STAT 428
Spring 2010
Homework 7
Apr 21
Homework 7
Due in class on Friday, Apr 30.
1.
Consider the simulation of bivariate normal random variables. Let
X
= (
X
1
, X
2
) and the
target distribution is
π
(
x
1
, x
2
)
∼
N
parenleftbiggparenleftbigg
0
0
parenrightbigg
,
parenleftbigg
1
0
.
5
0
.
5
1
parenrightbiggparenrightbigg
.
Design a Gibbs sampler to generate samples approximately from the target distribution. Give
your estimate of the mean of
X
2
. Plot the histogram of
X
2
based on your samples, and compare it
with the histogram based on i.i.d. samples from
N
(0
,
1) (which is the true marginal distribution
of
X
2
).
2.
Consider the onedimensional Ising model that we discussed in class. Let
x
= (
x
1
, . . . , x
d
),
where
x
i
is either +1 or

1. The target distribution is
π
(
x
)
∝
exp
braceleftBigg
μ
d

1
summationdisplay
i
=1
x
i
x
i
+1
bracerightBigg
.
Let
μ
= 2,
d
= 40.
(a) Design a Gibbs sampling algorithm to generate samples approximately from the target dis
tribution
π
(
x
), and implement your algorithm in R. Attach the R code.
(b) Suppose the output of your Gibbs sampling algorithm at step
t
is
x
(
t
)
= (
x
(
t
)
1
, . . . , x
(
t
)
d
).
Define the total magnetization
M
(
t
)
=
∑
d
i
=1
x
(
t
)
i
.
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 Spring '08
 Chen
 Normal Distribution, Standard Deviation, Probability theory, Gibbs, Gibbs Sampling

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