<s>
Bayesian	B-General_Concept
quadrature	I-General_Concept
views	O
numerical	B-Algorithm
integration	I-Algorithm
as	O
a	O
Bayesian	O
inference	O
task	O
,	O
where	O
function	O
evaluations	O
are	O
used	O
to	O
estimate	O
the	O
integral	O
of	O
that	O
function	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
it	O
is	O
sometimes	O
also	O
referred	O
to	O
as	O
"	O
Bayesian	O
probabilistic	O
numerical	B-Algorithm
integration	I-Algorithm
"	O
or	O
"	O
Bayesian	O
numerical	B-Algorithm
integration	I-Algorithm
"	O
.	O
</s>
<s>
The	O
name	O
"	O
Bayesian	O
cubature	B-Algorithm
"	O
is	O
also	O
sometimes	O
used	O
when	O
the	O
integrand	O
is	O
multi-dimensional	O
.	O
</s>
<s>
In	O
numerical	B-Algorithm
integration	I-Algorithm
,	O
function	O
evaluations	O
at	O
distinct	O
locations	O
in	O
are	O
used	O
to	O
estimate	O
the	O
integral	O
of	O
against	O
a	O
measure	O
:	O
i.e.	O
</s>
<s>
Bayesian	B-General_Concept
quadrature	I-General_Concept
consists	O
of	O
specifying	O
a	O
prior	O
distribution	O
over	O
,	O
conditioning	O
this	O
prior	O
on	O
to	O
obtain	O
a	O
posterior	O
distribution	O
,	O
then	O
computing	O
the	O
implied	O
posterior	O
distribution	O
on	O
.	O
</s>
<s>
The	O
most	O
common	O
choice	O
of	O
prior	O
distribution	O
for	O
is	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
as	O
this	O
permits	O
conjugate	O
inference	O
to	O
obtain	O
a	O
closed-form	O
posterior	O
distribution	O
on	O
.	O
</s>
<s>
Suppose	O
we	O
have	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
with	O
prior	O
mean	O
function	O
and	O
covariance	O
function	O
(	O
or	O
kernel	O
function	O
)	O
.	O
</s>
<s>
Then	O
,	O
the	O
posterior	O
distribution	O
on	O
is	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
with	O
mean	O
and	O
kernel	O
given	O
by	O
:	O
</s>
<s>
is	O
the	O
kernel	B-General_Concept
mean	I-General_Concept
embedding	I-General_Concept
of	O
and	O
denotes	O
the	O
integral	O
of	O
with	O
respect	O
to	O
both	O
inputs	O
.	O
</s>
<s>
The	O
estimation	O
of	O
kernel	O
hyperparameters	O
introduces	O
adaptivity	B-Algorithm
into	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
.	O
</s>
<s>
using	O
a	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
rule	O
based	O
on	O
a	O
zero-mean	O
Gaussian	B-General_Concept
process	I-General_Concept
prior	O
with	O
the	O
Matérn	O
covariance	O
function	O
of	O
smoothness	O
and	O
correlation	O
length	O
.	O
</s>
<s>
Convergence	O
of	O
the	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
point	O
estimate	O
and	O
concentration	O
of	O
the	O
posterior	O
mass	O
,	O
as	O
quantified	O
by	O
,	O
around	O
the	O
true	O
integral	O
as	O
is	O
evaluated	O
at	O
more	O
and	O
more	O
points	O
is	O
displayed	O
in	O
the	O
accompanying	O
animation	O
.	O
</s>
<s>
Since	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
is	O
an	O
example	O
of	O
probabilistic	O
numerics	O
,	O
it	O
inherits	O
certain	O
advantages	O
compared	O
with	O
traditional	O
numerical	B-Algorithm
integration	I-Algorithm
methods	I-Algorithm
:	O
</s>
<s>
It	O
provides	O
a	O
principled	O
way	O
to	O
incorporate	O
prior	O
knowledge	O
by	O
using	O
a	O
judicious	O
choice	O
of	O
prior	O
distributions	O
for	O
,	O
which	O
may	O
be	O
more	O
sophisticated	O
compared	O
to	O
the	O
standard	O
Gaussian	B-General_Concept
process	I-General_Concept
just	O
described	O
.	O
</s>
<s>
jointly	O
inferring	O
multiple	O
related	O
quantities	O
of	O
interest	O
or	O
using	O
active	B-General_Concept
learning	I-General_Concept
to	O
reduce	O
the	O
required	O
number	O
of	O
points	O
.	O
</s>
<s>
Despite	O
these	O
merits	O
,	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
methods	O
possess	O
the	O
following	O
limitations	O
:	O
</s>
<s>
for	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
with	O
Gaussian	B-General_Concept
processes	I-General_Concept
,	O
the	O
kernel	B-General_Concept
mean	I-General_Concept
embedding	I-General_Concept
has	O
no	O
closed-form	O
expression	O
for	O
a	O
general	O
kernel	O
and	O
measure	O
.	O
</s>
<s>
The	O
computational	O
cost	O
of	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
methods	O
based	O
on	O
Gaussian	B-General_Concept
processes	I-General_Concept
is	O
in	O
general	O
due	O
to	O
the	O
cost	O
of	O
inverting	O
matrices	O
,	O
which	O
may	O
defy	O
their	O
applications	O
to	O
large-scale	O
problems	O
.	O
</s>
<s>
The	O
most	O
commonly	O
used	O
prior	O
for	O
is	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
prior	O
.	O
</s>
<s>
This	O
is	O
mainly	O
due	O
to	O
the	O
advantage	O
provided	O
by	O
Gaussian	O
conjugacy	O
and	O
the	O
fact	O
that	O
Gaussian	B-General_Concept
processes	I-General_Concept
can	O
encode	O
a	O
wide	O
range	O
of	O
prior	O
knowledge	O
including	O
smoothness	O
,	O
periodicity	O
and	O
sparsity	O
through	O
a	O
careful	O
choice	O
of	O
prior	O
covariance	O
.	O
</s>
<s>
This	O
includes	O
multi-output	B-Algorithm
Gaussian	I-Algorithm
processes	I-Algorithm
,	O
which	O
are	O
particularly	O
useful	O
when	O
tackling	O
multiple	O
related	O
numerical	B-Algorithm
integration	I-Algorithm
tasks	O
simultaneously	O
or	O
sequentially	O
,	O
and	O
tree-based	O
priors	O
such	O
as	O
Bayesian	O
additive	O
regression	O
trees	O
,	O
which	O
are	O
well	O
suited	O
for	O
discontinuous	O
.	O
</s>
<s>
Additionally	O
,	O
Dirichlet	B-General_Concept
processes	I-General_Concept
priors	O
have	O
also	O
been	O
proposed	O
for	O
the	O
integration	O
measure	O
.	O
</s>
<s>
One	O
approach	O
consists	O
of	O
using	O
point	O
sets	O
from	O
other	O
quadrature	B-Algorithm
rules	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
taking	O
independent	O
and	O
identically	O
distributed	O
realisations	O
from	O
recovers	O
a	O
Bayesian	O
approach	O
to	O
Monte	B-Algorithm
Carlo	I-Algorithm
,	O
whereas	O
using	O
certain	O
deterministic	O
point	O
sets	O
such	O
as	O
low-discrepancy	O
sequences	O
or	O
lattices	O
recovers	O
a	O
Bayesian	O
alternative	O
to	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
.	O
</s>
<s>
who	O
exploited	O
symmetries	O
in	O
point	O
sets	O
to	O
obtain	O
scalable	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
estimators	O
.	O
</s>
<s>
Alternatively	O
,	O
points	O
can	O
also	O
be	O
selected	O
adaptively	O
following	O
principles	O
from	O
active	B-General_Concept
learning	I-General_Concept
and	O
Bayesian	O
experimental	O
design	O
so	O
as	O
to	O
directly	O
minimise	O
posterior	O
uncertainty	O
,	O
including	O
for	O
multi-output	B-Algorithm
Gaussian	I-Algorithm
processes	I-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
challenges	O
when	O
implementing	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
is	O
the	O
need	O
to	O
evaluate	O
the	O
function	O
and	O
the	O
constant	O
.	O
</s>
<s>
The	O
former	O
is	O
commonly	O
called	O
the	O
kernel	B-General_Concept
mean	I-General_Concept
,	O
and	O
is	O
a	O
quantity	O
which	O
is	O
key	O
to	O
the	O
computation	O
of	O
kernel-based	O
distances	O
such	O
as	O
the	O
maximum	O
mean	O
discrepancy	O
.	O
</s>
<s>
Unfortunately	O
,	O
the	O
kernel	B-General_Concept
mean	I-General_Concept
and	O
initial	O
error	O
can	O
only	O
be	O
computed	O
for	O
a	O
small	O
number	O
of	O
pairs	O
;	O
see	O
for	O
example	O
Table	O
1	O
in	O
.	O
</s>
<s>
There	O
have	O
been	O
a	O
number	O
of	O
theoretical	O
guarantees	O
derived	O
for	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
.	O
</s>
<s>
Most	O
of	O
the	O
results	O
apply	O
to	O
the	O
case	O
of	O
Monte	B-Algorithm
Carlo	I-Algorithm
or	O
deterministic	O
grid	O
point	O
sets	O
,	O
but	O
some	O
results	O
also	O
extend	O
to	O
adaptive	O
designs	O
.	O
</s>
<s>
:	O
Probabilistic	O
numerical	O
methods	O
in	O
Python	O
,	O
including	O
a	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
implementation	O
.	O
</s>
<s>
:	O
Bayesian	B-General_Concept
quadrature	I-General_Concept
with	O
QMC	O
point	O
sets	O
in	O
Python	O
.	O
</s>
