（x /θ）
α
x≥0，
0 x <0，
使用γ=θ重新参数化模型
α。假设我们获得以下数据；
set.seed（440）
x = rweibull（24，形状= 1/2，比例= 64）
（
γ，α）=（
αxα-1
γ
-x
α/γx≥0，
0 x <0，
注意：这里我们对γ=θ的推论感兴趣
α以α为条件。
a）[4分]显示γ的共轭先验是具有超参数的反伽马分布
由invΓ（λ，β）表示的（λ，β）。密度由下式给出
（
λ，β）=（β
λ
Γ（λ）
（1 / x）
λ+ 1 exp（-β/ x）x≥0，
0 x <0，
b）[6分]使用以下每个先验绘制先验和后验，然后计算可信度
γ的间隔
i）λ= 1和β= 10之前的反伽马，
ii）λ= 10且β= 1的先验反伽马，以及
iii）p（γ）= 1的不当先验
请注意，R包invgamma可能会有所帮助。
c) [4 Marks] Calculate a confidence interval using the log-likelihood ratio.
d) [2 Marks] Compare and discuss the intervals generated in b) and d).
e) In b) we can calculate the credible intervals exactly. Instead use MCMC to estimate the credible interval
from b iii).
i) [1 Mark] Construct a R function that generates a random walk MCMC algorithm to sample
from the posterior. The input is the random walk length, the standard deviation for the random
walk density σ, and the initial position.
ii) [8 Marks] Run a random walk MCMC using σ = 0.1, 1, 5, 20, T = 104
iterations and starting
state γ
(0) = 8.9. Then for each MCMC prov
ide
traceplot, autocorrelation,
summary table with the acceptance rate, estimate of the posterior mean, naive estimate of the
credible interval, and a measure of mixing using
1
T
X
T
t=1
(xt xt1)
2
and then comment on the results.
Question Two – 16 Marks
Here we are interested in Bayesian inference for α conditional on the scale scale parameter being known and
equal to θ = 64. Use an improper of p(α) = 1 for α ≥ 0.
set.seed(440)
x = rweibull(n=24, shape=1/2, scale=64)
a) [2 Mark] Write a R function that is proportional to the posterior on the log-scale.
b) Write R functions that generates a MCMC sample from the posterior using the following the proposals
or candidate densities;
i) [2 Marks] follows a gamma with shape equal to 1 and rate equal to 2
ii) [2 Marks] follows a gamma with shape equal to 1 and rate equal to 1/Xt
iii) [2 Marks] follows a N(Xt, 0.1), and
iv) [2 Marks] follows a N(α, b O(αb)
1
). i.e. an Gaussian independence sampler with mean equal to
the MLE and variance equal to the inverse observed information.
v) [6 Marks] Generate a sample path of length 10, 000 using each of the above MCMCs and the
initial value equal to 1/2.
Summarize and comment on the traceplots, autocorrelations, histogram of the posterior from
each MCMC.
Use a table to summarize the acceptance ratio, the estimate of the posterior average & variance
and the mixing criteria, 1
T
PT
t=1 (xt xt1)
2
.