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In-Depth Information
For
γ
c
=0
, only a main effect term may be added since no interaction terms are in
the model. Hence the probability distribution for
Ω
is:
P
(
AMT
)=1
,P
(
DMT
)=0
,
P
(
AIT
)=0
,P
(
DIT
)=0
.
(11)
For
γ
c
=1
, one of the
p
1
main effect terms not in the model may be added or the
one main effect term in the model may be dropped and no interaction terms are allowed
in this model. Hence the probability distribution for
Ω
is:
−
P
(
AMT
)=
p
−
1
,P
(
DMT
)=
1
p
,
p
P
(
AIT
)=0
,P
(
DIT
)=0
.
(12)
For
2
γ
c
<r
, no restrictions are involved. Hence, all actions in
Ω
are allowed.
Hence, the probability distribution for
Ω
is:
≤
γ
c
2
−φ
c
p
+
γ
c
2
γ
c
p
+
γ
c
2
p
−
P
(
AMT
)=
,P
(
AIT
)=
,
(13)
γ
c
p
+
γ
c
2
φ
c
p
+
γ
c
2
P
(
DMT
)=
,P
(
DIT
)=
.
For
γ
c
=
r
, due to the restriction that no more than
r
main effect terms may be in a
model at a single time, no main effect terms may be added. However, main effect terms
may be dropped and interaction terms may be added or dropped. Hence, the probability
distribution for
Ω
is:
r
2
−
φ
r
r
2
+
k
P
(
AMT
)=0
,P
(
AIT
)=
,
r
φ
c
P
(
DMT
)=
r
2
+
r
,P
(
DIT
)=
r
2
+
r
.
(14)
Since each model in
nbd
(
M
c
)
is equally likely to be sampled,
q
(
M
t
|
M
c
)
can easily
be calculated. For example, let
M
t
and
M
c
be such that
γ
t
=
γ
c
+1
where
γ
t
<r
and
γ
c
>
2
. Then this corresponds to the action AMT and the probability of candi-
date model
M
t
given that the current model is
M
c
is one out of the number of models
M
c
)=
p
+
γ
c
−
1
in
nbd
(
M
c
)
, specifically,
q
(
M
t
|
and similarly
q
(
M
c
|
M
t
)=
2
p
+
γ
t
2
−
1
. Hence the ratio of the probability of candidate models for this case
is:
