Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 5

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- Solution 5

Solution

Introduce

Z_{ij} \in {0, 1}

, where

Z_{ij} = 1

means that the observation came from the structural-zero component. Conditional on the parameters,

Pr (Z_{ij} = 1 ∣ Y_{ij}) = ⎩ ⎨ ⎧ \frac{π _{j}}{π _{j} + ( 1 - π _{j} ) e ^{- α_{i} β_{j}}}, 0, Y_{ij} = 0, Y_{ij} > 0.

(1)

Given

Z

, the nonstructural observations are independent Poisson variables. Using shape-rate parameterization and the stated unit-rate priors, the remaining Gibbs updates are

α_{i} ∣ - \sim Gamma (1 + \sum_{j} (1 - Z_{ij}) Y_{ij}, 1 + \sum_{j} (1 - Z_{ij}) β_{j}),

(2)

β_{j} ∣ - \sim Gamma (1 + \sum_{i} (1 - Z_{ij}) Y_{ij}, 1 + \sum_{i} (1 - Z_{ij}) α_{i}),

(3)

π_{j} ∣ - \sim Beta (1 + \sum_{i} Z_{ij}, 1 + n - \sum_{i} Z_{ij}) .

(4)

Alternating these four standard-distribution updates defines the requested Gibbs sampler for the Zero-inflated Poisson distribution posterior.

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 5

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