<s>
Empirical	B-General_Concept
Bayes	I-General_Concept
methods	I-General_Concept
are	O
procedures	O
for	O
statistical	O
inference	O
in	O
which	O
the	O
prior	O
probability	O
distribution	O
is	O
estimated	O
from	O
the	O
data	O
.	O
</s>
<s>
Despite	O
this	O
difference	O
in	O
perspective	O
,	O
empirical	B-General_Concept
Bayes	I-General_Concept
may	O
be	O
viewed	O
as	O
an	O
approximation	O
to	O
a	O
fully	O
Bayesian	O
treatment	O
of	O
a	O
hierarchical	B-General_Concept
model	I-General_Concept
wherein	O
the	O
parameters	O
at	O
the	O
highest	O
level	O
of	O
the	O
hierarchy	O
are	O
set	O
to	O
their	O
most	O
likely	O
values	O
,	O
instead	O
of	O
being	O
integrated	O
out	O
.	O
</s>
<s>
Empirical	B-General_Concept
Bayes	I-General_Concept
,	O
also	O
known	O
as	O
maximum	O
marginal	O
likelihood	O
,	O
represents	O
a	O
convenient	O
approach	O
for	O
setting	O
hyperparameters	B-General_Concept
,	O
but	O
has	O
been	O
mostly	O
supplanted	O
by	O
fully	O
Bayesian	O
hierarchical	O
analyses	O
since	O
the	O
2000s	O
with	O
the	O
increasing	O
availability	O
of	O
well-performing	O
computation	O
techniques	O
.	O
</s>
<s>
It	O
is	O
still	O
commonly	O
used	O
,	O
however	O
,	O
for	O
variational	O
methods	O
in	O
Deep	O
Learning	O
,	O
such	O
as	O
variational	B-Algorithm
autoencoders	I-Algorithm
,	O
where	O
latent	O
variable	O
spaces	O
are	O
high-dimensional	O
.	O
</s>
<s>
Empirical	B-General_Concept
Bayes	I-General_Concept
methods	I-General_Concept
can	O
be	O
seen	O
as	O
an	O
approximation	O
to	O
a	O
fully	O
Bayesian	O
treatment	O
of	O
a	O
hierarchical	B-General_Concept
Bayes	I-General_Concept
model	I-General_Concept
.	O
</s>
<s>
In	O
,	O
for	O
example	O
,	O
a	O
two-stage	O
hierarchical	B-General_Concept
Bayes	I-General_Concept
model	I-General_Concept
,	O
observed	O
data	O
are	O
assumed	O
to	O
be	O
generated	O
from	O
an	O
unobserved	O
set	O
of	O
parameters	O
according	O
to	O
a	O
probability	O
distribution	O
.	O
</s>
<s>
In	O
turn	O
,	O
the	O
parameters	O
can	O
be	O
considered	O
samples	O
drawn	O
from	O
a	O
population	O
characterised	O
by	O
hyperparameters	B-General_Concept
according	O
to	O
a	O
probability	O
distribution	O
.	O
</s>
<s>
In	O
the	O
hierarchical	B-General_Concept
Bayes	I-General_Concept
model	I-General_Concept
,	O
though	O
not	O
in	O
the	O
empirical	B-General_Concept
Bayes	I-General_Concept
approximation	O
,	O
the	O
hyperparameters	B-General_Concept
are	O
considered	O
to	O
be	O
drawn	O
from	O
an	O
unparameterized	O
distribution	O
.	O
</s>
<s>
Information	O
about	O
a	O
particular	O
quantity	O
of	O
interest	O
therefore	O
comes	O
not	O
only	O
from	O
the	O
properties	O
of	O
those	O
data	O
that	O
directly	O
depend	O
on	O
it	O
,	O
but	O
also	O
from	O
the	O
properties	O
of	O
the	O
population	O
of	O
parameters	O
as	O
a	O
whole	O
,	O
inferred	O
from	O
the	O
data	O
as	O
a	O
whole	O
,	O
summarised	O
by	O
the	O
hyperparameters	B-General_Concept
.	O
</s>
<s>
In	O
general	O
,	O
this	O
integral	O
will	O
not	O
be	O
tractable	O
analytically	O
or	O
symbolically	B-Algorithm
and	O
must	O
be	O
evaluated	O
by	O
numerical	O
methods	O
.	O
</s>
<s>
Example	O
stochastic	O
methods	O
are	O
Markov	B-General_Concept
Chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
and	O
Monte	B-Algorithm
Carlo	I-Algorithm
sampling	O
.	O
</s>
<s>
Deterministic	O
approximations	O
are	O
discussed	O
in	O
quadrature	B-Algorithm
.	O
</s>
<s>
These	O
suggest	O
an	O
iterative	O
scheme	O
,	O
qualitatively	O
similar	O
in	O
structure	O
to	O
a	O
Gibbs	B-Algorithm
sampler	I-Algorithm
,	O
to	O
evolve	O
successively	O
improved	O
approximations	O
to	O
and	O
.	O
</s>
<s>
With	O
this	O
approximation	O
,	O
the	O
above	O
iterative	O
scheme	O
becomes	O
the	O
EM	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
term	O
"	O
Empirical	B-General_Concept
Bayes	I-General_Concept
"	O
can	O
cover	O
a	O
wide	O
variety	O
of	O
methods	O
,	O
but	O
most	O
can	O
be	O
regarded	O
as	O
an	O
early	O
truncation	O
of	O
either	O
the	O
above	O
scheme	O
or	O
something	O
quite	O
like	O
it	O
.	O
</s>
<s>
This	O
shrinkage	O
effect	O
is	O
typical	O
of	O
empirical	B-General_Concept
Bayes	I-General_Concept
analyses	O
.	O
</s>
<s>
If	O
the	O
likelihood	O
and	O
its	O
prior	O
take	O
on	O
simple	O
parametric	O
forms	O
(	O
such	O
as	O
1	O
-	O
or	O
2-dimensional	O
likelihood	O
functions	O
with	O
simple	O
conjugate	O
priors	O
)	O
,	O
then	O
the	O
empirical	B-General_Concept
Bayes	I-General_Concept
problem	O
is	O
only	O
to	O
estimate	O
the	O
marginal	O
and	O
the	O
hyperparameters	B-General_Concept
using	O
the	O
complete	O
set	O
of	O
empirical	O
measurements	O
.	O
</s>
<s>
For	O
example	O
,	O
one	O
common	O
approach	O
,	O
called	O
parametric	O
empirical	B-General_Concept
Bayes	I-General_Concept
point	O
estimation	O
,	O
is	O
to	O
approximate	O
the	O
marginal	O
using	O
the	O
maximum	O
likelihood	O
estimate	O
(	O
MLE	O
)	O
,	O
or	O
a	O
Moments	O
expansion	O
,	O
which	O
allows	O
one	O
to	O
express	O
the	O
hyperparameters	B-General_Concept
in	O
terms	O
of	O
the	O
empirical	O
mean	O
and	O
variance	O
.	O
</s>
<s>
There	O
are	O
several	O
common	O
parametric	O
empirical	B-General_Concept
Bayes	I-General_Concept
models	O
,	O
including	O
the	O
Poisson	O
–	O
gamma	O
model	O
(	O
below	O
)	O
,	O
the	O
Beta-binomial	O
model	O
,	O
the	O
Gaussian	O
–	O
Gaussian	O
model	O
,	O
the	O
Dirichlet-multinomial	O
model	O
,	O
as	O
well	O
specific	O
models	O
for	O
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
(	O
see	O
below	O
)	O
and	O
Bayesian	B-General_Concept
multivariate	I-General_Concept
linear	I-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
More	O
advanced	O
approaches	O
include	O
hierarchical	B-General_Concept
Bayes	I-General_Concept
models	I-General_Concept
and	O
Bayesian	O
mixture	O
models	O
.	O
</s>
<s>
For	O
an	O
example	O
of	O
empirical	B-General_Concept
Bayes	I-General_Concept
estimation	O
using	O
a	O
Gaussian-Gaussian	O
model	O
,	O
see	O
Empirical	B-General_Concept
Bayes	I-General_Concept
estimators	I-General_Concept
.	O
</s>
<s>
To	O
apply	O
empirical	B-General_Concept
Bayes	I-General_Concept
,	O
we	O
will	O
approximate	O
the	O
marginal	O
using	O
the	O
maximum	O
likelihood	O
estimate	O
(	O
MLE	O
)	O
.	O
</s>
<s>
To	O
obtain	O
the	O
values	O
of	O
and	O
,	O
empirical	B-General_Concept
Bayes	I-General_Concept
prescribes	O
estimating	O
mean	O
and	O
variance	O
using	O
the	O
complete	O
set	O
of	O
empirical	O
data	O
.	O
</s>
<s>
This	O
turns	O
out	O
to	O
be	O
a	O
general	O
feature	O
of	O
empirical	B-General_Concept
Bayes	I-General_Concept
;	O
the	O
point	O
estimates	O
for	O
the	O
prior	O
(	O
i.e.	O
</s>
