Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 13J / a

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Solution

Conditional on the observed colleges, the stochastic block model assumes that the friendship indicators are independent random variables. For arbitrary linear predictors

θ_{ij}

, their Bernoulli distribution likelihood is

L (θ; y) = \prod_{1 \leq i < j \leq m} (\frac{e ^{θ_{ij}}}{1 + e ^{θ_{ij}}})^{y_{ij}} (\frac{1}{1 + e ^{θ_{ij}}})^{1 - y_{ij}} = \prod_{i < j} \frac{e ^{y_{ij} θ_{ij}}}{1 + e ^{θ_{ij}}} .

Thus the three requested likelihoods are

L_{1} (β; y) L_{2} (β; y) L_{3} (β; y) = i < j \prod \frac{e ^{y_{ij} β_{z_{i} z_{j}}}}{1 + e ^{β_{z_{i} z_{j}}}}, = i < j \prod \frac{e ^{y_{ij} (β_{z_{i}} + β_{z_{j}})}}{1 + e ^{β_{z_{i}} + β_{z_{j}}}}, = i < j \prod \frac{e ^{y_{ij} (β_{z_{i}} + β_{z_{j}} + β_{0} δ_{z_{i} z_{j}})}}{1 + e ^{β_{z_{i}} + β_{z_{j}} + β_{0} δ_{z_{i} z_{j}}}} .

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