Y
=
X
T
β
+
±
with
±
i
N
(0
,σ
2
). Thus
Y
i
∼
N
(
X
T
β,σ
2
). A
likelihood
for
y
1
,...y
n
is
lik
(
y
1
,...,y
n

β,σ
2
) =
n
Y
i
=1
φ
(
y
i

β,σ
2
)
The Bayesian program includes
prior
distributions for the parameters
β
. Remember Bayes’
rule:
π
(
β,σ
2

y
1
,...,y
n
) =
lik
(
y
1
,...,y
n

β,σ
2
)
π
(
β,σ
2
)
g
(
y
1
,...y
n
)
(1)
where
g
(
y
1
,...,y
n
) =
R
β,σ
2
lik
(
y
1
,...,y
n

β,σ
2
)
π
(
β,σ
2
)
dβdσ
2
The frequentist setup can be seen as a subset of the Bayesian, where the prior
π
(
β,σ
2
) is
”noninformative” or uniform over the support of the parameters.
A Bayesian setup augments
Y
=
X
T
β
+
±
with
β
∼
N
(
μ
β
,
Σ
β
)
for example. We call
μ
β
,
Σ
β
hyperparameters
in that they are parameters for the parameters
of interest (
β
).
2 Random and Mixed Eﬀects
Let’s look at a real setup using the ANOVA approach. Let
Y
ij
=
β
0
+
β
j
+
±
i
with
j
= 1
,...,K
— the levels of the factor and
i
= 1
...n
for each
j
. The ordinary setup has
±
i
∼
N
(0
,σ
2
Y
)
2