<s>
Within	O
bayesian	O
statistics	O
for	O
machine	O
learning	O
,	O
kernel	B-Algorithm
methods	I-Algorithm
arise	O
from	O
the	O
assumption	O
of	O
an	O
inner	O
product	O
space	O
or	O
similarity	O
structure	O
on	O
inputs	O
.	O
</s>
<s>
For	O
some	O
such	O
methods	O
,	O
such	O
as	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
(	O
SVMs	B-Algorithm
)	O
,	O
the	O
original	O
formulation	O
and	O
its	O
regularization	O
were	O
not	O
Bayesian	O
in	O
nature	O
.	O
</s>
<s>
In	O
Bayesian	O
probability	O
kernel	B-Algorithm
methods	I-Algorithm
are	O
a	O
key	O
component	O
of	O
Gaussian	B-General_Concept
processes	I-General_Concept
,	O
where	O
the	O
kernel	O
function	O
is	O
known	O
as	O
the	O
covariance	O
function	O
.	O
</s>
<s>
Kernel	B-Algorithm
methods	I-Algorithm
have	O
traditionally	O
been	O
used	O
in	O
supervised	B-General_Concept
learning	I-General_Concept
problems	O
where	O
the	O
input	O
space	O
is	O
usually	O
a	O
space	O
of	O
vectors	O
while	O
the	O
output	O
space	O
is	O
a	O
space	O
of	O
scalars	O
.	O
</s>
<s>
More	O
recently	O
these	O
methods	O
have	O
been	O
extended	O
to	O
problems	O
that	O
deal	O
with	O
multiple	B-Algorithm
outputs	I-Algorithm
such	O
as	O
in	O
multi-task	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
We	O
start	O
with	O
a	O
brief	O
review	O
of	O
the	O
main	O
ideas	O
underlying	O
kernel	B-Algorithm
methods	I-Algorithm
for	O
scalar	O
learning	O
,	O
and	O
briefly	O
introduce	O
the	O
concepts	O
of	O
regularization	O
and	O
Gaussian	B-General_Concept
processes	I-General_Concept
.	O
</s>
<s>
The	O
classical	O
supervised	B-General_Concept
learning	I-General_Concept
problem	O
requires	O
estimating	O
the	O
output	O
for	O
some	O
new	O
input	O
point	O
by	O
learning	O
a	O
scalar-valued	O
estimator	O
on	O
the	O
basis	O
of	O
a	O
training	O
set	O
consisting	O
of	O
input-output	O
pairs	O
,	O
.	O
</s>
<s>
where	O
is	O
the	O
kernel	B-Algorithm
matrix	I-Algorithm
with	O
entries	O
,	O
,	O
and	O
.	O
</s>
<s>
The	O
notion	O
of	O
a	O
kernel	O
plays	O
a	O
crucial	O
role	O
in	O
Bayesian	O
probability	O
as	O
the	O
covariance	O
function	O
of	O
a	O
stochastic	O
process	O
called	O
the	O
Gaussian	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
As	O
part	O
of	O
the	O
Bayesian	O
framework	O
,	O
the	O
Gaussian	B-General_Concept
process	I-General_Concept
specifies	O
the	O
prior	O
distribution	O
that	O
describes	O
the	O
prior	O
beliefs	O
about	O
the	O
properties	O
of	O
the	O
function	O
being	O
modeled	O
.	O
</s>
<s>
A	O
Gaussian	B-General_Concept
process	I-General_Concept
(	O
GP	O
)	O
is	O
a	O
stochastic	O
process	O
in	O
which	O
any	O
finite	O
number	O
of	O
random	O
variables	O
that	O
are	O
sampled	O
follow	O
a	O
joint	O
Normal	O
distribution	O
.	O
</s>
<s>
Let	O
a	O
function	O
follow	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
with	O
mean	O
function	O
and	O
kernel	O
function	O
,	O
</s>
<s>
Under	O
this	O
assumption	O
,	O
regularization	O
theory	O
and	O
Bayesian	O
theory	O
are	O
connected	O
through	O
Gaussian	B-General_Concept
process	I-General_Concept
prediction	O
.	O
</s>
<s>
We	O
can	O
now	O
build	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
by	O
assuming	O
to	O
be	O
distributed	O
according	O
to	O
a	O
multivariate	O
Gaussian	O
distribution	O
with	O
zero	O
mean	O
and	O
identity	O
covariance	O
matrix	O
,	O
</s>
