统计代考 | bayesian statistics代考 ICML 2021

Consider the following hierarchical model:
$$
\begin{aligned}
y_{i j} \mid \boldsymbol{\beta}, \phi & \sim \mathrm{Ga}\left(2, \beta_{i}\right), \quad i=1, \ldots, I ; j=1, \ldots, n_{i}, \text { independently, } \
\beta_{i} \mid \phi & \sim \mathrm{Ga}(3, \phi), \quad i=1, \ldots, I, \text { independently }, \
\phi & \sim \operatorname{Expo}(1)
\end{aligned}
$$
where $\beta=\left(\beta_{1}, \ldots, \beta_{I}\right)$.
(i) Show that the joint p.d.f of all the random quantities in the model is
$$
p(\boldsymbol{y}, \boldsymbol{\beta}, \phi)=2^{-t}\left(\prod_{i=1}^{l} \prod_{j=1}^{n_{i}} y_{i j}\right)\left(\prod_{i=1}^{L} \beta_{i}^{2 n_{i}+2}\right) \exp \left(-\left[\phi+\sum_{i=1}^{\ell} \beta_{i}\left(n_{i} \bar{y}{i}+\phi\right)\right]\right) \phi^{3 I} $$ where $y=\left(y{11}, \ldots, y_{1 m}, \ldots, y_{i 1}, \ldots, y_{I_{n_{7}}}\right)$ and $\bar{y}{i}=\sum{j=1}^{n_{i}} y_{i j} / n_{i}$.
MARKS]
(ii) Hence, or otherwise, find the full conditional distributions of the $\beta_{i} \mathrm{~s}$ and of $\phi$, identifying what probability distributions they are. $\quad$ [4,2 MARKS]
(iii) Describe an algorithm to perform Gibbs sampling of the joint posterior distribution of $\beta$ amd $\phi$, using the results derived in part (b). $[4$ MARKS $]$
(iv) Explain how you would estimate a central $95 \%$ posterior interval for $\beta_{1}$ from the output of the Gibbs sampler? When might it be more sensible to try instead to produce a $95 \%$ highest posterior density region (HPDR)? $\quad[2,2$ MARKS $]$
(v) If two parameters are very strongly negatively correlated in the posterior distribution, does the Gibbs sampler
A. work efficiently,
B. work inefficiently,
C. work just as efficiently as if they weren’t correlated.
D. any of the above depending on context?

The effectiveness of a proposed gene therapy for a genetic condition that affects the liver was explored in mice (prior to potential application in humans). In the $i$ th of six replicate experiments, $n_{i}(i=1, \ldots, 6)$ mice with the liver condition were administered with the gene therapy and after a certain period of time the number $y_{i}$ of mice with liver function improved by a certain amount was determined, with the following results:

\begin{tabular}{lrrrrrr}
\hline Experiment, $i$ & 1 & 2 & 3 & 4 & 5 & 6 \
Sample size, $n_{i}$ & 91 & 88 & 102 & 96 & 110 & 113 \
Number improved, $y_{i}$ & 24 & 26 & 7 & 25 & 18 & 18 \
\hline
\end{tabular}
To explore the effect of the treatment, a Bayesian hierarchical model was fitted in WinBuGS with the following model code:
model {