<s>
In	O
statistics	O
,	O
the	O
Bayesian	B-General_Concept
information	I-General_Concept
criterion	I-General_Concept
(	O
BIC	B-General_Concept
)	O
or	O
Schwarz	B-General_Concept
information	I-General_Concept
criterion	I-General_Concept
(	O
also	O
SIC	O
,	O
SBC	O
,	O
SBIC	O
)	O
is	O
a	O
criterion	O
for	O
model	O
selection	O
among	O
a	O
finite	O
set	O
of	O
models	O
;	O
models	O
with	O
lower	O
BIC	B-General_Concept
are	O
generally	O
preferred	O
.	O
</s>
<s>
When	O
fitting	O
models	O
,	O
it	O
is	O
possible	O
to	O
increase	O
the	O
maximum	O
likelihood	O
by	O
adding	O
parameters	O
,	O
but	O
doing	O
so	O
may	O
result	O
in	O
overfitting	B-Error_Name
.	O
</s>
<s>
Both	O
BIC	B-General_Concept
and	O
AIC	O
attempt	O
to	O
resolve	O
this	O
problem	O
by	O
introducing	O
a	O
penalty	O
term	O
for	O
the	O
number	O
of	O
parameters	O
in	O
the	O
model	O
;	O
the	O
penalty	O
term	O
is	O
larger	O
in	O
BIC	B-General_Concept
than	O
in	O
AIC	O
for	O
sample	O
sizes	O
greater	O
than	O
7	O
.	O
</s>
<s>
The	O
BIC	B-General_Concept
was	O
developed	O
by	O
Gideon	O
E	O
.	O
Schwarz	O
and	O
published	O
in	O
a	O
1978	O
paper	O
,	O
where	O
he	O
gave	O
a	O
Bayesian	O
argument	O
for	O
adopting	O
it	O
.	O
</s>
<s>
Konishi	O
and	O
Kitagawa	O
derive	O
the	O
BIC	B-General_Concept
to	O
approximate	O
the	O
distribution	O
of	O
the	O
data	O
,	O
integrating	O
out	O
the	O
parameters	O
using	O
Laplace	O
's	O
method	O
,	O
starting	O
with	O
the	O
following	O
model	O
evidence	O
:	O
</s>
<s>
where	O
BIC	B-General_Concept
is	O
defined	O
as	O
above	O
,	O
and	O
either	O
(	O
a	O
)	O
is	O
the	O
Bayesian	O
posterior	O
mode	O
or	O
(	O
b	O
)	O
uses	O
the	O
MLE	O
and	O
the	O
prior	O
has	O
nonzero	O
slope	O
at	O
the	O
MLE	O
.	O
</s>
<s>
When	O
picking	O
from	O
several	O
models	O
,	O
ones	O
with	O
lower	O
BIC	B-General_Concept
values	O
are	O
generally	O
preferred	O
.	O
</s>
<s>
The	O
BIC	B-General_Concept
is	O
an	O
increasing	O
function	O
of	O
the	O
error	O
variance	O
and	O
an	O
increasing	O
function	O
of	O
k	O
.	O
That	O
is	O
,	O
unexplained	O
variation	O
in	O
the	O
dependent	O
variable	O
and	O
the	O
number	O
of	O
explanatory	O
variables	O
increase	O
the	O
value	O
of	O
BIC	B-General_Concept
.	O
</s>
<s>
However	O
,	O
a	O
lower	O
BIC	B-General_Concept
does	O
not	O
necessarily	O
indicate	O
one	O
model	O
is	O
better	O
than	O
another	O
.	O
</s>
<s>
Because	O
it	O
involves	O
approximations	O
,	O
the	O
BIC	B-General_Concept
is	O
merely	O
a	O
heuristic	O
.	O
</s>
<s>
In	O
particular	O
,	O
differences	O
in	O
BIC	B-General_Concept
should	O
never	O
be	O
treated	O
like	O
transformed	O
Bayes	B-General_Concept
factors	I-General_Concept
.	O
</s>
<s>
It	O
is	O
important	O
to	O
keep	O
in	O
mind	O
that	O
the	O
BIC	B-General_Concept
can	O
be	O
used	O
to	O
compare	O
estimated	O
models	O
only	O
when	O
the	O
numerical	O
values	O
of	O
the	O
dependent	O
variable	O
are	O
identical	O
for	O
all	O
models	O
being	O
compared	O
.	O
</s>
<s>
The	O
models	O
being	O
compared	O
need	O
not	O
be	O
nested	O
,	O
unlike	O
the	O
case	O
when	O
models	O
are	O
being	O
compared	O
using	O
an	O
F-test	B-General_Concept
or	O
a	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
BIC	B-General_Concept
generally	O
penalizes	O
free	O
parameters	O
more	O
strongly	O
than	O
the	O
Akaike	O
information	O
criterion	O
,	O
though	O
it	O
depends	O
on	O
the	O
size	O
of	O
n	O
and	O
relative	O
magnitude	O
of	O
n	O
andk	O
.	O
</s>
<s>
It	O
is	O
closely	O
related	O
to	O
other	O
penalized	O
likelihood	O
criteria	O
such	O
as	O
Deviance	B-General_Concept
information	O
criterion	O
and	O
the	O
Akaike	O
information	O
criterion	O
.	O
</s>
<s>
the	O
BIC	B-General_Concept
cannot	O
handle	O
complex	O
collections	O
of	O
models	O
as	O
in	O
the	O
variable	B-General_Concept
selection	I-General_Concept
(	O
or	O
feature	B-General_Concept
selection	I-General_Concept
)	O
problem	O
in	O
high-dimension	O
.	O
</s>
<s>
deviance	B-General_Concept
as	O
:	O
</s>
