<s>
Laplace	B-General_Concept
's	I-General_Concept
approximation	I-General_Concept
provides	O
an	O
analytical	O
expression	O
for	O
a	O
posterior	O
probability	O
distribution	O
by	O
fitting	O
a	O
Gaussian	O
distribution	O
with	O
a	O
mean	O
equal	O
to	O
the	O
MAP	B-General_Concept
solution	O
and	O
a	O
variance	O
equal	O
to	O
the	O
observed	O
Fisher	O
information	O
.	O
</s>
<s>
The	O
approximation	O
is	O
justified	O
by	O
the	O
Bernstein	B-General_Concept
–	I-General_Concept
von	I-General_Concept
Mises	I-General_Concept
theorem	I-General_Concept
,	O
which	O
states	O
that	O
under	O
regularity	O
conditions	O
the	O
posterior	O
converges	O
to	O
a	O
Gaussian	O
in	O
large	O
samples	O
.	O
</s>
<s>
In	O
Laplace	B-General_Concept
's	I-General_Concept
approximation	I-General_Concept
we	O
approximate	O
the	O
joint	O
by	O
an	O
un-normalised	O
Gaussian	O
,	O
where	O
we	O
use	O
to	O
denote	O
approximate	O
density	O
,	O
for	O
un-normalised	O
density	O
and	O
is	O
a	O
constant	O
(	O
independent	O
of	O
)	O
.	O
</s>
<s>
where	O
is	O
the	O
location	O
of	O
a	O
mode	O
of	O
the	O
joint	O
target	O
density	O
,	O
also	O
known	O
as	O
the	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
or	O
MAP	B-General_Concept
point	O
and	O
is	O
the	O
positive	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
of	O
second	O
derivatives	O
of	O
the	O
negative	O
log	O
joint	O
target	O
density	O
at	O
the	O
mode	O
.	O
</s>
<s>
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
In	O
the	O
special	O
case	O
of	O
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
with	O
a	O
Gaussian	O
prior	O
,	O
the	O
approximation	O
is	O
exact	O
.	O
</s>
<s>
The	O
main	O
weaknesses	O
of	O
Laplace	B-General_Concept
's	I-General_Concept
approximation	I-General_Concept
are	O
that	O
it	O
is	O
symmetric	O
around	O
the	O
mode	O
and	O
that	O
it	O
is	O
very	O
local	O
:	O
the	O
entire	O
approximation	O
is	O
derived	O
from	O
properties	O
at	O
a	O
single	O
point	O
of	O
the	O
target	O
density	O
.	O
</s>
<s>
Laplace	O
's	O
method	O
is	O
widely	O
used	O
and	O
was	O
pioneered	O
in	O
the	O
context	O
of	O
neural	O
networks	O
by	O
David	O
MacKay	O
,	O
and	O
for	O
Gaussian	B-General_Concept
processes	I-General_Concept
by	O
Williams	O
and	O
Barber	O
.	O
</s>
