Description
(a) Consider the data (yi
, xi)
n
i=1 is following a high dimensional probit regression model
P(yi = 1|xi) = Φ(x
0
iβ),
where yi ∈ {0, 1}, xi ∈ Rp and β = (β1, …, βp)
0
. Consider p = 400, n = 1000.
(i) Simulate 5 datasets by drawing xi ∼ N(0, I) and β = (1.3, 4, −1, 1.6, 5, −2, 0394×1)
0
.
(ii) Compare parameter estimates from MCMC and Sequential Monte Carlo (SMC).
(iii) Provide the coverage and length of 95% credible intervals of β1, …, β6 for MCMC
and SMC.
(b) Suppose
yi = x1i + (x2i − 0.5)+ + x
2
3i + i
, i ∼ N(0, σ2
),
for i = 1, …., 5000. Consider, σ
2 = 0.5 and x1i
, x2i
, x3i ∼ N(0, 1).
(i) Use the above model to generate 5 datasets.
(ii) Run a predictive process model with the above data by writing your own code.
(iii) Comment on the model fit in terms of estimating the mean function and error
variance.
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