In empirical Bayes methods, the<unk>  is an analytic<unk>  where the likelihood function 
L
(
x
|
θ
)
{\displaystyle L(x|\theta )}
 is specifed by a<unk>  distribution
and the conjugate prior 
π
(
θ
|
α
,
β
)
{\displaystyle \pi (\theta |\alpha ,\beta )}
 is a<unk>  distribution
It is convenient to reparameterize the distributions so that the expected mean of the prior is a single parameter: Let
where
so that
The posterior distribution 
ρ
(
θ
|
x
)
{\displaystyle \rho (\theta |x)}
 is also a<unk>  distribution
while the marginal distribution 
m
(
x
|
μ
,
M
)
{\displaystyle m(x|\mu ,M)}
 is given by
Because the marginal is a complex, non-linear function of Gamma and Digamma functions, it is quite difficult to obtain a marginal maximum likelihood estimate (MMLE) for the mean and variance.  Instead, we use the method of iterated expectations to find the expected value of the marginal moments.
Let us write our<unk>  as a two-stage compund sampling<unk>  (for each event i out of ni possible events):
We can find iterated moment estimates for the mean and variance using the moments for the distributions in the two-stage<unk> :
We now seek a point estimate 
θ
~
{\displaystyle {\tilde {\theta }}}
 as a weighted average of the sample estimate 
θ
i
^
{\displaystyle {\hat {\theta _{i}}}}
 and an estimate for 
μ
^
{\displaystyle {\hat {\mu }}}
. The sample estimate 
θ
i
^
{\displaystyle {\hat {\theta _{i}}}}
 is given by
Therefore we need point estimates for  
μ
{\displaystyle \mu }
 and 
M
{\displaystyle M}
.   The estimated mean 
μ
^
{\displaystyle {\hat {\mu }}}
  is given as a weighted average
The hyperparameter 
M
{\displaystyle M}
 is obtained using the moment estimates for the variance of the two-stage<unk> :
We can now solve for 
M
^
{\displaystyle {\hat {M}}}
:
Given our point estimates for the prior, we may now plug in these values to find a point estimate for the posterior
We may write the posterior estimate as a weighted average:
where  
B
i
^
{\displaystyle {\hat {B_{i}}}}
 is called the Shrinkage Factor.
Beta-binomial model
Ancillary statistic
Linear transform model
Non-parametric statistics
Bayesian inference
Binomial test
Linear regression
Likelihood-ratio test