<s>
In	O
probability	O
theory	O
and	O
statistics	O
,	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
(	O
DP	O
)	O
is	O
one	O
of	O
the	O
most	O
popular	O
Bayesian	O
nonparametric	O
models	O
.	O
</s>
<s>
A	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
completely	O
defined	O
by	O
its	O
parameters	O
:	O
(	O
the	O
base	O
distribution	O
or	O
base	O
measure	O
)	O
is	O
an	O
arbitrary	O
distribution	O
and	O
(	O
the	O
concentration	O
parameter	O
)	O
is	O
a	O
positive	O
real	O
number	O
(	O
it	O
is	O
often	O
denoted	O
as	O
)	O
.	O
</s>
<s>
the	O
name	O
of	O
Bayesian	B-Application
bootstrap	I-Application
by	O
Rubin	O
;	O
in	O
fact	O
it	O
can	O
be	O
proven	O
that	O
the	O
Bayesian	B-Application
bootstrap	I-Application
is	O
asymptotically	O
equivalent	O
to	O
the	O
frequentist	O
bootstrap	O
introduced	O
by	O
Bradley	O
Efron	O
.	O
</s>
<s>
The	O
limiting	O
Dirichlet	B-General_Concept
process	I-General_Concept
has	O
been	O
criticized	O
on	O
diverse	O
grounds	O
.	O
</s>
<s>
The	O
imprecise	B-General_Concept
Dirichlet	I-General_Concept
process	I-General_Concept
has	O
been	O
proposed	O
to	O
overcome	O
these	O
issues	O
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
imprecise	B-General_Concept
Dirichlet	I-General_Concept
process	I-General_Concept
(	O
IDP	O
)	O
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
From	O
the	O
above	O
expressions	O
of	O
the	O
lower	O
and	O
upper	O
bounds	O
,	O
it	O
can	O
be	O
observed	O
that	O
the	O
range	B-Algorithm
of	O
under	O
the	O
IDP	O
is	O
the	O
same	O
as	O
the	O
original	O
range	B-Algorithm
of	O
.	O
</s>
<s>
where	O
is	O
the	O
empirical	B-General_Concept
distribution	I-General_Concept
function	I-General_Concept
.	O
</s>
<s>
In	O
this	O
respect	O
,	O
the	O
Imprecise	B-General_Concept
Dirichlet	I-General_Concept
Process	I-General_Concept
has	O
been	O
used	O
for	O
nonparametric	O
hypothesis	O
testing	O
,	O
see	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
IDP	O
test	O
suspends	O
the	O
judgment	O
when	O
the	O
option	O
which	O
minimizes	O
the	O
expected	O
loss	O
changes	O
depending	O
on	O
the	O
Dirichlet	B-General_Concept
Process	I-General_Concept
base	O
measure	O
we	O
focus	O
on	O
.	O
</s>
<s>
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
reduces	O
to	O
a	O
Dirichlet	O
distribution	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
Imprecise	B-General_Concept
Dirichlet	I-General_Concept
Process	I-General_Concept
reduces	O
to	O
the	O
Imprecise	O
Dirichlet	O
model	O
proposed	O
by	O
Walley	O
as	O
a	O
model	O
for	O
prior	O
(	O
near	O
)	O
-ignorance	O
for	O
chances	O
.	O
</s>
