<s>
Subset	B-Algorithm
simulation	I-Algorithm
is	O
a	O
method	O
used	O
in	O
reliability	O
engineering	O
to	O
compute	O
small	O
(	O
i.e.	O
,	O
rare	O
event	O
)	O
failure	O
probabilities	O
encountered	O
in	O
engineering	O
systems	O
.	O
</s>
<s>
The	O
generation	O
of	O
conditional	O
samples	O
is	O
not	O
trivial	O
but	O
can	O
be	O
performed	O
efficiently	O
using	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
(	O
MCMC	O
)	O
.	O
</s>
<s>
Subset	B-Algorithm
Simulation	I-Algorithm
takes	O
the	O
relationship	O
between	O
the	O
(	O
input	O
)	O
random	O
variables	O
and	O
the	O
(	O
output	O
)	O
response	O
quantity	O
of	O
interest	O
as	O
a	O
'	O
black	B-Device
box	I-Device
 '	O
.	O
</s>
<s>
This	O
can	O
be	O
attractive	O
for	O
complex	O
systems	O
where	O
it	O
is	O
difficult	O
to	O
use	O
other	O
variance	B-Algorithm
reduction	I-Algorithm
or	O
rare	O
event	O
sampling	O
techniques	O
that	O
require	O
prior	O
information	O
about	O
the	O
system	O
behaviour	O
.	O
</s>
<s>
For	O
problems	O
where	O
it	O
is	O
possible	O
to	O
incorporate	O
prior	O
information	O
into	O
the	O
reliability	O
algorithm	O
,	O
it	O
is	O
often	O
more	O
efficient	O
to	O
use	O
other	O
variance	B-Algorithm
reduction	I-Algorithm
techniques	O
such	O
as	O
importance	B-Algorithm
sampling	I-Algorithm
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
subset	B-Algorithm
simulation	I-Algorithm
is	O
more	O
efficient	O
than	O
traditional	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
,	O
but	O
less	O
efficient	O
than	O
line	B-Algorithm
sampling	I-Algorithm
,	O
when	O
applied	O
to	O
a	O
fracture	O
mechanics	O
test	O
problem	O
.	O
</s>
<s>
Using	O
direct	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
one	O
can	O
generate	O
i.i.d.	O
</s>
<s>
Subset	B-Algorithm
simulation	I-Algorithm
attempts	O
to	O
convert	O
a	O
rare	O
event	O
problem	O
into	O
more	O
frequent	O
ones	O
.	O
</s>
<s>
The	O
'	O
raw	O
idea	O
 '	O
of	O
subset	B-Algorithm
simulation	I-Algorithm
is	O
to	O
estimate	O
P(F )	O
by	O
estimating	O
and	O
the	O
conditional	O
probabilities	O
for	O
,	O
anticipating	O
efficiency	O
gain	O
when	O
these	O
probabilities	O
are	O
not	O
small	O
.	O
</s>
<s>
In	O
the	O
standard	O
algorithm	O
of	O
subset	B-Algorithm
simulation	I-Algorithm
the	O
first	O
issue	O
is	O
resolved	O
by	O
using	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
.	O
</s>
<s>
More	O
generic	O
and	O
flexible	O
version	O
of	O
the	O
simulation	O
algorithms	O
not	O
based	O
on	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
have	O
been	O
recently	O
developed	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
subset	B-Algorithm
simulation	I-Algorithm
in	O
fact	O
produces	O
a	O
set	O
of	O
estimates	O
for	O
b	O
that	O
corresponds	O
to	O
different	O
fixed	O
values	O
of	O
p	O
=P	O
(	O
Y>b	O
)	O
,	O
rather	O
than	O
estimates	O
of	O
probabilities	O
for	O
fixed	O
threshold	O
values	O
.	O
</s>
<s>
These	O
versions	O
of	O
subset	B-Algorithm
simulation	I-Algorithm
can	O
also	O
be	O
used	O
to	O
approximately	O
sample	O
from	O
the	O
distribution	O
of	O
X	O
given	O
the	O
failure	O
of	O
the	O
system	O
(	O
that	O
is	O
,	O
conditional	O
on	O
the	O
event	O
)	O
.	O
</s>
