<s>
In	O
Bayesian	O
statistics	O
,	O
a	O
credible	B-General_Concept
interval	I-General_Concept
is	O
an	O
interval	O
within	O
which	O
an	O
unobserved	O
parameter	O
value	O
falls	O
with	O
a	O
particular	O
probability	O
.	O
</s>
<s>
The	O
generalisation	O
to	O
multivariate	O
problems	O
is	O
the	O
credible	B-General_Concept
region	I-General_Concept
.	O
</s>
<s>
Credible	B-General_Concept
intervals	I-General_Concept
are	O
analogous	O
to	O
confidence	O
intervals	O
and	O
confidence	O
regions	O
in	O
frequentist	B-General_Concept
statistics	I-General_Concept
,	O
although	O
they	O
differ	O
on	O
a	O
philosophical	O
basis	O
:	O
Bayesian	O
intervals	O
treat	O
their	O
bounds	O
as	O
fixed	O
and	O
the	O
estimated	O
parameter	O
as	O
a	O
random	O
variable	O
,	O
whereas	O
frequentist	B-General_Concept
confidence	O
intervals	O
treat	O
their	O
bounds	O
as	O
random	O
variables	O
and	O
the	O
parameter	O
as	O
a	O
fixed	O
value	O
.	O
</s>
<s>
Also	O
,	O
Bayesian	O
credible	B-General_Concept
intervals	I-General_Concept
use	O
(	O
and	O
indeed	O
,	O
require	O
)	O
knowledge	O
of	O
the	O
situation-specific	O
prior	O
distribution	O
,	O
while	O
the	O
frequentist	B-General_Concept
confidence	O
intervals	O
do	O
not	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
an	O
experiment	O
that	O
determines	O
the	O
distribution	O
of	O
possible	O
values	O
of	O
the	O
parameter	O
,	O
if	O
the	O
subjective	O
probability	O
that	O
lies	O
between	O
35	O
and	O
45	O
is	O
0.95	O
,	O
then	O
is	O
a	O
95%	O
credible	B-General_Concept
interval	I-General_Concept
.	O
</s>
<s>
Credible	B-General_Concept
intervals	I-General_Concept
are	O
not	O
unique	O
on	O
a	O
posterior	O
distribution	O
.	O
</s>
<s>
Methods	O
for	O
defining	O
a	O
suitable	O
credible	B-General_Concept
interval	I-General_Concept
include	O
:	O
</s>
<s>
Choosing	O
the	O
narrowest	O
interval	O
,	O
which	O
for	O
a	O
unimodal	O
distribution	O
will	O
involve	O
choosing	O
those	O
values	O
of	O
highest	O
probability	O
density	O
including	O
the	O
mode	O
(	O
the	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
)	O
.	O
</s>
<s>
This	O
is	O
sometimes	O
called	O
the	O
highest	B-General_Concept
posterior	I-General_Concept
density	I-General_Concept
interval	I-General_Concept
(	O
HPDI	B-General_Concept
)	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
frame	O
the	O
choice	O
of	O
a	O
credible	B-General_Concept
interval	I-General_Concept
within	O
decision	O
theory	O
and	O
,	O
in	O
that	O
context	O
,	O
a	O
smallest	O
interval	O
will	O
always	O
be	O
a	O
highest	O
probability	O
density	O
set	O
.	O
</s>
<s>
Credible	B-General_Concept
intervals	I-General_Concept
can	O
also	O
be	O
estimated	O
through	O
the	O
use	O
of	O
simulation	O
techniques	O
such	O
as	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
.	O
</s>
<s>
A	O
frequentist	B-General_Concept
95%	O
confidence	O
interval	O
means	O
that	O
with	O
a	O
large	O
number	O
of	O
repeated	O
samples	O
,	O
95%	O
of	O
such	O
calculated	O
confidence	O
intervals	O
would	O
include	O
the	O
true	O
value	O
of	O
the	O
parameter	O
.	O
</s>
<s>
In	O
frequentist	B-General_Concept
terms	O
,	O
the	O
parameter	O
is	O
fixed	O
(	O
cannot	O
be	O
considered	O
to	O
have	O
a	O
distribution	O
of	O
possible	O
values	O
)	O
and	O
the	O
confidence	O
interval	O
is	O
random	O
(	O
as	O
it	O
depends	O
on	O
the	O
random	O
sample	O
)	O
.	O
</s>
<s>
Bayesian	O
credible	B-General_Concept
intervals	I-General_Concept
can	O
be	O
quite	O
different	O
from	O
frequentist	B-General_Concept
confidence	O
intervals	O
for	O
two	O
reasons	O
:	O
</s>
<s>
credible	B-General_Concept
intervals	I-General_Concept
incorporate	O
problem-specific	O
contextual	O
information	O
from	O
the	O
prior	O
distribution	O
whereas	O
confidence	O
intervals	O
are	O
based	O
only	O
on	O
the	O
data	O
;	O
</s>
<s>
credible	B-General_Concept
intervals	I-General_Concept
and	O
confidence	O
intervals	O
treat	O
nuisance	O
parameters	O
in	O
radically	O
different	O
ways	O
.	O
</s>
<s>
For	O
the	O
case	O
of	O
a	O
single	O
parameter	O
and	O
data	O
that	O
can	O
be	O
summarised	O
in	O
a	O
single	O
sufficient	O
statistic	O
,	O
it	O
can	O
be	O
shown	O
that	O
the	O
credible	B-General_Concept
interval	I-General_Concept
and	O
the	O
confidence	O
interval	O
will	O
coincide	O
if	O
the	O
unknown	O
parameter	O
is	O
a	O
location	O
parameter	O
(	O
i.e.	O
</s>
