<s>
In	O
probability	O
theory	O
and	O
statistics	O
,	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
is	O
a	O
stochastic	O
process	O
(	O
a	O
collection	O
of	O
random	O
variables	O
indexed	O
by	O
time	O
or	O
space	O
)	O
,	O
such	O
that	O
every	O
finite	O
collection	O
of	O
those	O
random	O
variables	O
has	O
a	O
multivariate	O
normal	O
distribution	O
,	O
i.e.	O
</s>
<s>
The	O
distribution	O
of	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
is	O
the	O
joint	O
distribution	O
of	O
all	O
those	O
(	O
infinitely	O
many	O
)	O
random	O
variables	O
,	O
and	O
as	O
such	O
,	O
it	O
is	O
a	O
distribution	O
over	O
functions	O
with	O
a	O
continuous	O
domain	O
,	O
e.g.	O
</s>
<s>
The	O
concept	O
of	O
Gaussian	B-General_Concept
processes	I-General_Concept
is	O
named	O
after	O
Carl	O
Friedrich	O
Gauss	O
because	O
it	O
is	O
based	O
on	O
the	O
notion	O
of	O
the	O
Gaussian	O
distribution	O
(	O
normal	O
distribution	O
)	O
.	O
</s>
<s>
Gaussian	B-General_Concept
processes	I-General_Concept
can	O
be	O
seen	O
as	O
an	O
infinite-dimensional	O
generalization	O
of	O
multivariate	O
normal	O
distributions	O
.	O
</s>
<s>
Gaussian	B-General_Concept
processes	I-General_Concept
are	O
useful	O
in	O
statistical	O
modelling	O
,	O
benefiting	O
from	O
properties	O
inherited	O
from	O
the	O
normal	O
distribution	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
a	O
random	O
process	O
is	O
modelled	O
as	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
,	O
the	O
distributions	O
of	O
various	O
derived	O
quantities	O
can	O
be	O
obtained	O
explicitly	O
.	O
</s>
<s>
While	O
exact	O
models	O
often	O
scale	O
poorly	O
as	O
the	O
amount	O
of	O
data	O
increases	O
,	O
multiple	O
approximation	B-Algorithm
methods	I-Algorithm
have	O
been	O
developed	O
which	O
often	O
retain	O
good	O
accuracy	O
while	O
drastically	O
reducing	O
computation	O
time	O
.	O
</s>
<s>
For	O
general	O
stochastic	O
processes	O
strict-sense	O
stationarity	B-Algorithm
implies	O
wide-sense	O
stationarity	B-Algorithm
but	O
not	O
every	O
wide-sense	O
stationary	B-Algorithm
stochastic	I-Algorithm
process	I-Algorithm
is	O
strict-sense	O
stationary	B-Algorithm
.	O
</s>
<s>
However	O
,	O
for	O
a	O
Gaussian	B-General_Concept
stochastic	I-General_Concept
process	I-General_Concept
the	O
two	O
concepts	O
are	O
equivalent	O
.	O
</s>
<s>
A	O
Gaussian	B-General_Concept
stochastic	I-General_Concept
process	I-General_Concept
is	O
strict-sense	O
stationary	B-Algorithm
if	O
,	O
and	O
only	O
if	O
,	O
it	O
is	O
wide-sense	O
stationary	B-Algorithm
.	O
</s>
<s>
There	O
is	O
an	O
explicit	O
representation	O
for	O
stationary	B-Algorithm
Gaussian	B-General_Concept
processes	I-General_Concept
.	O
</s>
<s>
A	O
key	O
fact	O
of	O
Gaussian	B-General_Concept
processes	I-General_Concept
is	O
that	O
they	O
can	O
be	O
completely	O
defined	O
by	O
their	O
second-order	O
statistics	O
.	O
</s>
<s>
Thus	O
,	O
if	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
is	O
assumed	O
to	O
have	O
mean	O
zero	O
,	O
defining	O
the	O
covariance	O
function	O
completely	O
defines	O
the	O
process	O
 '	O
behaviour	O
.	O
</s>
<s>
Basic	O
aspects	O
that	O
can	O
be	O
defined	O
through	O
the	O
covariance	O
function	O
are	O
the	O
process	O
 '	O
stationarity	B-Algorithm
,	O
isotropy	O
,	O
smoothness	O
and	O
periodicity	O
.	O
</s>
<s>
Stationarity	B-Algorithm
refers	O
to	O
the	O
process	O
 '	O
behaviour	O
regarding	O
the	O
separation	O
of	O
any	O
two	O
points	O
and	O
.	O
</s>
<s>
If	O
the	O
process	O
is	O
stationary	B-Algorithm
,	O
the	O
covariance	O
function	O
depends	O
only	O
on	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Ornstein	O
–	O
Uhlenbeck	O
process	O
is	O
stationary	B-Algorithm
.	O
</s>
<s>
A	O
process	O
that	O
is	O
concurrently	O
stationary	B-Algorithm
and	O
isotropic	O
is	O
considered	O
to	O
be	O
homogeneous	B-General_Concept
;	O
in	O
practice	O
these	O
properties	O
reflect	O
the	O
differences	O
(	O
or	O
rather	O
the	O
lack	O
of	O
them	O
)	O
in	O
the	O
behaviour	O
of	O
the	O
process	O
given	O
the	O
location	O
of	O
the	O
observer	O
.	O
</s>
<s>
Ultimately	O
Gaussian	B-General_Concept
processes	I-General_Concept
translate	O
as	O
taking	O
priors	O
on	O
functions	O
and	O
the	O
smoothness	O
of	O
these	O
priors	O
can	O
be	O
induced	O
by	O
the	O
covariance	O
function	O
.	O
</s>
<s>
The	O
parameter	O
is	O
the	O
characteristic	O
length-scale	O
of	O
the	O
process	O
(	O
practically	O
,	O
"	O
how	O
close	O
"	O
two	O
points	O
and	O
have	O
to	O
be	O
to	O
influence	O
each	O
other	O
significantly	O
)	O
,	O
is	O
the	O
Kronecker	O
delta	O
and	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
noise	O
fluctuations	O
.	O
</s>
<s>
A	O
popular	O
choice	O
for	O
is	O
to	O
provide	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
(	O
MAP	O
)	O
estimates	O
of	O
it	O
with	O
some	O
chosen	O
prior	O
.	O
</s>
<s>
This	O
approach	O
is	O
also	O
known	O
as	O
maximum	O
likelihood	O
II	O
,	O
evidence	O
maximization	O
,	O
or	O
empirical	B-General_Concept
Bayes	I-General_Concept
.	O
</s>
<s>
For	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
,	O
continuity	O
in	O
probability	O
is	O
equivalent	O
to	O
mean-square	O
continuity	O
,	O
</s>
<s>
In	O
contrast	O
,	O
sample	O
continuity	O
was	O
challenging	O
even	O
for	O
stationary	B-Algorithm
Gaussian	B-General_Concept
processes	I-General_Concept
(	O
as	O
probably	O
noted	O
first	O
by	O
Andrey	O
Kolmogorov	O
)	O
,	O
and	O
more	O
challenging	O
for	O
more	O
general	O
processes	O
.	O
</s>
<s>
For	O
a	O
stationary	B-Algorithm
Gaussian	B-General_Concept
process	I-General_Concept
some	O
conditions	O
on	O
its	O
spectrum	O
are	O
sufficient	O
for	O
sample	O
continuity	O
,	O
but	O
fail	O
to	O
be	O
necessary	O
.	O
</s>
<s>
(	O
the	O
right-hand	O
side	O
does	O
not	O
depend	O
on	O
due	O
to	O
stationarity	B-Algorithm
)	O
.	O
</s>
<s>
A	O
Wiener	O
process	O
(	O
also	O
known	O
as	O
Brownian	O
motion	O
)	O
is	O
the	O
integral	O
of	O
a	O
white	O
noise	O
generalized	O
Gaussian	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
It	O
is	O
not	O
stationary	B-Algorithm
,	O
but	O
it	O
has	O
stationary	B-Algorithm
increments	O
.	O
</s>
<s>
The	O
Ornstein	O
–	O
Uhlenbeck	O
process	O
is	O
a	O
stationary	B-Algorithm
Gaussian	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
The	O
Brownian	O
bridge	O
is	O
(	O
like	O
the	O
Ornstein	O
–	O
Uhlenbeck	O
process	O
)	O
an	O
example	O
of	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
whose	O
increments	O
are	O
not	O
independent	O
.	O
</s>
<s>
The	O
fractional	O
Brownian	O
motion	O
is	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
whose	O
covariance	O
function	O
is	O
a	O
generalisation	O
of	O
that	O
of	O
the	O
Wiener	O
process	O
.	O
</s>
<s>
Driscoll	O
's	O
zero-one	O
law	O
is	O
a	O
result	O
characterizing	O
the	O
sample	O
functions	O
generated	O
by	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
Let	O
be	O
a	O
mean-zero	O
Gaussian	B-General_Concept
process	I-General_Concept
with	O
non-negative	O
definite	O
covariance	O
function	O
.	O
</s>
<s>
As	O
such	O
,	O
almost	O
all	O
sample	O
paths	O
of	O
a	O
mean-zero	O
Gaussian	B-General_Concept
process	I-General_Concept
with	O
positive	O
definite	O
kernel	O
will	O
lie	O
outside	O
of	O
the	O
Hilbert	O
space	O
.	O
</s>
<s>
the	O
case	O
where	O
the	O
output	O
of	O
the	O
Gaussian	B-General_Concept
process	I-General_Concept
corresponds	O
to	O
a	O
magnetic	O
field	O
;	O
here	O
,	O
the	O
real	O
magnetic	O
field	O
is	O
bound	O
by	O
Maxwell	O
's	O
equations	O
and	O
a	O
way	O
to	O
incorporate	O
this	O
constraint	O
into	O
the	O
Gaussian	B-General_Concept
process	I-General_Concept
formalism	O
would	O
be	O
desirable	O
as	O
this	O
would	O
likely	O
improve	O
the	O
accuracy	O
of	O
the	O
algorithm	O
.	O
</s>
<s>
A	O
method	O
on	O
how	O
to	O
incorporate	O
linear	O
constraints	O
into	O
Gaussian	B-General_Concept
processes	I-General_Concept
already	O
exists	O
:	O
</s>
<s>
Hence	O
,	O
linear	O
constraints	O
can	O
be	O
encoded	O
into	O
the	O
mean	O
and	O
covariance	O
function	O
of	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
A	O
Gaussian	B-General_Concept
process	I-General_Concept
can	O
be	O
used	O
as	O
a	O
prior	O
probability	O
distribution	O
over	O
functions	O
in	O
Bayesian	O
inference	O
.	O
</s>
<s>
Given	O
any	O
set	O
of	O
N	O
points	O
in	O
the	O
desired	O
domain	O
of	O
your	O
functions	O
,	O
take	O
a	O
multivariate	O
Gaussian	O
whose	O
covariance	O
matrix	B-Architecture
parameter	O
is	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
your	O
N	O
points	O
with	O
some	O
desired	O
kernel	O
,	O
and	O
sample	O
from	O
that	O
Gaussian	O
.	O
</s>
<s>
For	O
solution	O
of	O
the	O
multi-output	O
prediction	O
problem	O
,	O
Gaussian	B-General_Concept
process	I-General_Concept
regression	O
for	O
vector-valued	O
function	O
was	O
developed	O
.	O
</s>
<s>
This	O
approach	O
was	O
elaborated	O
in	O
detail	O
for	O
the	O
matrix-valued	O
Gaussian	B-General_Concept
processes	I-General_Concept
and	O
generalised	O
to	O
processes	O
with	O
'	O
heavier	O
tails	O
 '	O
like	O
Student-t	O
processes	O
.	O
</s>
<s>
Inference	O
of	O
continuous	O
values	O
with	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
prior	O
is	O
known	O
as	O
Gaussian	B-General_Concept
process	I-General_Concept
regression	O
,	O
or	O
kriging	O
;	O
extending	O
Gaussian	B-General_Concept
process	I-General_Concept
regression	O
to	O
multiple	B-Algorithm
target	I-Algorithm
variables	I-Algorithm
is	O
known	O
as	O
cokriging	O
.	O
</s>
<s>
Gaussian	B-General_Concept
processes	I-General_Concept
are	O
thus	O
useful	O
as	O
a	O
powerful	O
non-linear	O
multivariate	O
interpolation	B-Algorithm
tool	O
.	O
</s>
<s>
Gaussian	B-General_Concept
processes	I-General_Concept
are	O
also	O
commonly	O
used	O
to	O
tackle	O
numerical	O
analysis	O
problems	O
such	O
as	O
numerical	O
integration	O
,	O
solving	O
differential	O
equations	O
,	O
or	O
optimisation	O
in	O
the	O
field	O
of	O
probabilistic	O
numerics	O
.	O
</s>
<s>
Gaussian	B-General_Concept
processes	I-General_Concept
can	O
also	O
be	O
used	O
in	O
the	O
context	O
of	O
mixture	O
of	O
experts	O
models	O
,	O
for	O
example	O
.	O
</s>
<s>
The	O
underlying	O
rationale	O
of	O
such	O
a	O
learning	O
framework	O
consists	O
in	O
the	O
assumption	O
that	O
a	O
given	O
mapping	O
cannot	O
be	O
well	O
captured	O
by	O
a	O
single	O
Gaussian	B-General_Concept
process	I-General_Concept
model	O
.	O
</s>
<s>
Instead	O
,	O
the	O
observation	O
space	O
is	O
divided	O
into	O
subsets	O
,	O
each	O
of	O
which	O
is	O
characterized	O
by	O
a	O
different	O
mapping	O
function	O
;	O
each	O
of	O
these	O
is	O
learned	O
via	O
a	O
different	O
Gaussian	B-General_Concept
process	I-General_Concept
component	O
in	O
the	O
postulated	O
mixture	O
.	O
</s>
<s>
In	O
the	O
natural	O
sciences	O
,	O
Gaussian	B-General_Concept
processes	I-General_Concept
have	O
found	O
use	O
as	O
probabilistic	O
models	O
of	O
astronomical	O
time	O
series	O
and	O
as	O
predictors	O
of	O
molecular	O
properties	O
.	O
</s>
<s>
When	O
concerned	O
with	O
a	O
general	O
Gaussian	B-General_Concept
process	I-General_Concept
regression	O
problem	O
(	O
Kriging	O
)	O
,	O
it	O
is	O
assumed	O
that	O
for	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
observed	O
at	O
coordinates	O
,	O
the	O
vector	O
of	O
values	O
is	O
just	O
one	O
sample	O
from	O
a	O
multivariate	O
Gaussian	O
distribution	O
of	O
dimension	O
equal	O
to	O
number	O
of	O
observed	O
coordinates	O
.	O
</s>
<s>
Therefore	O
,	O
under	O
the	O
assumption	O
of	O
a	O
zero-mean	O
distribution	O
,	O
,	O
where	O
is	O
the	O
covariance	O
matrix	B-Architecture
between	O
all	O
possible	O
pairs	O
for	O
a	O
given	O
set	O
of	O
hyperparameters	O
θ	O
.	O
</s>
<s>
and	O
maximizing	O
this	O
marginal	O
likelihood	O
towards	O
provides	O
the	O
complete	O
specification	O
of	O
the	O
Gaussian	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
A	O
known	O
bottleneck	O
in	O
Gaussian	B-General_Concept
process	I-General_Concept
prediction	O
is	O
that	O
the	O
computational	O
complexity	O
of	O
inference	O
and	O
likelihood	O
evaluation	O
is	O
cubic	O
in	O
the	O
number	O
of	O
points	O
|x|	O
,	O
and	O
as	O
such	O
can	O
become	O
unfeasible	O
for	O
larger	O
data	O
sets	O
.	O
</s>
<s>
Works	O
on	O
sparse	O
Gaussian	B-General_Concept
processes	I-General_Concept
,	O
that	O
usually	O
are	O
based	O
on	O
the	O
idea	O
of	O
building	O
a	O
representative	O
set	O
for	O
the	O
given	O
process	O
f	O
,	O
try	O
to	O
circumvent	O
this	O
issue	O
.	O
</s>
<s>
Bayesian	O
neural	B-Architecture
networks	I-Architecture
are	O
a	O
particular	O
type	O
of	O
Bayesian	O
network	O
that	O
results	O
from	O
treating	O
deep	B-Algorithm
learning	I-Algorithm
and	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
models	O
probabilistically	O
,	O
and	O
assigning	O
a	O
prior	O
distribution	O
to	O
their	O
parameters	O
.	O
</s>
<s>
Computation	O
in	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
is	O
usually	O
organized	O
into	O
sequential	O
layers	O
of	O
artificial	B-Algorithm
neurons	I-Algorithm
.	O
</s>
<s>
As	O
layer	O
width	O
grows	O
large	O
,	O
many	O
Bayesian	O
neural	B-Architecture
networks	I-Architecture
reduce	O
to	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
with	O
a	O
closed	O
form	O
compositional	O
kernel	O
.	O
</s>
<s>
This	O
Gaussian	B-General_Concept
process	I-General_Concept
is	O
called	O
the	O
Neural	B-Architecture
Network	I-Architecture
Gaussian	B-General_Concept
Process	I-General_Concept
(	O
NNGP	O
)	O
.	O
</s>
<s>
It	O
allows	O
predictions	O
from	O
Bayesian	O
neural	B-Architecture
networks	I-Architecture
to	O
be	O
more	O
efficiently	O
evaluated	O
,	O
and	O
provides	O
an	O
analytic	O
tool	O
to	O
understand	O
deep	B-Algorithm
learning	I-Algorithm
models	O
.	O
</s>
<s>
In	O
practical	O
applications	O
,	O
Gaussian	B-General_Concept
process	I-General_Concept
models	O
are	O
often	O
evaluated	O
on	O
a	O
grid	O
leading	O
to	O
multivariate	O
normal	O
distributions	O
.	O
</s>
<s>
Using	O
these	O
models	O
for	O
prediction	O
or	O
parameter	O
estimation	O
using	O
maximum	O
likelihood	O
requires	O
evaluating	O
a	O
multivariate	O
Gaussian	O
density	O
,	O
which	O
involves	O
calculating	O
the	O
determinant	O
and	O
the	O
inverse	O
of	O
the	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
This	O
drawback	O
led	O
to	O
the	O
development	O
of	O
multiple	O
approximation	B-Algorithm
methods	I-Algorithm
.	O
</s>
