<s>
Integrated	B-General_Concept
nested	I-General_Concept
Laplace	I-General_Concept
approximations	I-General_Concept
(	O
INLA	O
)	O
is	O
a	O
method	O
for	O
approximate	O
Bayesian	O
inference	O
based	O
on	O
Laplace	O
's	O
method	O
.	O
</s>
<s>
It	O
is	O
designed	O
for	O
a	O
class	O
of	O
models	O
called	O
latent	O
Gaussian	O
models	O
(	O
LGMs	O
)	O
,	O
for	O
which	O
it	O
can	O
be	O
a	O
fast	O
and	O
accurate	O
alternative	O
for	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
methods	I-General_Concept
to	O
compute	O
posterior	O
marginal	O
distributions	O
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
combine	O
INLA	O
with	O
a	O
finite	B-Application
element	I-Application
method	I-Application
solution	O
of	O
a	O
stochastic	O
partial	O
differential	O
equation	O
to	O
study	O
e.g.	O
</s>
<s>
The	O
INLA	O
method	O
is	O
implemented	O
in	O
the	O
R-INLA	O
R	B-Language
package	O
.	O
</s>
<s>
The	O
hyperparameters	B-General_Concept
of	O
the	O
model	O
are	O
denoted	O
by	O
.	O
</s>
<s>
where	O
is	O
a	O
-dependent	O
sparse	O
precision	B-General_Concept
matrix	I-General_Concept
and	O
is	O
its	O
determinant	O
.	O
</s>
<s>
The	O
precision	B-General_Concept
matrix	I-General_Concept
is	O
sparse	O
due	O
to	O
the	O
GMRF	O
assumption	O
.	O
</s>
<s>
The	O
prior	O
distribution	O
for	O
the	O
hyperparameters	B-General_Concept
need	O
not	O
be	O
Gaussian	O
.	O
</s>
<s>
However	O
,	O
the	O
number	O
of	O
hyperparameters	B-General_Concept
,	O
,	O
is	O
assumed	O
to	O
be	O
small	O
(	O
say	O
,	O
less	O
than	O
15	O
)	O
.	O
</s>
<s>
The	O
trick	O
in	O
the	O
Laplace	B-General_Concept
approximation	I-General_Concept
above	O
is	O
the	O
fact	O
that	O
the	O
Gaussian	O
approximation	O
is	O
applied	O
on	O
the	O
full	O
conditional	O
of	O
in	O
the	O
denominator	O
since	O
it	O
is	O
usually	O
close	O
to	O
a	O
Gaussian	O
due	O
to	O
the	O
GMRF	O
property	O
of	O
.	O
</s>
<s>
The	O
second	O
important	O
property	O
of	O
a	O
GMRF	O
,	O
the	O
sparsity	O
of	O
the	O
precision	B-General_Concept
matrix	I-General_Concept
,	O
is	O
required	O
for	O
efficient	O
computation	O
of	O
for	O
each	O
value	O
.	O
</s>
<s>
Obtaining	O
the	O
approximate	O
distribution	O
is	O
more	O
involved	O
,	O
and	O
the	O
INLA	O
method	O
provides	O
three	O
options	O
for	O
this	O
:	O
Gaussian	O
approximation	O
,	O
Laplace	B-General_Concept
approximation	I-General_Concept
,	O
or	O
the	O
simplified	O
Laplace	B-General_Concept
approximation	I-General_Concept
.	O
</s>
