<s>
In	O
other	O
words	O
,	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
a	O
probability	O
distribution	O
whose	O
range	O
is	O
itself	O
a	O
set	O
of	O
probability	O
distributions	O
.	O
</s>
<s>
The	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
specified	O
by	O
a	O
base	O
distribution	O
and	O
a	O
positive	O
real	O
number	O
called	O
the	O
concentration	O
parameter	O
(	O
also	O
known	O
as	O
scaling	O
parameter	O
)	O
.	O
</s>
<s>
The	O
base	O
distribution	O
is	O
the	O
expected	O
value	O
of	O
the	O
process	O
,	O
i.e.	O
,	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
draws	O
distributions	O
"	O
around	O
"	O
the	O
base	O
distribution	O
the	O
way	O
a	O
normal	O
distribution	O
draws	O
real	O
numbers	O
around	O
its	O
mean	O
.	O
</s>
<s>
However	O
,	O
even	O
if	O
the	O
base	O
distribution	O
is	O
continuous	O
,	O
the	O
distributions	O
drawn	O
from	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
are	O
almost	O
surely	O
discrete	O
.	O
</s>
<s>
The	O
Dirichlet	B-General_Concept
process	I-General_Concept
can	O
also	O
be	O
seen	O
as	O
the	O
infinite-dimensional	O
generalization	O
of	O
the	O
Dirichlet	O
distribution	O
.	O
</s>
<s>
In	O
the	O
same	O
way	O
as	O
the	O
Dirichlet	O
distribution	O
is	O
the	O
conjugate	O
prior	O
for	O
the	O
categorical	O
distribution	O
,	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
the	O
conjugate	O
prior	O
for	O
infinite	O
,	O
nonparametric	B-General_Concept
discrete	O
distributions	O
.	O
</s>
<s>
The	O
Dirichlet	B-General_Concept
process	I-General_Concept
was	O
formally	O
introduced	O
by	O
Thomas	O
Ferguson	O
in	O
1973	O
.	O
</s>
<s>
It	O
has	O
since	O
been	O
applied	O
in	O
data	B-Application
mining	I-Application
and	O
machine	O
learning	O
,	O
among	O
others	O
for	O
natural	B-Language
language	I-Language
processing	I-Language
,	O
computer	B-Application
vision	I-Application
and	O
bioinformatics	O
.	O
</s>
<s>
This	O
distribution	O
(	O
over	O
distributions	O
)	O
is	O
called	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
(	O
)	O
.	O
</s>
<s>
This	O
is	O
a	O
common	O
phenomenon	O
in	O
the	O
context	O
of	O
Bayesian	O
non-parametric	B-General_Concept
statistics	I-General_Concept
where	O
a	O
typical	O
task	O
is	O
to	O
learn	O
distributions	O
on	O
function	O
spaces	O
,	O
which	O
involve	O
effectively	O
infinitely	O
many	O
parameters	O
.	O
</s>
<s>
Given	O
a	O
measurable	O
set	O
S	O
,	O
a	O
base	O
probability	O
distribution	O
H	O
and	O
a	O
positive	O
real	O
number	O
,	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
a	O
stochastic	O
process	O
whose	O
sample	O
path	O
(	O
or	O
realization	O
,	O
i.e.	O
</s>
<s>
There	O
are	O
several	O
equivalent	O
views	O
of	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
Besides	O
the	O
formal	O
definition	O
above	O
,	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
can	O
be	O
defined	O
implicitly	O
through	O
de	O
Finetti	O
's	O
theorem	O
as	O
described	O
in	O
the	O
first	O
section	O
;	O
this	O
is	O
often	O
called	O
the	O
Chinese	B-General_Concept
restaurant	I-General_Concept
process	I-General_Concept
.	O
</s>
<s>
A	O
third	O
alternative	O
is	O
the	O
stick-breaking	O
process	O
,	O
which	O
defines	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
constructively	O
by	O
writing	O
a	O
distribution	O
sampled	O
from	O
the	O
process	O
as	O
,	O
where	O
are	O
samples	O
from	O
the	O
base	O
distribution	O
,	O
is	O
an	O
indicator	O
function	O
centered	O
on	O
(	O
zero	O
everywhere	O
except	O
for	O
)	O
and	O
the	O
are	O
defined	O
by	O
a	O
recursive	O
scheme	O
that	O
repeatedly	O
samples	O
from	O
the	O
beta	O
distribution	O
.	O
</s>
<s>
A	O
widely	O
employed	O
metaphor	O
for	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
based	O
on	O
the	O
so-called	O
Chinese	B-General_Concept
restaurant	I-General_Concept
process	I-General_Concept
.	O
</s>
<s>
This	O
probability	O
distribution	O
over	O
the	O
tables	O
is	O
a	O
random	O
sample	O
of	O
the	O
probabilities	O
of	O
observations	O
drawn	O
from	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
with	O
scaling	O
parameter	O
.	O
</s>
<s>
If	O
one	O
associates	O
draws	O
from	O
the	O
base	O
measure	O
with	O
every	O
table	O
,	O
the	O
resulting	O
distribution	O
over	O
the	O
sample	O
space	O
is	O
a	O
random	O
sample	O
of	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
The	O
Chinese	B-General_Concept
restaurant	I-General_Concept
process	I-General_Concept
is	O
related	O
to	O
the	O
Pólya	O
urn	O
sampling	O
scheme	O
which	O
yields	O
samples	O
from	O
finite	O
Dirichlet	O
distributions	O
.	O
</s>
<s>
The	O
Dirichlet	B-General_Concept
process	I-General_Concept
exhibits	O
a	O
self-reinforcing	O
property	O
:	O
The	O
more	O
often	O
a	O
given	O
value	O
has	O
been	O
sampled	O
in	O
the	O
past	O
,	O
the	O
more	O
likely	O
it	O
is	O
to	O
be	O
sampled	O
again	O
.	O
</s>
<s>
A	O
third	O
approach	O
to	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
the	O
so-called	O
stick-breaking	O
process	O
view	O
.	O
</s>
<s>
Remember	O
that	O
draws	O
from	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
are	O
distributions	O
over	O
a	O
set	O
.	O
</s>
<s>
The	O
locations	O
are	O
independent	O
and	O
identically	O
distributed	O
according	O
to	O
,	O
the	O
base	O
distribution	O
of	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
Yet	O
another	O
way	O
to	O
visualize	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
and	O
Chinese	B-General_Concept
restaurant	I-General_Concept
process	I-General_Concept
is	O
as	O
a	O
modified	O
Pólya	O
urn	O
scheme	O
sometimes	O
called	O
the	O
Blackwell-MacQueen	O
sampling	O
scheme	O
.	O
</s>
<s>
The	O
resulting	O
distribution	O
over	O
colours	O
is	O
the	O
same	O
as	O
the	O
distribution	O
over	O
tables	O
in	O
the	O
Chinese	B-General_Concept
restaurant	I-General_Concept
process	I-General_Concept
.	O
</s>
<s>
Furthermore	O
,	O
when	O
we	O
draw	O
a	O
black	O
ball	O
,	O
if	O
rather	O
than	O
generating	O
a	O
new	O
colour	O
,	O
we	O
instead	O
pick	O
a	O
random	O
value	O
from	O
a	O
base	O
distribution	O
and	O
use	O
that	O
value	O
to	O
label	O
the	O
new	O
ball	O
,	O
the	O
resulting	O
distribution	O
over	O
labels	O
will	O
be	O
the	O
same	O
as	O
the	O
distribution	O
over	O
the	O
values	O
in	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
The	O
Dirichlet	B-General_Concept
Process	I-General_Concept
can	O
be	O
used	O
as	O
a	O
prior	O
distribution	O
to	O
estimate	O
the	O
probability	O
distribution	O
that	O
generates	O
the	O
data	O
.	O
</s>
<s>
The	O
Dirichlet	B-General_Concept
Process	I-General_Concept
distribution	O
satisfies	O
prior	O
conjugacy	O
,	O
posterior	O
consistency	O
,	O
and	O
the	O
Bernstein	B-General_Concept
–	I-General_Concept
von	I-General_Concept
Mises	I-General_Concept
theorem	I-General_Concept
.	O
</s>
<s>
In	O
this	O
model	O
,	O
the	O
posterior	O
distribution	O
is	O
again	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
This	O
means	O
that	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
a	O
conjugate	O
prior	O
for	O
this	O
model	O
.	O
</s>
<s>
If	O
we	O
take	O
the	O
frequentist	B-General_Concept
view	O
of	O
probability	O
,	O
we	O
believe	O
there	O
is	O
a	O
true	O
probability	O
distribution	O
that	O
generated	O
the	O
data	O
.	O
</s>
<s>
Then	O
it	O
turns	O
out	O
that	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
consistent	O
in	O
the	O
weak	O
topology	O
,	O
which	O
means	O
that	O
for	O
every	O
weak	O
neighbourhood	O
of	O
,	O
the	O
posterior	O
probability	O
of	O
converges	O
to	O
.	O
</s>
<s>
In	O
order	O
to	O
interpret	O
the	O
credible	O
sets	O
as	O
confidence	O
sets	O
,	O
a	O
Bernstein	B-General_Concept
–	I-General_Concept
von	I-General_Concept
Mises	I-General_Concept
theorem	I-General_Concept
is	O
needed	O
.	O
</s>
<s>
In	O
case	O
of	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
we	O
compare	O
the	O
posterior	O
distribution	O
with	O
the	O
empirical	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
This	O
implies	O
that	O
credible	O
sets	O
you	O
construct	O
are	O
asymptotic	O
confidence	O
sets	O
,	O
and	O
the	O
Bayesian	O
inference	O
based	O
on	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
asymptotically	O
also	O
valid	O
frequentist	B-General_Concept
inference	I-General_Concept
.	O
</s>
<s>
To	O
understand	O
what	O
Dirichlet	O
processes	O
are	O
and	O
the	O
problem	O
they	O
solve	O
we	O
consider	O
the	O
example	O
of	O
data	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
By	O
looking	O
at	O
how	O
votes	O
were	O
cast	O
in	O
previous	O
years	O
on	O
similar	O
pieces	O
of	O
legislation	O
one	O
could	O
fit	O
a	O
predictive	O
model	O
using	O
a	O
simple	O
clustering	B-Algorithm
algorithm	I-Algorithm
such	O
as	O
k-means	B-Algorithm
.	O
</s>
<s>
By	O
using	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
prior	O
for	O
the	O
distribution	O
of	O
cluster	O
means	O
we	O
circumvent	O
the	O
need	O
to	O
explicitly	O
specify	O
ahead	O
of	O
time	O
how	O
many	O
clusters	O
there	O
are	O
,	O
although	O
the	O
concentration	O
parameter	O
still	O
controls	O
it	O
implicitly	O
.	O
</s>
<s>
To	O
understand	O
the	O
connection	O
to	O
Dirichlet	B-General_Concept
process	I-General_Concept
priors	O
we	O
rewrite	O
this	O
model	O
in	O
an	O
equivalent	O
but	O
more	O
suggestive	O
form	O
:	O
</s>
<s>
With	O
this	O
in	O
hand	O
we	O
can	O
better	O
understand	O
the	O
computational	O
merits	O
of	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
It	O
is	O
,	O
however	O
,	O
possible	O
to	O
draw	O
samples	O
from	O
this	O
posterior	O
using	O
a	O
modified	O
Gibbs	B-Algorithm
sampler	I-Algorithm
.	O
</s>
<s>
This	O
is	O
the	O
critical	O
fact	O
that	O
makes	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
prior	O
useful	O
for	O
inference	O
.	O
</s>
<s>
Dirichlet	O
processes	O
are	O
frequently	O
used	O
in	O
Bayesian	O
nonparametric	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
"	O
Nonparametric	B-General_Concept
"	O
here	O
does	O
not	O
mean	O
a	O
parameter-less	O
model	O
,	O
rather	O
a	O
model	O
in	O
which	O
representations	O
grow	O
as	O
more	O
data	O
are	O
observed	O
.	O
</s>
<s>
Bayesian	O
nonparametric	B-General_Concept
models	I-General_Concept
have	O
gained	O
considerable	O
popularity	O
in	O
the	O
field	O
of	O
machine	O
learning	O
because	O
of	O
the	O
above-mentioned	O
flexibility	O
,	O
especially	O
in	O
unsupervised	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
In	O
a	O
Bayesian	O
nonparametric	B-General_Concept
model	O
,	O
the	O
prior	O
and	O
posterior	O
distributions	O
are	O
not	O
parametric	O
distributions	O
,	O
but	O
stochastic	O
processes	O
.	O
</s>
<s>
Additionally	O
,	O
the	O
nonparametric	B-General_Concept
nature	O
of	O
this	O
model	O
makes	O
it	O
an	O
ideal	O
candidate	O
for	O
clustering	O
problems	O
where	O
the	O
distinct	O
number	O
of	O
clusters	O
is	O
unknown	O
beforehand	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
Dirichlet	B-General_Concept
process	I-General_Concept
has	O
also	O
been	O
used	O
for	O
developing	O
a	O
mixture	O
of	O
expert	O
models	O
,	O
in	O
the	O
context	O
of	O
supervised	O
learning	O
algorithms	O
(	O
regression	O
or	O
classification	O
settings	O
)	O
.	O
</s>
<s>
As	O
draws	O
from	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
are	O
discrete	O
,	O
an	O
important	O
use	O
is	O
as	O
a	O
prior	O
probability	O
in	O
infinite	O
mixture	O
models	O
.	O
</s>
<s>
The	O
generative	O
process	O
is	O
therefore	O
that	O
a	O
sample	O
is	O
drawn	O
from	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
,	O
and	O
for	O
each	O
data	O
point	O
,	O
in	O
turn	O
,	O
a	O
value	O
is	O
drawn	O
from	O
this	O
sample	O
distribution	O
and	O
used	O
as	O
the	O
component	O
distribution	O
for	O
that	O
data	O
point	O
.	O
</s>
<s>
The	O
infinite	O
nature	O
of	O
these	O
models	O
also	O
lends	O
them	O
to	O
natural	B-Language
language	I-Language
processing	I-Language
applications	O
,	O
where	O
it	O
is	O
often	O
desirable	O
to	O
treat	O
the	O
vocabulary	O
as	O
an	O
infinite	O
,	O
discrete	O
set	O
.	O
</s>
<s>
The	O
Dirichlet	B-General_Concept
Process	I-General_Concept
can	O
also	O
be	O
used	O
for	O
nonparametric	B-General_Concept
hypothesis	O
testing	O
,	O
i.e.	O
</s>
<s>
to	O
develop	O
Bayesian	O
nonparametric	B-General_Concept
versions	O
of	O
the	O
classical	O
nonparametric	B-General_Concept
hypothesis	O
tests	O
,	O
e.g.	O
</s>
<s>
sign	B-General_Concept
test	I-General_Concept
,	O
Wilcoxon	B-General_Concept
rank-sum	I-General_Concept
test	I-General_Concept
,	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
,	O
etc	O
.	O
</s>
<s>
For	O
instance	O
,	O
Bayesian	O
nonparametric	B-General_Concept
versions	O
of	O
the	O
Wilcoxon	B-General_Concept
rank-sum	I-General_Concept
test	I-General_Concept
and	O
the	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
have	O
been	O
developed	O
by	O
using	O
the	O
imprecise	B-General_Concept
Dirichlet	I-General_Concept
process	I-General_Concept
,	O
a	O
prior	O
ignorance	O
Dirichlet	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
The	O
hierarchical	B-General_Concept
Dirichlet	I-General_Concept
process	I-General_Concept
extends	O
the	O
ordinary	O
Dirichlet	B-General_Concept
process	I-General_Concept
for	O
modelling	O
grouped	O
data	O
.	O
</s>
