Homework
Homework 1
1.
M. D. Computing describes the use of Bayes’ theorem and the use of
conditional probability in medical diagnosis. Prior probabilities of
diseases are based on the physician’s assessment of such things as
geographical location, seasonal influence, occurrence of epidemics,
and so forth. Assume that a patient is believed to have one of two
diseases, denoted
1
D
and
2
D
with
6
.
0
1
D
P
and
4
.
0
2
D
P
that medical research has determined the probability associated with
each symptom that may accompany the diseases. Suppose that given
diseases
1
D
and
2
D
, the probabilities that the patient will have
symptoms
1
S
,
2
S
or
3
S
are as follows.
1
S
2
S
3
S
1
D
0.15
0.1
0.15
2
D
0.80
0.15
0.03
After a certain symptom is found to be present, the medical diagnosis
may be aided by finding the revised probabilities of each particular
disease. Compute the posterior probabilities of each disease given the
following medical findings.
(a) The patient has symptom
1
S
(b) The patient has symptoms
2
S
or
3
S
(c) For the patient with symptom
1
S
in part (a), suppose we also find
symptom
2
S
. What are the revised probabilities of
1
D
and
2
2. Let
2
1
,
~
,
,
,
N
X
X
X
n
, with
2
known. For the prior distribution
for
, we shall assume the normal distribution
2
,
N
(a) Find the maximum likelihood estimate.
(b) Find the posterior distribution of
and posterior mode as estimate
of .
(c) Compare the estimates in (a) and (b).
and
.
.
D
.
2
.
.
1
3. Let
P
X
X
X
n
~
,
,
,
1
,
!

i
x
i
x
e
x
f
i
For the prior distribution for
, we shall assume the gamma distribution
e
G
1
1
,
,
~
.
(a) Find the maximum likelihood estimate.
(b)Find the posterior distribution of
and posterior mean as estimate
of .
(c) Compare the estimates in (a) and (b).
2
.
Solution 1:
3
676
.
0
222
.
0
15
.
0
)

(
3
2
3
2
1
3
2
1
S
S
P
S
S
D
P
S
S
D
P
and
324
.
0
222
.
0
072
.
0
)

(
3
2
3
2
2
3
2
2
S
S
P
S
S
D
P
S
S
D
P
.
(c)
We can use
1
1

S
D
P
and
1
2

S
D
P
as new revised probability for the
diseases. That is,
2195
.
0

reviesed
,
1
1
1
S
D
P
D
P
and
7805
.
0

reviesed
,
1
2
2
S
D
P
D
P
.