<s>
In	O
computational	B-Algorithm
statistics	I-Algorithm
,	O
the	O
Metropolis-adjusted	B-Algorithm
Langevin	I-Algorithm
algorithm	I-Algorithm
(	O
MALA	O
)	O
or	O
Langevin	O
Monte	O
Carlo	O
(	O
LMC	O
)	O
is	O
a	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
(	O
MCMC	O
)	O
method	O
for	O
obtaining	O
random	B-Algorithm
samples	I-Algorithm
–	O
sequences	O
of	O
random	O
observations	O
–	O
from	O
a	O
probability	O
distribution	O
for	O
which	O
direct	O
sampling	O
is	O
difficult	O
.	O
</s>
<s>
these	O
proposals	O
are	O
accepted	O
or	O
rejected	O
using	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
,	O
which	O
uses	O
evaluations	O
of	O
the	O
target	O
probability	O
density	O
(	O
but	O
not	O
its	O
gradient	O
)	O
.	O
</s>
<s>
Informally	O
,	O
the	O
Langevin	O
dynamics	O
drive	O
the	O
random	O
walk	O
towards	O
regions	O
of	O
high	O
probability	O
in	O
the	O
manner	O
of	O
a	O
gradient	O
flow	O
,	O
while	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
accept/reject	O
mechanism	O
improves	O
the	O
mixing	O
and	O
convergence	O
properties	O
of	O
this	O
random	O
walk	O
.	O
</s>
<s>
the	O
manifold	B-Architecture
variant	O
of	O
Girolami	O
and	O
Calderhead	O
(	O
2011	O
)	O
.	O
</s>
<s>
The	O
method	O
is	O
equivalent	O
to	O
using	O
the	O
Hamiltonian	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
(	O
hybrid	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
)	O
algorithm	O
with	O
only	O
a	O
single	O
discrete	O
time	O
step	O
.	O
</s>
<s>
where	O
each	O
is	O
an	O
independent	O
draw	O
from	O
a	O
multivariate	O
normal	O
distribution	O
on	O
with	O
mean	O
0	O
and	O
covariance	O
matrix	O
equal	O
to	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
is	O
normally	O
distributed	O
with	O
mean	O
and	O
covariance	O
equal	O
to	O
times	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
combined	O
dynamics	O
of	O
the	O
Langevin	O
diffusion	O
and	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
satisfy	O
the	O
detailed	O
balance	O
conditions	O
necessary	O
for	O
the	O
existence	O
of	O
a	O
unique	O
,	O
invariant	O
,	O
stationary	O
distribution	O
.	O
</s>
<s>
Compared	O
to	O
naive	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
,	O
MALA	O
has	O
the	O
advantage	O
that	O
it	O
usually	O
proposes	O
moves	O
into	O
regions	O
of	O
higher	O
probability	O
,	O
which	O
are	O
then	O
more	O
likely	O
to	O
be	O
accepted	O
.	O
</s>
