STAT 8640
BAYESIAN STATISTICS
SOLUTION for HW5
1
3.11
(a) Repeat the computations and plots with new prior.
The prior distribution of (, ) is
1
1
1 2
2 2
1 2
p
exp
p(, ) =
(
) +(
) 2(
)(
)
2(1 2 )
Sketch of Solutions for Homework #4
Exer. 2, p. 113
The inverse of the information matrix is
ni x2 ai
4
i
i=1 (1+ai )2
4
ni xi ai
i=1 (1+ai )2
4
ni xi ai
i=1 (1+ai )2
4
ni ai
i=1 (1+ai )2
1
,
B
where
Sketch of Solutions for Homework #5
Exer. 1, p. 440
One term of the likelihood is
1
L(yi |i , ) L(yi |i , ) L(yi |i , ) + (i i )L (yi |i , ) + (i i )2 L (yi |i , ),
2
where the derivatives are partial
Sketch of Solutions for Homework #6
Problem 1
Below are plots of sample autocorrelation functions obtained after generating 10,000 values
from each of the two poposal distributions. There is very litt
Sketch of Solutions for Homework #7
Exercise 1, p. 152
20
10
10
0
theta1
30
40
50
(a) I used MCMC to generate values from the posterior and thereby estimate the quantities of
interest. The only reason
STAT 6 32
October 1 1, 2 010
Midterm
E xam f l-
Instructions: W rite a ll y our a nswerso n t he t est p aper p rovided. C ircle w hat y ou t hink i s
o
the c orrect a nswer o n m ultiple c hoice q ue
S TAT 6 32
November 1 9' 2 o1o
M idt
e rrn , ,xarn f t2
Instructions: W rite a ll y our a nswerso n t he t est p aper p rovided. C ircle w hat y ou t hink
is t he c orrect a nswer o n m ultiple c hoic
STAT 632
October 11, 2010
Midterm Exam #1
Instructions: Write all your answers on the test paper provided. Circle what you think is
the correct answer on multiple choice questions, and be specic on sh
STAT 632
November 19, 2010
Midterm Exam #2
Instructions: Write all your answers on the test paper provided. Circle what you think
is the correct answer on multiple choice questions, be as specic as po
STAT 8640
BAYESIAN STATISTICS
SOLUTION for HW9
1
1.
(a) posterior mean and variance
We can write the model as
i < 0 + xi1 1 + xi2 2 + xi3 3
yi N (i , 2 )
0 N (20, 10000)
j N (1, 100), f or j = 1, 2, 3
STAT 8640
BAYESIAN STATISTICS
SOLUTION for HW1
1.3
(a)
Let
A = cfw_brown-eyed parents
B = cfw_brown-eyed children
C = cfw_children are heterozygotes
A1 = cfw_parents are (Xx, Xx)
A2 = cfw_parents are
STAT 8640
BAYESIAN STATISTICS
SOLUTION for HW4
1
35 points in total
3.1
(a)(5 points) Marginal posterior distribution of .
Label the prior distribution p() as Dirichlet(a1 , . . . , an ), so p(1 , . .
STAT 8640
10.1
BAYESIAN STATISTICS
SOLUTION for HW7
1
Number of simulation draws
a) Approximate standard deviation
2
1 |y N (8, 42 ), according to the large sample Central Limiting Theorem, 1 |y N (8,
STAT 8640
BAYESIAN STATISTICS
SOLUTION for HW3
1
y
2.12 p(y | ) = e y!
log p(y | ) = + y log log y!
log p(y | ) = y
Then,
2
J() = E
log p(y | )
y 2
1
= E
= 2 V ar(y | )
1
1
=
=
2
So the Jeffreys p
STAT 8640
BAYESIAN STATISTICS
SOLUTION for HW6
1
5.2
(a) Are the parameters exchangeable?
Yes, they are exchangeable. The joint distribution is
1 X Y
J
2J
Y
2J
p(1 , . . . , 2J ) =
N (p(j) |1, 1)
N (p
Sketch of Solutions for Homework #3
Exer. 1, p. 95
(a) First show that the joint posterior of 1 and 2 is Dirichlet with parameters y1 + 1 , y2 + 2
and n y1 y2 + 3 + + J , where 1 , . . . , J are the p
Sketch of Solutions for Homework #2
Exer. 16, p. 70
(a) The marginal distribution of Y is obtained by determining
1
m(y ) =
f (y |) () d.
0
(b) Let n > 1. Use proof by contradiction. Suppose that at l
Homework #1: Answers for Selected Exercises
Exer. 1, p. 67
In this case our data consist of the knowledge that Y < 3, where Y is the number of
heads in 10 tosses. The posterior is thus (|Y < 3). Use B
S TAT 6 32
November , 2 011
9
Midterm E lxarn f f2
o
Instructions: W rite a ll y our a nswers n t he t est p aper p rovided. C ircle w hat y ou t hink
is the correct answeron multiple choicequestions,
STAT 632-600
December 9, 2008
Final Exam
Instructions: Answer each question on the test paper provided. Circle the correct answer
on multiple choice questions. Point values are indicated in parenthese
Homework #1: Answers for Selected Exercises
Exer. 1, p. 67
In this case our data consist of the knowledge that Y < 3, where Y is the number of
heads in 10 tosses. The posterior is thus (|Y < 3). Use B
Sketch of Solutions for Homework #2
Exer. 16, p. 70
(a) The marginal distribution of Y is obtained by determining
1
m(y ) =
f (y |) () d.
0
(b) Let n > 1. Use proof by contradiction. Suppose that at l
Sketch of Solutions for Homework #3
Exer. 1, p. 95
(a) First show that the joint posterior of 1 and 2 is Dirichlet with parameters y1 + 1 , y2 + 2
and n y1 y2 + 3 + + J , where 1 , . . . , J are the p
Sketch of Solutions for Homework #4
Exer. 2, p. 113
The inverse of the information matrix is
ni x2 ai
4
i
i=1 (1+ai )2
4
ni xi ai
i=1 (1+ai )2
4
ni xi ai
i=1 (1+ai )2
4
ni ai
i=1 (1+ai )2
1
,
B
where
Sketch of Solutions for Homework #5
Exer. 1, p. 440
One term of the likelihood is
1
L(yi |i , ) L(yi |i , ) L(yi |i , ) + (i i )L (yi |i , ) + (i i )2 L (yi |i , ),
2
where the derivatives are partial
Sketch of Solutions for Homework #6
Problem 1
Below are plots of sample autocorrelation functions obtained after generating 10,000 values
from each of the two poposal distributions. There is very litt
Sketch of Solutions for Homework #7
Exercise 1, p. 152
20
10
10
0
theta1
30
40
50
(a) I used MCMC to generate values from the posterior and thereby estimate the quantities of
interest. The only reason
STAT 6 32
October 1 1, 2 010
Midterm
E xam f l-
Instructions: W rite a ll y our a nswerso n t he t est p aper p rovided. C ircle w hat y ou t hink i s
o
the c orrect a nswer o n m ultiple c hoice q ue