<s>
Importance	B-Algorithm
sampling	I-Algorithm
is	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
for	O
evaluating	O
properties	O
of	O
a	O
particular	O
distribution	O
,	O
while	O
only	O
having	O
samples	O
generated	O
from	O
a	O
different	O
distribution	O
than	O
the	O
distribution	O
of	O
interest	O
.	O
</s>
<s>
Importance	B-Algorithm
sampling	I-Algorithm
is	O
also	O
related	O
to	O
umbrella	B-Algorithm
sampling	I-Algorithm
in	O
computational	B-Algorithm
physics	I-Algorithm
.	O
</s>
<s>
The	O
basic	O
idea	O
of	O
importance	B-Algorithm
sampling	I-Algorithm
is	O
to	O
sample	O
the	O
states	O
from	O
a	O
different	O
distribution	O
to	O
lower	O
the	O
variance	O
of	O
the	O
estimation	O
of	O
E[X;P],	O
or	O
when	O
sampling	O
from	O
P	O
is	O
difficult	O
.	O
</s>
<s>
However	O
this	O
theoretical	O
best	O
case	O
L*	O
gives	O
us	O
an	O
insight	O
into	O
what	O
importance	B-Algorithm
sampling	I-Algorithm
does	O
:	O
</s>
<s>
therefore	O
,	O
a	O
good	O
probability	O
change	O
P(L )	O
in	O
importance	B-Algorithm
sampling	I-Algorithm
will	O
redistribute	O
the	O
law	O
of	O
X	O
so	O
that	O
its	O
samples	O
 '	O
frequencies	O
are	O
sorted	O
directly	O
according	O
to	O
their	O
weights	O
in	O
E[X;P]	O
.	O
</s>
<s>
Importance	B-Algorithm
sampling	I-Algorithm
is	O
often	O
used	O
as	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
integrator	I-Algorithm
.	O
</s>
<s>
Importance	B-Algorithm
sampling	I-Algorithm
is	O
a	O
variance	B-Algorithm
reduction	I-Algorithm
technique	O
that	O
can	O
be	O
used	O
in	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
idea	O
behind	O
importance	B-Algorithm
sampling	I-Algorithm
is	O
that	O
certain	O
values	O
of	O
the	O
input	O
random	O
variables	O
in	O
a	O
simulation	O
have	O
more	O
impact	O
on	O
the	O
parameter	O
being	O
estimated	O
than	O
others	O
.	O
</s>
<s>
Hence	O
,	O
the	O
basic	O
methodology	O
in	O
importance	B-Algorithm
sampling	I-Algorithm
is	O
to	O
choose	O
a	O
distribution	O
which	O
"	O
encourages	O
"	O
the	O
important	O
values	O
.	O
</s>
<s>
However	O
,	O
the	O
simulation	O
outputs	O
are	O
weighted	O
to	O
correct	O
for	O
the	O
use	O
of	O
the	O
biased	O
distribution	O
,	O
and	O
this	O
ensures	O
that	O
the	O
new	O
importance	B-Algorithm
sampling	I-Algorithm
estimator	O
is	O
unbiased	O
.	O
</s>
<s>
The	O
weight	O
is	O
given	O
by	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
,	O
that	O
is	O
,	O
the	O
Radon	O
–	O
Nikodym	O
derivative	B-Algorithm
of	O
the	O
true	O
underlying	O
distribution	O
with	O
respect	O
to	O
the	O
biased	O
simulation	O
distribution	O
.	O
</s>
<s>
The	O
fundamental	O
issue	O
in	O
implementing	O
importance	B-Algorithm
sampling	I-Algorithm
simulation	O
is	O
the	O
choice	O
of	O
the	O
biased	O
distribution	O
which	O
encourages	O
the	O
important	O
regions	O
of	O
the	O
input	O
variables	O
.	O
</s>
<s>
Choosing	O
or	O
designing	O
a	O
good	O
biased	O
distribution	O
is	O
the	O
"	O
art	O
"	O
of	O
importance	B-Algorithm
sampling	I-Algorithm
.	O
</s>
<s>
The	O
rewards	O
for	O
a	O
good	O
distribution	O
can	O
be	O
huge	O
run-time	O
savings	O
;	O
the	O
penalty	O
for	O
a	O
bad	O
distribution	O
can	O
be	O
longer	O
run	O
times	O
than	O
for	O
a	O
general	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
without	O
importance	B-Algorithm
sampling	I-Algorithm
.	O
</s>
<s>
Consider	O
to	O
be	O
the	O
sample	O
and	O
to	O
be	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
,	O
where	O
is	O
the	O
probability	O
density	O
(	O
mass	O
)	O
function	O
of	O
the	O
desired	O
distribution	O
and	O
is	O
the	O
probability	O
density	O
(	O
mass	O
)	O
function	O
of	O
the	O
biased/proposal/sample	O
distribution	O
.	O
</s>
<s>
Consider	O
estimating	O
by	O
simulation	O
the	O
probability	O
of	O
an	O
event	O
,	O
where	O
is	O
a	O
random	O
variable	O
with	O
distribution	O
and	O
probability	O
density	O
function	O
,	O
where	O
prime	O
denotes	O
derivative	B-Algorithm
.	O
</s>
<s>
Importance	B-Algorithm
sampling	I-Algorithm
is	O
concerned	O
with	O
the	O
determination	O
and	O
use	O
of	O
an	O
alternate	O
density	O
function	O
(	O
for	O
)	O
,	O
usually	O
referred	O
to	O
as	O
a	O
biasing	O
density	O
,	O
for	O
the	O
simulation	O
experiment	O
.	O
</s>
<s>
is	O
a	O
likelihood	B-General_Concept
ratio	I-General_Concept
and	O
is	O
referred	O
to	O
as	O
the	O
weighting	O
function	O
.	O
</s>
<s>
This	O
is	O
the	O
importance	B-Algorithm
sampling	I-Algorithm
estimator	O
of	O
and	O
is	O
unbiased	O
.	O
</s>
<s>
Now	O
,	O
the	O
importance	B-Algorithm
sampling	I-Algorithm
problem	O
then	O
focuses	O
on	O
finding	O
a	O
biasing	O
density	O
such	O
that	O
the	O
variance	O
of	O
the	O
importance	B-Algorithm
sampling	I-Algorithm
estimator	O
is	O
less	O
than	O
the	O
variance	O
of	O
the	O
general	O
Monte	O
Carlo	O
estimate	O
.	O
</s>
<s>
Although	O
there	O
are	O
many	O
kinds	O
of	O
biasing	O
methods	O
,	O
the	O
following	O
two	O
methods	O
are	O
most	O
widely	O
used	O
in	O
the	O
applications	O
of	O
importance	B-Algorithm
sampling	I-Algorithm
.	O
</s>
<s>
In	O
importance	B-Algorithm
sampling	I-Algorithm
by	O
scaling	O
,	O
the	O
simulation	O
density	O
is	O
chosen	O
as	O
the	O
density	O
function	O
of	O
the	O
scaled	O
random	O
variable	O
,	O
where	O
usually	O
for	O
tail	O
probability	O
estimation	O
.	O
</s>
<s>
The	O
consequence	O
of	O
this	O
is	O
a	O
decreasing	O
importance	B-Algorithm
sampling	I-Algorithm
gain	O
for	O
increasing	O
,	O
and	O
is	O
called	O
the	O
dimensionality	O
effect	O
.	O
</s>
<s>
A	O
modern	O
version	O
of	O
importance	B-Algorithm
sampling	I-Algorithm
by	O
scaling	O
is	O
e.g.	O
</s>
<s>
where	O
is	O
the	O
amount	O
of	O
shift	O
and	O
is	O
to	O
be	O
chosen	O
to	O
minimize	O
the	O
variance	O
of	O
the	O
importance	B-Algorithm
sampling	I-Algorithm
estimator	O
.	O
</s>
<s>
The	O
fundamental	O
problem	O
with	O
importance	B-Algorithm
sampling	I-Algorithm
is	O
that	O
designing	O
good	O
biased	O
distributions	O
becomes	O
more	O
complicated	O
as	O
the	O
system	O
complexity	O
increases	O
.	O
</s>
<s>
In	O
principle	O
,	O
the	O
importance	B-Algorithm
sampling	I-Algorithm
ideas	O
remain	O
the	O
same	O
in	O
these	O
situations	O
,	O
but	O
the	O
design	O
becomes	O
much	O
harder	O
.	O
</s>
<s>
Then	O
importance	B-Algorithm
sampling	I-Algorithm
strategies	O
are	O
used	O
to	O
target	O
each	O
of	O
the	O
simpler	O
subproblems	O
.	O
</s>
<s>
In	O
order	O
to	O
identify	O
successful	O
importance	B-Algorithm
sampling	I-Algorithm
techniques	O
,	O
it	O
is	O
useful	O
to	O
be	O
able	O
to	O
quantify	O
the	O
run-time	O
savings	O
due	O
to	O
the	O
use	O
of	O
the	O
importance	B-Algorithm
sampling	I-Algorithm
approach	O
.	O
</s>
<s>
The	O
performance	O
measure	O
commonly	O
used	O
is	O
,	O
and	O
this	O
can	O
be	O
interpreted	O
as	O
the	O
speed-up	O
factor	O
by	O
which	O
the	O
importance	B-Algorithm
sampling	I-Algorithm
estimator	O
achieves	O
the	O
same	O
precision	O
as	O
the	O
MC	O
estimator	O
.	O
</s>
<s>
Other	O
useful	O
concepts	O
in	O
quantifying	O
an	O
importance	B-Algorithm
sampling	I-Algorithm
estimator	O
are	O
the	O
variance	O
bounds	O
and	O
the	O
notion	O
of	O
asymptotic	O
efficiency	O
.	O
</s>
<s>
An	O
associated	O
issue	O
is	O
the	O
fact	O
that	O
the	O
ratio	O
overestimates	O
the	O
run-time	O
savings	O
due	O
to	O
importance	B-Algorithm
sampling	I-Algorithm
since	O
it	O
does	O
not	O
include	O
the	O
extra	O
computing	O
time	O
required	O
to	O
compute	O
the	O
weight	O
function	O
.	O
</s>
<s>
Perhaps	O
a	O
more	O
serious	O
overhead	O
to	O
importance	B-Algorithm
sampling	I-Algorithm
is	O
the	O
time	O
taken	O
to	O
devise	O
and	O
program	O
the	O
technique	O
and	O
analytically	O
derive	O
the	O
desired	O
weight	O
function	O
.	O
</s>
<s>
In	O
an	O
adaptive	O
setting	O
,	O
the	O
proposal	O
distributions	O
,	O
,	O
and	O
are	O
updated	O
each	O
iteration	O
of	O
the	O
adaptive	O
importance	B-Algorithm
sampling	I-Algorithm
algorithm	O
.	O
</s>
