<s>
In	O
Bayesian	O
statistics	O
,	O
a	O
hyperparameter	B-General_Concept
is	O
a	O
parameter	O
of	O
a	O
prior	O
distribution	O
;	O
the	O
term	O
is	O
used	O
to	O
distinguish	O
them	O
from	O
parameters	O
of	O
the	O
model	O
for	O
the	O
underlying	O
system	O
under	O
analysis	O
.	O
</s>
<s>
α	O
and	O
β	O
are	O
parameters	O
of	O
the	O
prior	O
distribution	O
(	O
beta	O
distribution	O
)	O
,	O
hence	O
hyperparameters	B-General_Concept
.	O
</s>
<s>
One	O
may	O
take	O
a	O
single	O
value	O
for	O
a	O
given	O
hyperparameter	B-General_Concept
,	O
or	O
one	O
can	O
iterate	O
and	O
take	O
a	O
probability	O
distribution	O
on	O
the	O
hyperparameter	B-General_Concept
itself	O
,	O
called	O
a	O
hyperprior	O
.	O
</s>
<s>
One	O
often	O
uses	O
a	O
prior	O
which	O
comes	O
from	O
a	O
parametric	O
family	O
of	O
probability	O
distributions	O
–	O
this	O
is	O
done	O
partly	O
for	O
explicitness	O
(	O
so	O
one	O
can	O
write	O
down	O
a	O
distribution	O
,	O
and	O
choose	O
the	O
form	O
by	O
varying	O
the	O
hyperparameter	B-General_Concept
,	O
rather	O
than	O
trying	O
to	O
produce	O
an	O
arbitrary	O
function	O
)	O
,	O
and	O
partly	O
so	O
that	O
one	O
can	O
vary	O
the	O
hyperparameter	B-General_Concept
,	O
particularly	O
in	O
the	O
method	O
of	O
conjugate	O
priors	O
,	O
or	O
for	O
sensitivity	O
analysis	O
.	O
</s>
<s>
When	O
using	O
a	O
conjugate	O
prior	O
,	O
the	O
posterior	O
distribution	O
will	O
be	O
from	O
the	O
same	O
family	O
,	O
but	O
will	O
have	O
different	O
hyperparameters	B-General_Concept
,	O
which	O
reflect	O
the	O
added	O
information	O
from	O
the	O
data	O
:	O
in	O
subjective	O
terms	O
,	O
one	O
's	O
beliefs	O
have	O
been	O
updated	O
.	O
</s>
<s>
For	O
a	O
general	O
prior	O
distribution	O
,	O
this	O
is	O
computationally	O
very	O
involved	O
,	O
and	O
the	O
posterior	O
may	O
have	O
an	O
unusual	O
or	O
hard	O
to	O
describe	O
form	O
,	O
but	O
with	O
a	O
conjugate	O
prior	O
,	O
there	O
is	O
generally	O
a	O
simple	O
formula	O
relating	O
the	O
values	O
of	O
the	O
hyperparameters	B-General_Concept
of	O
the	O
posterior	O
to	O
those	O
of	O
the	O
prior	O
,	O
and	O
thus	O
the	O
computation	O
of	O
the	O
posterior	O
distribution	O
is	O
very	O
easy	O
.	O
</s>
<s>
Hyperparameters	B-General_Concept
address	O
this	O
by	O
allowing	O
one	O
to	O
easily	O
vary	O
them	O
and	O
see	O
how	O
the	O
posterior	O
distribution	O
(	O
and	O
various	O
statistics	O
of	O
it	O
,	O
such	O
as	O
credible	B-General_Concept
intervals	I-General_Concept
)	O
vary	O
:	O
one	O
can	O
see	O
how	O
sensitive	O
one	O
's	O
conclusions	O
are	O
to	O
one	O
's	O
prior	O
assumptions	O
,	O
and	O
the	O
process	O
is	O
called	O
sensitivity	O
analysis	O
.	O
</s>
<s>
Similarly	O
,	O
one	O
may	O
use	O
a	O
prior	O
distribution	O
with	O
a	O
range	O
for	O
a	O
hyperparameter	B-General_Concept
,	O
perhaps	O
reflecting	O
uncertainty	O
in	O
the	O
correct	O
prior	O
to	O
take	O
,	O
and	O
reflect	O
this	O
in	O
a	O
range	O
for	O
final	O
uncertainty	O
.	O
</s>
<s>
Instead	O
of	O
using	O
a	O
single	O
value	O
for	O
a	O
given	O
hyperparameter	B-General_Concept
,	O
one	O
can	O
instead	O
consider	O
a	O
probability	O
distribution	O
of	O
the	O
hyperparameter	B-General_Concept
itself	O
;	O
this	O
is	O
called	O
a	O
"	O
hyperprior.	O
"	O
</s>
