<s>
In	O
statistics	O
and	O
machine	O
learning	O
,	O
the	O
hierarchical	B-General_Concept
Dirichlet	I-General_Concept
process	I-General_Concept
(	O
HDP	O
)	O
is	O
a	O
nonparametric	B-General_Concept
Bayesian	O
approach	O
to	O
clustering	O
grouped	B-General_Concept
data	I-General_Concept
.	O
</s>
<s>
It	O
uses	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
for	O
each	O
group	O
of	O
data	O
,	O
with	O
the	O
Dirichlet	O
processes	O
for	O
all	O
groups	O
sharing	O
a	O
base	O
distribution	O
which	O
is	O
itself	O
drawn	O
from	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
.	O
</s>
<s>
The	O
base	O
distribution	O
being	O
drawn	O
from	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
is	O
important	O
,	O
because	O
draws	O
from	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
are	O
atomic	O
probability	O
measures	O
,	O
and	O
the	O
atoms	O
will	O
appear	O
in	O
all	O
group-level	O
Dirichlet	O
processes	O
.	O
</s>
<s>
The	O
HDP	O
is	O
a	O
model	O
for	O
grouped	B-General_Concept
data	I-General_Concept
.	O
</s>
<s>
The	O
th	O
group	O
is	O
associated	O
with	O
a	O
random	O
probability	O
measure	O
which	O
has	O
distribution	O
given	O
by	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
:	O
</s>
<s>
In	O
turn	O
,	O
the	O
common	O
base	O
distribution	O
is	O
Dirichlet	B-General_Concept
process	I-General_Concept
distributed	O
:	O
</s>
<s>
To	O
understand	O
how	O
the	O
HDP	O
implements	O
a	O
clustering	O
model	O
,	O
and	O
how	O
clusters	O
become	O
shared	O
across	O
groups	O
,	O
recall	O
that	O
draws	O
from	O
a	O
Dirichlet	B-General_Concept
process	I-General_Concept
are	O
atomic	O
probability	O
measures	O
with	O
probability	O
one	O
.	O
</s>
<s>
The	O
HDP	O
mixture	O
model	O
is	O
a	O
natural	O
nonparametric	B-General_Concept
generalization	O
of	O
Latent	O
Dirichlet	O
allocation	O
,	O
where	O
the	O
number	O
of	O
topics	O
can	O
be	O
unbounded	O
and	O
learnt	O
from	O
data	O
.	O
</s>
<s>
The	O
HDP	O
is	O
also	O
a	O
core	O
component	O
of	O
the	O
infinite	O
hidden	O
Markov	O
model	O
,	O
which	O
is	O
a	O
nonparametric	B-General_Concept
generalization	O
of	O
the	O
hidden	O
Markov	O
model	O
allowing	O
the	O
number	O
of	O
states	O
to	O
be	O
unbounded	O
and	O
learnt	O
from	O
data	O
.	O
</s>
<s>
The	O
Dirichlet	O
processes	O
can	O
be	O
replaced	O
by	O
Pitman-Yor	O
processes	O
and	O
Gamma	O
processes	O
,	O
resulting	O
in	O
the	O
Hierarchical	O
Pitman-Yor	B-General_Concept
process	I-General_Concept
and	O
Hierarchical	O
Gamma	O
process	O
.	O
</s>
<s>
Such	O
an	O
arrangement	O
has	O
been	O
exploited	O
in	O
the	O
sequence	O
memoizer	O
,	O
a	O
Bayesian	O
nonparametric	B-General_Concept
model	O
for	O
sequences	O
which	O
has	O
a	O
multi-level	O
hierarchy	O
of	O
Pitman-Yor	O
processes	O
.	O
</s>
