<s>
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
is	O
a	O
type	O
of	O
conditional	O
modeling	O
in	O
which	O
the	O
mean	O
of	O
one	O
variable	O
is	O
described	O
by	O
a	O
linear	O
combination	O
of	O
other	O
variables	O
,	O
with	O
the	O
goal	O
of	O
obtaining	O
the	O
posterior	O
probability	O
of	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
(	O
as	O
well	O
as	O
other	O
parameters	O
describing	O
the	O
distribution	O
of	O
the	O
regressand	O
)	O
and	O
ultimately	O
allowing	O
the	O
out-of-sample	O
prediction	O
of	O
the	O
regressand	O
(	O
often	O
labelled	O
)	O
conditional	O
on	O
observed	O
values	O
of	O
the	O
regressors	O
(	O
usually	O
)	O
.	O
</s>
<s>
Consider	O
a	O
standard	O
linear	B-General_Concept
regression	I-General_Concept
problem	O
,	O
in	O
which	O
for	O
we	O
specify	O
the	O
mean	O
of	O
the	O
conditional	O
distribution	O
of	O
given	O
a	O
predictor	O
vector	O
:	O
</s>
<s>
The	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
solution	O
is	O
used	O
to	O
estimate	O
the	O
coefficient	O
vector	O
using	O
the	O
Moore	O
–	O
Penrose	O
pseudoinverse	O
:	O
</s>
<s>
where	O
is	O
the	O
design	B-Algorithm
matrix	I-Algorithm
,	O
each	O
row	O
of	O
which	O
is	O
a	O
predictor	O
vector	O
;	O
and	O
is	O
the	O
column	O
-vector	O
.	O
</s>
<s>
This	O
is	O
a	O
frequentist	B-General_Concept
approach	O
,	O
and	O
it	O
assumes	O
that	O
there	O
are	O
enough	O
measurements	O
to	O
say	O
something	O
meaningful	O
about	O
.	O
</s>
<s>
where	O
is	O
the	O
number	O
of	O
regression	B-General_Concept
coefficients	I-General_Concept
.	O
</s>
<s>
The	O
model	O
evidence	O
of	O
the	O
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
model	O
presented	O
in	O
this	O
section	O
can	O
be	O
used	O
to	O
compare	O
competing	O
linear	O
models	O
by	O
Bayesian	B-General_Concept
model	I-General_Concept
comparison	I-General_Concept
.	O
</s>
<s>
However	O
,	O
it	O
is	O
possible	O
to	O
approximate	O
the	O
posterior	O
by	O
an	O
approximate	O
Bayesian	O
inference	O
method	O
such	O
as	O
Monte	B-Algorithm
Carlo	I-Algorithm
sampling	I-Algorithm
or	O
variational	O
Bayes	O
.	O
</s>
<s>
A	O
similar	O
analysis	O
can	O
be	O
performed	O
for	O
the	O
general	O
case	O
of	O
the	O
multivariate	O
regression	O
and	O
part	O
of	O
this	O
provides	O
for	O
Bayesian	O
estimation	B-General_Concept
of	I-General_Concept
covariance	I-General_Concept
matrices	I-General_Concept
:	O
see	O
Bayesian	B-General_Concept
multivariate	I-General_Concept
linear	I-General_Concept
regression	I-General_Concept
.	O
</s>
