<s>
In	O
mathematics	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
is	O
a	O
technique	O
for	O
numerical	B-Algorithm
integration	I-Algorithm
using	O
random	B-Error_Name
numbers	I-Error_Name
.	O
</s>
<s>
It	O
is	O
a	O
particular	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
that	O
numerically	O
computes	O
a	O
definite	B-Algorithm
integral	I-Algorithm
.	O
</s>
<s>
This	O
method	O
is	O
particularly	O
useful	O
for	O
higher-dimensional	O
integrals	B-Algorithm
.	O
</s>
<s>
There	O
are	O
different	O
methods	O
to	O
perform	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
,	O
such	O
as	O
uniform	O
sampling	O
,	O
stratified	O
sampling	O
,	O
importance	B-Algorithm
sampling	I-Algorithm
,	O
sequential	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
(	O
also	O
known	O
as	O
a	O
particle	B-Algorithm
filter	I-Algorithm
)	O
,	O
and	O
mean-field	O
particle	O
methods	O
.	O
</s>
<s>
In	O
numerical	B-Algorithm
integration	I-Algorithm
,	O
methods	O
such	O
as	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
use	O
a	O
deterministic	B-General_Concept
approach	I-General_Concept
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
,	O
on	O
the	O
other	O
hand	O
,	O
employs	O
a	O
non-deterministic	O
approach	O
:	O
each	O
realization	O
provides	O
a	O
different	O
outcome	O
.	O
</s>
<s>
This	O
result	O
does	O
not	O
depend	O
on	O
the	O
number	O
of	O
dimensions	O
of	O
the	O
integral	O
,	O
which	O
is	O
the	O
promised	O
advantage	O
of	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
against	O
most	O
deterministic	O
methods	O
that	O
depend	O
exponentially	O
on	O
the	O
dimension	O
.	O
</s>
<s>
While	O
the	O
naive	O
Monte	O
Carlo	O
works	O
for	O
simple	O
examples	O
,	O
an	O
improvement	O
over	O
deterministic	B-General_Concept
algorithms	I-General_Concept
can	O
only	O
be	O
accomplished	O
with	O
algorithms	O
that	O
use	O
problem-specific	O
sampling	O
distributions	O
.	O
</s>
<s>
In	O
particular	O
,	O
stratified	O
sampling	O
—	O
dividing	O
the	O
region	O
in	O
sub-domains	O
—	O
and	O
importance	B-Algorithm
sampling	I-Algorithm
—	O
sampling	O
from	O
non-uniform	O
distributions	O
—	O
are	O
two	O
examples	O
of	O
such	O
techniques	O
.	O
</s>
<s>
A	O
paradigmatic	O
example	O
of	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
is	O
the	O
estimation	O
of	O
π	O
.	O
</s>
<s>
Made	O
in	O
Python	B-Language
.	O
</s>
<s>
from	O
using	O
the	O
Monte-Carlo	B-Algorithm
method	I-Algorithm
in	O
Mathematica	B-Language
:	O
</s>
<s>
Recursive	O
stratified	O
sampling	O
is	O
a	O
generalization	O
of	O
one-dimensional	O
adaptive	B-Algorithm
quadratures	I-Algorithm
to	O
multi-dimensional	O
integrals	B-Algorithm
.	O
</s>
<s>
The	O
MISER	B-Algorithm
algorithm	I-Algorithm
is	O
based	O
on	O
recursive	O
stratified	O
sampling	O
.	O
</s>
<s>
This	O
technique	O
aims	O
to	O
reduce	O
the	O
overall	O
integration	O
error	O
by	O
concentrating	O
integration	B-Algorithm
points	I-Algorithm
in	O
the	O
regions	O
of	O
highest	O
variance	O
.	O
</s>
<s>
The	O
MISER	B-Algorithm
algorithm	I-Algorithm
proceeds	O
by	O
bisecting	O
the	O
integration	O
region	O
along	O
one	O
coordinate	O
axis	O
to	O
give	O
two	O
sub-regions	O
at	O
each	O
step	O
.	O
</s>
<s>
This	O
recursive	O
allocation	O
of	O
integration	B-Algorithm
points	I-Algorithm
continues	O
down	O
to	O
a	O
user-specified	O
depth	O
where	O
each	O
sub-region	O
is	O
integrated	O
using	O
a	O
plain	O
Monte	O
Carlo	O
estimate	O
.	O
</s>
<s>
Importance	B-Algorithm
sampling	I-Algorithm
provides	O
a	O
very	O
important	O
tool	O
to	O
perform	O
Monte-Carlo	B-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
The	O
main	O
result	O
of	O
importance	B-Algorithm
sampling	I-Algorithm
to	O
this	O
method	O
is	O
that	O
the	O
uniform	O
sampling	O
of	O
is	O
a	O
particular	O
case	O
of	O
a	O
more	O
generic	O
choice	O
,	O
on	O
which	O
the	O
samples	O
are	O
drawn	O
from	O
any	O
distribution	O
.	O
</s>
<s>
The	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
is	O
one	O
of	O
the	O
most	O
used	O
algorithms	O
to	O
generate	O
from	O
,	O
thus	O
providing	O
an	O
efficient	O
way	O
of	O
computing	O
integrals	B-Algorithm
.	O
</s>
<s>
VEGAS	O
incorporates	O
a	O
number	O
of	O
additional	O
features	O
,	O
and	O
combines	O
both	O
stratified	O
sampling	O
and	O
importance	B-Algorithm
sampling	I-Algorithm
.	O
</s>
