Σ(j=1 to ni) (yij - μi)² ).

Question 11

“What is the full conditional distribution for the overall mean hyperparameter μ0 in the school data example?”

Answer 11

N( (Σ(i=1 to 38) μi / σ0² + μ00 / σ00²) / (38 / σ0² + 1 / σ00²), 1 / (38 / σ0² + 1 / σ00²) ).

Question 12

“What is the full conditional distribution for the inverse hypervariance σ0⁻² in the school data example?”

Answer 12

Ga( a0 + 19, b0 + (1/2)Σ(i=1 to 38) (μi - μ0)² ). (Note: 19 = 38/2)

Question 13

“What specific hyperparameter values were used in the school data analysis (a, b, μ00, σ00², a0, b0)?”

Answer 13

a = 0.001, b = 0.001, μ00 = 0, σ00² = 100², a0 = 0.001, b0 = 0.001.

Question 14

“What is the posterior median estimate for μ0 in the school data example?”

Answer 14

0.11.

Question 15

“What is the 95% credible interval for μ0 in the school data example?”

Answer 15

(-0.14, 0.11).

Question 16

“What is the posterior median estimate for σ² in the school data example?”

Answer 16

0.91.

Question 17

“What is the 95% credible interval for σ² in the school data example?”

Answer 17

(0.87, 0.99).

Question 18

“What is the posterior median estimate for σ0² in the school data example?”

Answer 18

0.10.

Question 19

“What is the 95% credible interval for σ0² in the school data example?”

Answer 19

(0.05, 0.17).

Question 20

“What complication can arise in implementing a Gibbs sampler, especially in hierarchical models?”

Answer 20

Some of the full conditional distributions might not be known distributions (i.e., cannot be sampled from directly).

Question 21

“What approach is typically used within a Gibbs sampler to handle unknown full conditional distributions?”

Answer 21

A Metropolis-Hastings step is used to sample from that specific unknown conditional distribution.

Question 22

“What is the name given to the approach where Metropolis-Hastings steps are used for some updates within a Gibbs sampler?”

Answer 22

Metropolis-within-Gibbs.

Question 23

“In the surgical outcomes example, what does ri represent?”

Answer 23

The number of deaths in hospital i.

Question 24

“What model is used for the number of deaths ri given ni operations and mortality rate θi?”

Answer 24

ri | ni, θi ~ Binomial(ni, θi).

Question 25

"What model is used for the hospital-specific mortality rate θi given hyperparameters μ and φ?"

Answer 25

θi | μ, φ ~ Beta(μφ, (1-μ)φ).

Question 26

"What prior distribution is used for the hyperparameter μ in the surgical outcomes example?"

Answer 26

μ ~ Beta(1, 1) (which is Uniform(0,1)).

Question 27

"What prior distribution is used for the hyperparameter φ in the surgical outcomes example?"

Answer 27

φ ~ Gamma(1, 0.01).

Question 28

"What is the full conditional distribution for θi | μ, φ, θ-i, y in the surgical outcomes example?"

Answer 28

Beta(μφ + ri, (1-μ)φ + ni - ri).

Question 29

"Is the full conditional distribution for μ | θ1,...,θ12, φ a standard distribution?"

Answer 29

No, it is proportional to [Π θi^(μφ) * (1-θi)^((1-μ)φ)] / [Γ(μφ)Γ((1-μ)φ)]^12 (and the prior for μ).

Question 30

"Is the full conditional distribution for φ | θ1,...,θ12, μ a standard distribution?"

Answer 30

No, it is proportional to [Γ(φ) / (Γ(μφ)Γ((1-μ)φ))]^12 * [Π θi^(μφ) * (1-θi)^((1-μ)φ)] * φ^(a-1)exp(-bφ) (using general Gamma prior notation, slide used a=1, b=0.01).

Question 31

"In the surgical outcomes Gibbs sampler, how are the θi's updated?"

Answer 31

Directly sampled from their Beta full conditional distribution.

Question 32

"In the surgical outcomes Gibbs sampler, how are μ and φ updated?"

Answer 32

Using Metropolis-Hastings random walk steps, as their full conditionals are not standard distributions.

Question 33

"In the power plant pumps example, what distribution does the number of failures Yj follow given failure rate θj and time tj?"

Answer 33

Yj | θj ~ Poisson(θj * tj).

Question 34

"In the power plant pumps example, what prior distribution is assumed for the failure rate θj given hyperparameters α and β?"

Answer 34

θj | α, β ~ Gamma(α, β).

Question 35

"What hyperpriors were assumed for α and β in the power plant pumps example by George et al. (1993)?"

Answer 35

α ~ Exp(1) and β ~ Gamma(a1, b1) with a1=0.1, b1=1.

Question 36

"What is the full conditional distribution for θj in the power plant pumps example?"

Answer 36

Gamma(yj + α, tj + β).

Question 37

"Is the full conditional distribution for α in the power plant pumps example a standard distribution?"

Answer 37

No, it is proportional to [β^α / Γ(α)]^10 * (Π θj^(α-1)) * exp(-α).

Question 38

"What is the full conditional distribution for β in the power plant pumps example?"

Answer 38

Gamma(a1 + 10α, b1 + Σ(j=1 to 10) θj).

Question 39

"How is the posterior sampling performed in the power plant pumps example, given the non-standard conditional for α?"

Answer 39

Using Metropolis-within-Gibbs: Update θj's and β directly from their Gamma conditionals, update α using a Metropolis-Hastings step.