<s>
In	O
statistics	O
,	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
(	O
MSE	O
)	O
or	O
mean	B-Algorithm
squared	I-Algorithm
deviation	I-Algorithm
(	O
MSD	O
)	O
of	O
an	O
estimator	O
(	O
of	O
a	O
procedure	O
for	O
estimating	O
an	O
unobserved	O
quantity	O
)	O
measures	O
the	O
average	O
of	O
the	O
squares	O
of	O
the	O
errors	O
—	O
that	O
is	O
,	O
the	O
average	O
squared	O
difference	O
between	O
the	O
estimated	O
values	O
and	O
the	O
actual	O
value	O
.	O
</s>
<s>
In	O
machine	O
learning	O
,	O
specifically	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
,	O
MSE	O
may	O
refer	O
to	O
the	O
empirical	O
risk	O
(	O
the	O
average	O
loss	O
on	O
an	O
observed	O
data	O
set	O
)	O
,	O
as	O
an	O
estimate	O
of	O
the	O
true	O
MSE	O
(	O
the	O
true	O
risk	O
:	O
the	O
average	O
loss	O
on	O
the	O
actual	O
population	O
distribution	O
)	O
.	O
</s>
<s>
In	O
an	O
analogy	O
to	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
taking	O
the	O
square	O
root	O
of	O
MSE	O
yields	O
the	O
root-mean-square	B-General_Concept
error	I-General_Concept
or	O
root-mean-square	B-General_Concept
deviation	I-General_Concept
(	O
RMSE	B-General_Concept
or	O
RMSD	B-General_Concept
)	O
,	O
which	O
has	O
the	O
same	O
units	O
as	O
the	O
quantity	O
being	O
estimated	O
;	O
for	O
an	O
unbiased	O
estimator	O
,	O
the	O
RMSE	B-General_Concept
is	O
the	O
square	O
root	O
of	O
the	O
variance	O
,	O
known	O
as	O
the	O
standard	B-General_Concept
error	I-General_Concept
.	O
</s>
<s>
By	O
substituting	O
with	O
,	O
,	O
we	O
haveBut	O
in	O
real	O
modeling	O
case	O
,	O
MSE	O
could	O
be	O
described	O
as	O
the	O
addition	O
of	O
model	O
variance	O
,	O
model	O
bias	O
,	O
and	O
irreducible	O
uncertainty	O
(	O
see	O
Bias	B-General_Concept
–	I-General_Concept
variance	I-General_Concept
tradeoff	I-General_Concept
)	O
.	O
</s>
<s>
The	O
mean	O
of	O
the	O
distance	O
from	O
each	O
point	O
to	O
the	O
predicted	O
regression	O
model	O
can	O
be	O
calculated	O
,	O
and	O
shown	O
as	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
.	O
</s>
<s>
One	O
example	O
of	O
a	O
linear	B-General_Concept
regression	I-General_Concept
using	O
this	O
method	O
is	O
the	O
least	B-Algorithm
squares	I-Algorithm
method	I-Algorithm
—	O
which	O
evaluates	O
appropriateness	O
of	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
to	O
model	O
bivariate	O
dataset	O
,	O
but	O
whose	O
limitation	O
is	O
related	O
to	O
known	O
distribution	O
of	O
the	O
data	O
.	O
</s>
<s>
The	O
term	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
is	O
sometimes	O
used	O
to	O
refer	O
to	O
the	O
unbiased	O
estimate	O
of	O
error	O
variance	O
:	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
divided	O
by	O
the	O
number	O
of	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
In	O
regression	O
analysis	O
,	O
"	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
"	O
,	O
often	O
referred	O
to	O
as	O
mean	B-General_Concept
squared	I-General_Concept
prediction	I-General_Concept
error	I-General_Concept
or	O
"	O
out-of-sample	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
"	O
,	O
can	O
also	O
refer	O
to	O
the	O
mean	O
value	O
of	O
the	O
squared	B-General_Concept
deviations	I-General_Concept
of	O
the	O
predictions	O
from	O
the	O
true	O
values	O
,	O
over	O
an	O
out-of-sample	O
test	O
space	O
,	O
generated	O
by	O
a	O
model	O
estimated	O
over	O
a	O
particular	O
sample	O
space	O
.	O
</s>
<s>
where	O
is	O
the	O
population	B-General_Concept
variance	I-General_Concept
.	O
</s>
<s>
where	O
is	O
the	O
fourth	O
central	B-General_Concept
moment	I-General_Concept
of	O
the	O
distribution	O
or	O
population	O
,	O
and	O
is	O
the	O
excess	O
kurtosis	O
.	O
</s>
<s>
However	O
,	O
one	O
can	O
use	O
other	O
estimators	O
for	O
which	O
are	O
proportional	O
to	O
,	O
and	O
an	O
appropriate	O
choice	O
can	O
always	O
give	O
a	O
lower	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
.	O
</s>
<s>
True	O
value	O
Estimator	O
Mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
=	O
the	O
unbiased	O
estimator	O
of	O
the	O
population	O
mean	O
,	O
=	O
the	O
unbiased	O
estimator	O
of	O
the	O
population	B-General_Concept
variance	I-General_Concept
,	O
=	O
the	O
biased	O
estimator	O
of	O
the	O
population	B-General_Concept
variance	I-General_Concept
,	O
=	O
the	O
biased	O
estimator	O
of	O
the	O
population	B-General_Concept
variance	I-General_Concept
,	O
</s>
<s>
Both	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
and	O
linear	B-General_Concept
regression	I-General_Concept
techniques	O
estimate	O
the	O
MSE	O
as	O
part	O
of	O
the	O
analysis	O
and	O
use	O
the	O
estimated	O
MSE	O
to	O
determine	O
the	O
statistical	B-General_Concept
significance	I-General_Concept
of	O
the	O
factors	O
or	O
predictors	O
under	O
study	O
.	O
</s>
<s>
In	O
one-way	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
,	O
MSE	O
can	O
be	O
calculated	O
by	O
the	O
division	O
of	O
the	O
sum	O
of	O
squared	O
errors	O
and	O
the	O
degree	O
of	O
freedom	O
.	O
</s>
<s>
Minimizing	O
MSE	O
is	O
a	O
key	O
criterion	O
in	O
selecting	O
estimators	O
:	O
see	O
minimum	B-General_Concept
mean-square	I-General_Concept
error	I-General_Concept
.	O
</s>
<s>
Carl	O
Friedrich	O
Gauss	O
,	O
who	O
introduced	O
the	O
use	O
of	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
,	O
was	O
aware	O
of	O
its	O
arbitrariness	O
and	O
was	O
in	O
agreement	O
with	O
objections	O
to	O
it	O
on	O
these	O
grounds	O
.	O
</s>
<s>
The	O
mathematical	O
benefits	O
of	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
are	O
particularly	O
evident	O
in	O
its	O
use	O
at	O
analyzing	O
the	O
performance	O
of	O
linear	B-General_Concept
regression	I-General_Concept
,	O
as	O
it	O
allows	O
one	O
to	O
partition	O
the	O
variation	O
in	O
a	O
dataset	O
into	O
variation	O
explained	O
by	O
the	O
model	O
and	O
variation	O
explained	O
by	O
randomness	O
.	O
</s>
<s>
The	O
use	O
of	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
without	O
question	O
has	O
been	O
criticized	O
by	O
the	O
decision	O
theorist	O
James	O
Berger	O
.	O
</s>
<s>
Mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
is	O
the	O
negative	O
of	O
the	O
expected	O
value	O
of	O
one	O
specific	O
utility	O
function	O
,	O
the	O
quadratic	O
utility	O
function	O
,	O
which	O
may	O
not	O
be	O
the	O
appropriate	O
utility	O
function	O
to	O
use	O
under	O
a	O
given	O
set	O
of	O
circumstances	O
.	O
</s>
<s>
There	O
are	O
,	O
however	O
,	O
some	O
scenarios	O
where	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
can	O
serve	O
as	O
a	O
good	O
approximation	O
to	O
a	O
loss	O
function	O
occurring	O
naturally	O
in	O
an	O
application	O
.	O
</s>
<s>
Like	O
variance	O
,	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
has	O
the	O
disadvantage	O
of	O
heavily	O
weighting	O
outliers	O
.	O
</s>
<s>
This	O
property	O
,	O
undesirable	O
in	O
many	O
applications	O
,	O
has	O
led	O
researchers	O
to	O
use	O
alternatives	O
such	O
as	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
error	I-General_Concept
,	O
or	O
those	O
based	O
on	O
the	O
median	O
.	O
</s>
