<s>
In	O
machine	O
learning	O
,	O
a	O
probabilistic	B-General_Concept
classifier	I-General_Concept
is	O
a	O
classifier	B-General_Concept
that	O
is	O
able	O
to	O
predict	O
,	O
given	O
an	O
observation	O
of	O
an	O
input	O
,	O
a	O
probability	O
distribution	O
over	O
a	O
set	O
of	O
classes	O
,	O
rather	O
than	O
only	O
outputting	O
the	O
most	O
likely	O
class	O
that	O
the	O
observation	O
should	O
belong	O
to	O
.	O
</s>
<s>
Probabilistic	B-General_Concept
classifiers	I-General_Concept
provide	O
classification	O
that	O
can	O
be	O
useful	O
in	O
its	O
own	O
right	O
or	O
when	O
combining	O
classifiers	B-General_Concept
into	O
ensembles	B-Algorithm
.	O
</s>
<s>
Formally	O
,	O
an	O
"	O
ordinary	O
"	O
classifier	B-General_Concept
is	O
some	O
rule	O
,	O
or	O
function	O
,	O
that	O
assigns	O
to	O
a	O
sample	O
a	O
class	O
label	O
:	O
</s>
<s>
The	O
samples	O
come	O
from	O
some	O
set	O
(	O
e.g.	O
,	O
the	O
set	O
of	O
all	O
documents	B-Algorithm
,	O
or	O
the	O
set	O
of	O
all	O
images	O
)	O
,	O
while	O
the	O
class	O
labels	O
form	O
a	O
finite	O
set	O
defined	O
prior	O
to	O
training	O
.	O
</s>
<s>
Probabilistic	B-General_Concept
classifiers	I-General_Concept
generalize	O
this	O
notion	O
of	O
classifiers	B-General_Concept
:	O
instead	O
of	O
functions	O
,	O
they	O
are	O
conditional	O
distributions	O
,	O
meaning	O
that	O
for	O
a	O
given	O
,	O
they	O
assign	O
probabilities	O
to	O
all	O
(	O
and	O
these	O
probabilities	O
sum	O
to	O
one	O
)	O
.	O
</s>
<s>
Binary	B-General_Concept
probabilistic	B-General_Concept
classifiers	I-General_Concept
are	O
also	O
called	O
binary	B-General_Concept
regression	O
models	O
in	O
statistics	O
.	O
</s>
<s>
In	O
econometrics	O
,	O
probabilistic	B-General_Concept
classification	I-General_Concept
in	O
general	O
is	O
called	O
discrete	O
choice	O
.	O
</s>
<s>
Some	O
classification	O
models	O
,	O
such	O
as	O
naive	B-General_Concept
Bayes	I-General_Concept
,	O
logistic	O
regression	O
and	O
multilayer	B-Algorithm
perceptrons	I-Algorithm
(	O
when	O
trained	O
under	O
an	O
appropriate	O
loss	O
function	O
)	O
are	O
naturally	O
probabilistic	O
.	O
</s>
<s>
Other	O
models	O
such	O
as	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
are	O
not	O
,	O
but	O
methods	O
exist	O
to	O
turn	O
them	O
into	O
probabilistic	B-General_Concept
classifiers	I-General_Concept
.	O
</s>
<s>
Some	O
models	O
,	O
such	O
as	O
logistic	O
regression	O
,	O
are	O
conditionally	O
trained	O
:	O
they	O
optimize	O
the	O
conditional	O
probability	O
directly	O
on	O
a	O
training	O
set	O
(	O
see	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
)	O
.	O
</s>
<s>
Other	O
classifiers	B-General_Concept
,	O
such	O
as	O
naive	B-General_Concept
Bayes	I-General_Concept
,	O
are	O
trained	O
generatively	O
:	O
at	O
training	O
time	O
,	O
the	O
class-conditional	O
distribution	O
and	O
the	O
class	O
prior	O
are	O
found	O
,	O
and	O
the	O
conditional	O
distribution	O
is	O
derived	O
using	O
Bayes	O
 '	O
rule	O
.	O
</s>
<s>
Not	O
all	O
classification	O
models	O
are	O
naturally	O
probabilistic	O
,	O
and	O
some	O
that	O
are	O
,	O
notably	O
naive	B-General_Concept
Bayes	I-General_Concept
classifiers	I-General_Concept
,	O
decision	B-Algorithm
trees	I-Algorithm
and	O
boosting	B-Algorithm
methods	O
,	O
produce	O
distorted	O
class	O
probability	O
distributions	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
decision	B-Algorithm
trees	I-Algorithm
,	O
where	O
is	O
the	O
proportion	O
of	O
training	O
samples	O
with	O
label	O
in	O
the	O
leaf	O
where	O
ends	O
up	O
,	O
these	O
distortions	O
come	O
about	O
because	O
learning	O
algorithms	O
such	O
as	O
C4.5	B-Algorithm
or	O
CART	O
explicitly	O
aim	O
to	O
produce	O
homogeneous	O
leaves	O
(	O
giving	O
probabilities	O
close	O
to	O
zero	O
or	O
one	O
,	O
and	O
thus	O
high	O
bias	O
)	O
while	O
using	O
few	O
samples	O
to	O
estimate	O
the	O
relevant	O
proportion	O
(	O
high	O
variance	B-General_Concept
)	O
.	O
</s>
<s>
Calibration	B-General_Concept
can	O
be	O
assessed	O
using	O
a	O
calibration	B-General_Concept
plot	O
(	O
also	O
called	O
a	O
reliability	O
diagram	O
)	O
.	O
</s>
<s>
A	O
calibration	B-General_Concept
plot	O
shows	O
the	O
proportion	O
of	O
items	O
in	O
each	O
class	O
for	O
bands	O
of	O
predicted	O
probability	O
or	O
score	O
(	O
such	O
as	O
a	O
distorted	O
probability	O
distribution	O
or	O
the	O
"	O
signed	O
distance	O
to	O
the	O
hyperplane	O
"	O
in	O
a	O
support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
)	O
.	O
</s>
<s>
Deviations	O
from	O
the	O
identity	O
function	O
indicate	O
a	O
poorly-calibrated	O
classifier	B-General_Concept
for	O
which	O
the	O
predicted	O
probabilities	O
or	O
scores	O
can	O
not	O
be	O
used	O
as	O
probabilities	O
.	O
</s>
<s>
In	O
this	O
case	O
one	O
can	O
use	O
a	O
method	O
to	O
turn	O
these	O
scores	O
into	O
properly	O
calibrated	B-General_Concept
class	B-General_Concept
membership	I-General_Concept
probabilities	I-General_Concept
.	O
</s>
<s>
For	O
the	O
binary	B-General_Concept
case	O
,	O
a	O
common	O
approach	O
is	O
to	O
apply	O
Platt	B-General_Concept
scaling	I-General_Concept
,	O
which	O
learns	O
a	O
logistic	O
regression	O
model	O
on	O
the	O
scores	O
.	O
</s>
<s>
An	O
alternative	O
method	O
using	O
isotonic	B-General_Concept
regression	I-General_Concept
is	O
generally	O
superior	O
to	O
Platt	O
's	O
method	O
when	O
sufficient	O
training	O
data	O
is	O
available	O
.	O
</s>
<s>
In	O
the	O
multiclass	B-General_Concept
case	O
,	O
one	O
can	O
use	O
a	O
reduction	O
to	O
binary	B-General_Concept
tasks	O
,	O
followed	O
by	O
univariate	O
calibration	B-General_Concept
with	O
an	O
algorithm	O
as	O
described	O
above	O
and	O
further	O
application	O
of	O
the	O
pairwise	O
coupling	O
algorithm	O
by	O
Hastie	O
and	O
Tibshirani	O
.	O
</s>
<s>
Commonly	O
used	O
loss	O
functions	O
for	O
probabilistic	B-General_Concept
classification	I-General_Concept
include	O
log	O
loss	O
and	O
the	O
Brier	O
score	O
between	O
the	O
predicted	O
and	O
the	O
true	O
probability	O
distributions	O
.	O
</s>
<s>
MoRPE	O
is	O
a	O
trainable	O
probabilistic	B-General_Concept
classifier	I-General_Concept
that	O
uses	O
isotonic	B-General_Concept
regression	I-General_Concept
for	O
probability	O
calibration	B-General_Concept
.	O
</s>
<s>
It	O
solves	O
the	O
multiclass	B-General_Concept
case	O
by	O
reduction	O
to	O
binary	B-General_Concept
tasks	O
.	O
</s>
