CS229 Practice Midterm
1
CS 229, Autumn 2007
Practice Midterm
Notes:
1. The midterm will have about 56 long questions, and about 810 short questions. Space
will be provided on the actual midterm for you to write your answers.
2. The midterm is meant to be educational, and as such some questions could be quite
challenging. Use your time wisely to answer as much as you can!
1.
[13 points] Generalized Linear Models
Recall that generalized linear models assume that the response variable
y
(conditioned on
x
) is distributed according to a member of the exponential family:
P
(
y
;
η
) =
b
(
y
) exp(
ηT
(
y
)
−
a
(
η
))
,
where
η
=
θ
T
x
. For this problem, we will assume
η
∈
R
.
(a)
[10 points]
Given a training set
{
(
x
(
i
)
, y
(
i
)
)
}
m
i
=1
, the loglikelihood is given by
ℓ
(
θ
) =
m
summationdisplay
i
=1
log
p
(
y
(
i
)

x
(
i
)
;
θ
)
.
Give a set conditions on
b
(
y
),
T
(
y
), and
a
(
η
) which ensure that the loglikelihood is
a concave function of
θ
(and thus has a unique maximum). Your conditions must be
reasonable, and should be as weak as possible. (E.g., the answer “any
b
(
y
),
T
(
y
), and
a
(
η
) so that
ℓ
(
θ
) is concave” is not reasonable. Similarly, overly narrow conditions,
including ones that apply only to specific GLIMs, are also not reasonable.)
(b)
[3 points]
When the response variable is distributed according to a Normal distribu
tion (with unit variance), we have
b
(
y
) =
1
√
2
π
e

y
2
2
,
T
(
y
) =
y
, and
a
(
η
) =
η
2
2
. Verify
that the condition(s) you gave in part (a) hold for this setting.
2.
[15 points] Bayesian linear regression
Consider Bayesian linear regression using a Gaussian prior on the parameters
θ
∈
R
n
+1
.
Thus, in our prior,
θ
∼ N
(
vector
0
, τ
2
I
n
), where
τ
2
∈
R
, and
I
n
+1
is the
n
+ 1by
n
+ 1 identity
matrix. Also let the conditional distribution of
y
(
i
)
given
x
(
i
)
and
θ
be
N
(
θ
T
x
(
i
)
, σ
2
), as
in our usual linear leastsquares model.
1
Let a set of
m
IID training examples be given
(with
x
(
i
)
∈
R
n
+1
). Recall that the MAP estimate of the parameters
θ
is given by:
θ
MAP
= arg max
θ
parenleftBigg
m
productdisplay
i
=1
p
(
y
(
i
)

x
(
i
)
, θ
)
parenrightBigg
p
(
θ
)
Find, in closed form, the MAP estimate of the parameters
θ
. For this problem, you should
treat
τ
2
and
σ
2
as fixed, known, constants. [Hint: Your solution should involve deriving
something that looks a bit like the Normal equations.]
1
Equivalently,
y
(
i
)
=
θ
T
x
(
i
)
+
ε
(
i
)
, where the
ε
(
i
)
’s are distributed IID
N
(0
, σ
2
).
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CS229 Practice Midterm
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3.
[18 points] Kernels
In this problem, you will prove that certain functions
K
give valid kernels. Be careful to
justify every step in your proofs. Specifically, if you use a result proved either in the lecture
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