<s>
In	O
statistical	B-General_Concept
classification	I-General_Concept
,	O
the	O
Bayes	B-General_Concept
classifier	I-General_Concept
minimizes	O
the	O
probability	O
of	O
misclassification	O
.	O
</s>
<s>
A	O
classifier	B-General_Concept
is	O
a	O
rule	O
that	O
assigns	O
to	O
an	O
observation	O
X	O
=	O
x	O
a	O
guess	O
or	O
estimate	O
of	O
what	O
the	O
unobserved	O
label	O
Y	O
=	O
r	O
actually	O
was	O
.	O
</s>
<s>
In	O
theoretical	O
terms	O
,	O
a	O
classifier	B-General_Concept
is	O
a	O
measurable	O
function	O
,	O
with	O
the	O
interpretation	O
that	O
C	O
classifies	O
the	O
point	O
x	O
to	O
the	O
class	O
C(x )	O
.	O
</s>
<s>
The	O
Bayes	B-General_Concept
classifier	I-General_Concept
is	O
a	O
useful	O
benchmark	O
in	O
statistical	B-General_Concept
classification	I-General_Concept
.	O
</s>
<s>
A	O
classifier	B-General_Concept
is	O
said	O
to	O
be	O
consistent	O
if	O
the	O
excess	O
risk	O
converges	O
to	O
zero	O
as	O
the	O
size	O
of	O
the	O
training	O
data	O
set	O
tends	O
to	O
infinity	O
.	O
</s>
<s>
Proof	O
that	O
the	O
Bayes	B-General_Concept
classifier	I-General_Concept
is	O
optimal	O
and	O
Bayes	O
error	O
rate	O
is	O
minimal	O
proceeds	O
as	O
follows	O
.	O
</s>
<s>
is	O
a	O
Bayes	B-General_Concept
classifier	I-General_Concept
,	O
</s>
<s>
The	O
general	O
case	O
that	O
the	O
Bayes	B-General_Concept
classifier	I-General_Concept
minimises	O
classification	O
error	O
when	O
each	O
element	O
can	O
belong	O
to	O
either	O
of	O
n	O
categories	O
proceeds	O
by	O
towering	O
expectations	O
as	O
follows	O
.	O
</s>
