<s>
Bayesian	O
approach	O
to	O
multivariate	O
linear	B-General_Concept
regression	I-General_Concept
,	O
i.e.	O
</s>
<s>
linear	B-General_Concept
regression	I-General_Concept
where	O
the	O
predicted	O
outcome	O
is	O
a	O
vector	O
of	O
correlated	O
random	O
variables	O
rather	O
than	O
a	O
single	O
scalar	O
random	O
variable	O
.	O
</s>
<s>
A	O
more	O
general	O
treatment	O
of	O
this	O
approach	O
can	O
be	O
found	O
in	O
the	O
article	O
MMSE	B-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
Equivalently	O
,	O
it	O
can	O
be	O
viewed	O
as	O
a	O
single	O
regression	O
problem	O
where	O
the	O
outcome	O
is	O
a	O
row	O
vector	O
and	O
the	O
regression	B-General_Concept
coefficient	I-General_Concept
vectors	O
are	O
stacked	O
next	O
to	O
each	O
other	O
,	O
as	O
follows	O
:	O
</s>
<s>
The	O
design	B-Algorithm
matrix	I-Algorithm
X	O
is	O
an	O
matrix	O
with	O
the	O
observations	O
stacked	O
vertically	O
,	O
as	O
in	O
the	O
standard	O
linear	B-General_Concept
regression	I-General_Concept
setup	O
:	O
</s>
<s>
The	O
classical	O
,	O
frequentists	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
solution	O
is	O
to	O
simply	O
estimate	O
the	O
matrix	O
of	O
regression	B-General_Concept
coefficients	I-General_Concept
using	O
the	O
Moore-Penrose	O
pseudoinverse	B-Algorithm
:	O
</s>
<s>
As	O
with	O
the	O
univariate	O
case	O
of	O
linear	B-General_Concept
Bayesian	I-General_Concept
regression	I-General_Concept
,	O
we	O
will	O
find	O
that	O
we	O
can	O
specify	O
a	O
natural	O
conditional	O
conjugate	O
prior	O
(	O
which	O
is	O
scale	O
dependent	O
)	O
.	O
</s>
<s>
Using	O
the	O
same	O
technique	O
as	O
with	O
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
,	O
we	O
decompose	O
the	O
exponential	O
term	O
using	O
a	O
matrix-form	O
of	O
the	O
sum-of-squares	O
technique	O
.	O
</s>
<s>
Here	O
,	O
however	O
,	O
we	O
will	O
also	O
need	O
to	O
use	O
the	O
Matrix	O
Differential	O
Calculus	O
(	O
Kronecker	O
product	O
and	O
vectorization	B-Algorithm
transformations	O
)	O
.	O
</s>
<s>
This	O
is	O
accomplished	O
using	O
the	O
vectorization	B-Algorithm
transformation	O
,	O
which	O
converts	O
the	O
likelihood	O
from	O
a	O
function	O
of	O
the	O
matrices	O
to	O
a	O
function	O
of	O
the	O
vectors	O
.	O
</s>
