<s>
In	O
statistics	O
,	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
(	O
MCMC	O
)	O
methods	O
comprise	O
a	O
class	O
of	O
algorithms	O
for	O
sampling	O
from	O
a	O
probability	O
distribution	O
.	O
</s>
<s>
Various	O
algorithms	O
exist	O
for	O
constructing	O
chains	O
,	O
including	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
MCMC	B-General_Concept
methods	I-General_Concept
are	O
primarily	O
used	O
for	O
calculating	O
numerical	B-General_Concept
approximations	I-General_Concept
of	O
multi-dimensional	O
integrals	O
,	O
for	O
example	O
in	O
Bayesian	O
statistics	O
,	O
computational	B-Algorithm
physics	I-Algorithm
,	O
computational	O
biology	O
and	O
computational	O
linguistics	O
.	O
</s>
<s>
In	O
Bayesian	O
statistics	O
,	O
the	O
recent	O
development	O
of	O
MCMC	B-General_Concept
methods	I-General_Concept
has	O
made	O
it	O
possible	O
to	O
compute	O
large	O
hierarchical	O
models	O
that	O
require	O
integrations	O
over	O
hundreds	O
to	O
thousands	O
of	O
unknown	O
parameters	O
.	O
</s>
<s>
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
methods	I-General_Concept
create	O
samples	O
from	O
a	O
continuous	O
random	O
variable	O
,	O
with	O
probability	O
density	O
proportional	O
to	O
a	O
known	O
function	O
.	O
</s>
<s>
Random	B-General_Concept
walk	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
methods	O
are	O
a	O
kind	O
of	O
random	O
simulation	B-Application
or	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
However	O
,	O
whereas	O
the	O
random	B-Algorithm
samples	I-Algorithm
of	O
the	O
integrand	O
used	O
in	O
a	O
conventional	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
are	O
statistically	O
independent	O
,	O
those	O
used	O
in	O
MCMC	O
are	O
autocorrelated	O
.	O
</s>
<s>
While	O
MCMC	B-General_Concept
methods	I-General_Concept
were	O
created	O
to	O
address	O
multi-dimensional	O
problems	O
better	O
than	O
generic	O
Monte	O
Carlo	O
algorithms	O
,	O
when	O
the	O
number	O
of	O
dimensions	O
rises	O
they	O
too	O
tend	O
to	O
suffer	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
:	O
regions	O
of	O
higher	O
probability	O
tend	O
to	O
stretch	O
and	O
get	O
lost	O
in	O
an	O
increasing	O
volume	O
of	O
space	O
that	O
contributes	O
little	O
to	O
the	O
integral	O
.	O
</s>
<s>
More	O
sophisticated	O
methods	O
such	O
as	O
Hamiltonian	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
and	O
the	O
Wang	B-Algorithm
and	I-Algorithm
Landau	I-Algorithm
algorithm	I-Algorithm
use	O
various	O
ways	O
of	O
reducing	O
this	O
autocorrelation	O
,	O
while	O
managing	O
to	O
keep	O
the	O
process	O
in	O
the	O
regions	O
that	O
give	O
a	O
higher	O
contribution	O
to	O
the	O
integral	O
.	O
</s>
<s>
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
:	O
This	O
method	O
generates	O
a	O
Markov	O
chain	O
using	O
a	O
proposal	O
density	O
for	O
new	O
steps	O
and	O
a	O
method	O
for	O
rejecting	O
some	O
of	O
the	O
proposed	O
moves	O
.	O
</s>
<s>
It	O
is	O
actually	O
a	O
general	O
framework	O
which	O
includes	O
as	O
special	O
cases	O
the	O
very	O
first	O
and	O
simpler	O
MCMC	O
(	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
)	O
and	O
many	O
more	O
recent	O
alternatives	O
listed	O
below	O
.	O
</s>
<s>
Gibbs	B-Algorithm
sampling	I-Algorithm
:	O
When	O
target	O
distribution	O
is	O
multi-dimensional	O
,	O
Gibbs	B-Algorithm
sampling	I-Algorithm
algorithm	O
updates	O
each	O
coordinate	O
from	O
its	O
full	O
conditional	O
distribution	O
given	O
other	O
coordinates	O
.	O
</s>
<s>
Gibbs	B-Algorithm
sampling	I-Algorithm
can	O
be	O
viewed	O
as	O
a	O
special	O
case	O
of	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
with	O
acceptance	O
rate	O
uniformly	O
equal	O
to	O
1	O
.	O
</s>
<s>
Gibbs	B-Algorithm
sampling	I-Algorithm
is	O
popular	O
partly	O
because	O
it	O
does	O
not	O
require	O
any	O
'	O
tuning	O
 '	O
.	O
</s>
<s>
Algorithm	O
structure	O
of	O
the	O
Gibbs	B-Algorithm
sampling	I-Algorithm
highly	O
resembles	O
that	O
of	O
the	O
coordinate	O
ascent	O
variational	O
inference	O
in	O
that	O
both	O
algorithms	O
utilize	O
the	O
full-conditional	O
distributions	O
in	O
the	O
updating	O
procedure	O
.	O
</s>
<s>
Metropolis-adjusted	B-Algorithm
Langevin	I-Algorithm
algorithm	I-Algorithm
and	O
other	O
methods	O
that	O
rely	O
on	O
the	O
gradient	O
(	O
and	O
possibly	O
second	O
derivative	O
)	O
of	O
the	O
log	O
target	O
density	O
to	O
propose	O
steps	O
that	O
are	O
more	O
likely	O
to	O
be	O
in	O
the	O
direction	O
of	O
higher	O
probability	O
density	O
.	O
</s>
<s>
Pseudo-marginal	B-Algorithm
Metropolis	I-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
:	O
This	O
method	O
replaces	O
the	O
evaluation	O
of	O
the	O
density	O
of	O
the	O
target	O
distribution	O
with	O
an	O
unbiased	O
estimate	O
and	O
is	O
useful	O
when	O
the	O
target	O
density	O
is	O
not	O
available	O
analytically	O
,	O
e.g.	O
</s>
<s>
Slice	B-Algorithm
sampling	I-Algorithm
:	O
This	O
method	O
depends	O
on	O
the	O
principle	O
that	O
one	O
can	O
sample	O
from	O
a	O
distribution	O
by	O
sampling	O
uniformly	O
from	O
the	O
region	O
under	O
the	O
plot	O
of	O
its	O
density	O
function	O
.	O
</s>
<s>
Multiple-try	B-Algorithm
Metropolis	I-Algorithm
:	O
This	O
method	O
is	O
a	O
variation	O
of	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
that	O
allows	O
multiple	O
trials	O
at	O
each	O
point	O
.	O
</s>
<s>
By	O
making	O
it	O
possible	O
to	O
take	O
larger	O
steps	O
at	O
each	O
iteration	O
,	O
it	O
helps	O
address	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
</s>
<s>
Reversible-jump	B-Algorithm
:	O
This	O
method	O
is	O
a	O
variant	O
of	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
that	O
allows	O
proposals	O
that	O
change	O
the	O
dimensionality	O
of	O
the	O
space	O
.	O
</s>
<s>
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
methods	I-General_Concept
that	O
change	O
dimensionality	O
have	O
long	O
been	O
used	O
in	O
statistical	O
physics	O
applications	O
,	O
where	O
for	O
some	O
problems	O
a	O
distribution	O
that	O
is	O
a	O
grand	O
canonical	O
ensemble	O
is	O
used	O
(	O
e.g.	O
,	O
when	O
the	O
number	O
of	O
molecules	O
in	O
a	O
box	O
is	O
variable	O
)	O
.	O
</s>
<s>
But	O
the	O
reversible-jump	B-Algorithm
variant	O
is	O
useful	O
when	O
doing	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
or	O
Gibbs	B-Algorithm
sampling	I-Algorithm
over	O
nonparametric	B-General_Concept
Bayesian	O
models	O
such	O
as	O
those	O
involving	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
or	O
Chinese	B-General_Concept
restaurant	I-General_Concept
process	I-General_Concept
,	O
where	O
the	O
number	O
of	O
mixing	O
components/clusters/etc	O
.	O
</s>
<s>
Hamiltonian	O
(	O
or	O
hybrid	O
)	O
Monte	O
Carlo	O
(	O
HMC	O
)	O
:	O
Tries	O
to	O
avoid	O
random	O
walk	O
behaviour	O
by	O
introducing	O
an	O
auxiliary	O
momentum	B-Algorithm
vector	O
and	O
implementing	O
Hamiltonian	O
dynamics	O
,	O
so	O
the	O
potential	O
energy	O
function	O
is	O
the	O
target	O
density	O
.	O
</s>
<s>
The	O
momentum	B-Algorithm
samples	O
are	O
discarded	O
after	O
sampling	O
.	O
</s>
<s>
The	O
end	O
result	O
of	O
hybrid	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
is	O
that	O
proposals	O
move	O
across	O
the	O
sample	O
space	O
in	O
larger	O
steps	O
;	O
they	O
are	O
therefore	O
less	O
correlated	O
and	O
converge	O
to	O
the	O
target	O
distribution	O
more	O
rapidly	O
.	O
</s>
<s>
Interacting	O
MCMC	O
methodologies	O
are	O
a	O
class	O
of	O
mean-field	O
particle	O
methods	O
for	O
obtaining	O
random	B-Algorithm
samples	I-Algorithm
from	O
a	O
sequence	O
of	O
probability	O
distributions	O
with	O
an	O
increasing	O
level	O
of	O
sampling	O
complexity	O
.	O
</s>
<s>
These	O
probabilistic	O
models	O
include	O
path	O
space	O
state	O
models	O
with	O
increasing	O
time	O
horizon	O
,	O
posterior	O
distributions	O
w.r.t.	O
</s>
<s>
In	O
principle	O
,	O
any	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
sampler	O
can	O
be	O
turned	O
into	O
an	O
interacting	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
sampler	O
.	O
</s>
<s>
These	O
interacting	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
samplers	O
can	O
be	O
interpreted	O
as	O
a	O
way	O
to	O
run	O
in	O
parallel	O
a	O
sequence	O
of	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
samplers	O
.	O
</s>
<s>
For	O
instance	O
,	O
interacting	O
simulated	B-Algorithm
annealing	I-Algorithm
algorithms	I-Algorithm
are	O
based	O
on	O
independent	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
moves	O
interacting	O
sequentially	O
with	O
a	O
selection-resampling	O
type	O
mechanism	O
.	O
</s>
<s>
In	O
contrast	O
to	O
traditional	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
methods	I-General_Concept
,	O
the	O
precision	O
parameter	O
of	O
this	O
class	O
of	O
interacting	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
samplers	O
is	O
only	O
related	O
to	O
the	O
number	O
of	O
interacting	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
samplers	O
.	O
</s>
<s>
These	O
advanced	O
particle	O
methodologies	O
belong	O
to	O
the	O
class	O
of	O
Feynman	O
–	O
Kac	O
particle	O
models	O
,	O
also	O
called	O
Sequential	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
or	O
particle	B-Algorithm
filter	I-Algorithm
methods	O
in	O
Bayesian	O
inference	O
and	O
signal	O
processing	O
communities	O
.	O
</s>
<s>
Interacting	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
methods	I-General_Concept
can	O
also	O
be	O
interpreted	O
as	O
a	O
mutation-selection	O
genetic	B-Algorithm
particle	I-Algorithm
algorithm	I-Algorithm
with	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
mutations	O
.	O
</s>
<s>
The	O
advantage	O
of	O
low-discrepancy	O
sequences	O
in	O
lieu	O
of	O
random	O
numbers	O
for	O
simple	O
independent	O
Monte	B-Algorithm
Carlo	I-Algorithm
sampling	I-Algorithm
is	O
well	O
known	O
.	O
</s>
<s>
This	O
procedure	O
,	O
known	O
as	O
Quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
(	O
QMC	O
)	O
,	O
yields	O
an	O
integration	O
error	O
that	O
decays	O
at	O
a	O
superior	O
rate	O
to	O
that	O
obtained	O
by	O
IID	O
sampling	O
,	O
by	O
the	O
Koksma	O
–	O
Hlawka	O
inequality	O
.	O
</s>
<s>
The	O
Array	O
–	O
RQMC	O
method	O
combines	O
randomized	O
quasi	O
–	O
Monte	O
Carlo	O
and	O
Markov	O
chain	O
simulation	B-Application
by	O
simulating	O
chains	O
simultaneously	O
in	O
a	O
way	O
that	O
the	O
empirical	O
distribution	O
of	O
the	O
states	O
at	O
any	O
given	O
step	O
is	O
a	O
better	O
approximation	O
of	O
the	O
true	O
distribution	O
of	O
the	O
chain	O
than	O
with	O
ordinary	O
MCMC	O
.	O
</s>
<s>
Typically	O
,	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
sampling	O
can	O
only	O
approximate	O
the	O
target	O
distribution	O
,	O
as	O
there	O
is	O
always	O
some	O
residual	O
effect	O
of	O
the	O
starting	O
position	O
.	O
</s>
<s>
More	O
sophisticated	O
Markov	O
chain	O
Monte	O
Carlo-based	O
algorithms	O
such	O
as	O
coupling	B-Algorithm
from	I-Algorithm
the	I-Algorithm
past	I-Algorithm
can	O
produce	O
exact	O
samples	O
,	O
at	O
the	O
cost	O
of	O
additional	O
computation	O
and	O
an	O
unbounded	O
(	O
though	O
finite	O
in	O
expectation	O
)	O
running	O
time	O
.	O
</s>
<s>
Many	O
random	B-General_Concept
walk	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
methods	O
move	O
around	O
the	O
equilibrium	O
distribution	O
in	O
relatively	O
small	O
steps	O
,	O
with	O
no	O
tendency	O
for	O
the	O
steps	O
to	O
proceed	O
in	O
the	O
same	O
direction	O
.	O
</s>
<s>
See	O
for	O
a	O
discussion	O
of	O
the	O
theory	O
related	O
to	O
convergence	O
and	O
stationarity	O
of	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
parallel	O
Monte	B-Algorithm
Carlo	I-Algorithm
software	I-Algorithm
available	O
in	O
multiple	O
programming	O
languages	O
including	O
C	B-Language
,	O
C++	B-Language
,	O
Fortran	B-Application
,	O
MATLAB	B-Language
,	O
and	O
Python	B-Language
.	O
</s>
<s>
software	O
for	O
creation	O
of	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
available	O
in	O
Python	B-Language
.	O
</s>
<s>
Packages	O
that	O
use	O
dialects	O
of	O
the	O
BUGS	B-Algorithm
model	O
language	O
:	O
</s>
<s>
Python	B-Language
(	O
programming	O
language	O
)	O
with	O
the	O
packages	O
,	O
,	O
PyMC3	B-Application
,	O
and	O
vandal	O
.	O
</s>
<s>
R	B-Language
(	O
programming	O
language	O
)	O
with	O
the	O
packages	O
adaptMCMC	O
,	O
atmcmc	O
,	O
BRugs	O
,	O
mcmc	O
,	O
MCMCpack	O
,	O
ramcmc	O
,	O
rjags	O
,	O
rstan	O
,	O
etc	O
.	O
</s>
