Last updated: 2020-09-21
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Assume \[z_i(\tau)|r,\theta_i(\tau)\sim NB(r,\frac{\exp(\theta_i(\tau))}{r+\exp(\theta_i(\tau))}),\]
its mean and variance are \(\exp(\theta_i(\tau))\) and \(\exp(\theta_i(\tau))(1+\exp(\theta_i(\tau))/r)\).
This model can also be viewed as Poisson-gamma hierarchical model, where \(z_i(\tau)|\lambda_i(\tau)\sim Poisson(\lambda_i(\tau))\) and \(\lambda_i(\tau)|r,\theta_i(\tau)\sim Gamma(r,r\exp(\theta_i(\tau)))\).
If \(r\to+\infty\), then \(z_i(\tau)|\theta_i(\tau)\sim Poisson(\exp(\theta_i(\tau)))\).
To model the functional data, let \(\theta_i(\tau) = \mu_i(\tau)+\epsilon_i(\tau)\), \(\epsilon_i(\tau)\sim N(0,\sigma^2)\), and \(\mu_i(\tau) = \sum_k f_k(\tau)\beta_{k,i}\). The conditional expectation of \(z_i(\tau)\) is then \(E(z_i(\tau)|\mu,\sigma^2,r) = \exp(\mu_i(\tau))\exp(\sigma^2/2)\).
The author incorporates autoregressive model into basis coefficients: \(\beta_{k,i} = \mu_k+\phi_k(\beta_{k,i-1}-\mu_k)+\eta_{k,i}\) where \(\eta_{k,i}\sim N(0,\sigma^2_{\eta_{k,i}})\), and introduces ordered shrinkage via a multiplicative gamma process (MGP) prior on \(\mu_k,\eta_{k,i}\).
Modelling basis as unknown and produce data-adaptive basis? Kowal et al.(2017)
A Polya‐Gamma data augmentation scheme is adopted to sample from the full conditional distribution of \(\theta_i(\tau)\).
imputation
sample \(r\) using slice sampler.
Parameter expansion, sample from a polya-gamma dsitrobution
sample \(\theta_i(\tau)\)
Gaussian smoothing on \(\theta_i(\tau)\)