<s>
In	O
statistics	O
,	O
an	O
expectation	O
–	O
maximization	O
(	O
EM	O
)	O
algorithm	O
is	O
an	O
iterative	B-Algorithm
method	I-Algorithm
to	O
find	O
(	O
local	O
)	O
maximum	O
likelihood	O
or	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
(	O
MAP	O
)	O
estimates	O
of	O
parameters	O
in	O
statistical	O
models	O
,	O
where	O
the	O
model	O
depends	O
on	O
unobserved	O
latent	O
variables	O
.	O
</s>
<s>
The	O
EM	B-Algorithm
algorithm	I-Algorithm
was	O
explained	O
and	O
given	O
its	O
name	O
in	O
a	O
classic	O
1977	O
paper	O
by	O
Arthur	O
Dempster	O
,	O
Nan	O
Laird	O
,	O
and	O
Donald	O
Rubin	O
.	O
</s>
<s>
The	O
EM	B-Algorithm
algorithm	I-Algorithm
is	O
used	O
to	O
find	O
(	O
local	O
)	O
maximum	O
likelihood	O
parameters	O
of	O
a	O
statistical	O
model	O
in	O
cases	O
where	O
the	O
equations	O
cannot	O
be	O
solved	O
directly	O
.	O
</s>
<s>
Finding	O
a	O
maximum	O
likelihood	O
solution	O
typically	O
requires	O
taking	O
the	O
derivatives	B-Algorithm
of	O
the	O
likelihood	O
function	O
with	O
respect	O
to	O
all	O
the	O
unknown	O
values	O
,	O
the	O
parameters	O
and	O
the	O
latent	O
variables	O
,	O
and	O
simultaneously	O
solving	O
the	O
resulting	O
equations	O
.	O
</s>
<s>
The	O
EM	B-Algorithm
algorithm	I-Algorithm
proceeds	O
from	O
the	O
observation	O
that	O
there	O
is	O
a	O
way	O
to	O
solve	O
these	O
two	O
sets	O
of	O
equations	O
numerically	O
.	O
</s>
<s>
Additionally	O
,	O
it	O
can	O
be	O
proven	O
that	O
the	O
derivative	B-Algorithm
of	O
the	O
likelihood	O
is	O
(	O
arbitrarily	O
close	O
to	O
)	O
zero	O
at	O
that	O
point	O
,	O
which	O
in	O
turn	O
means	O
that	O
the	O
point	O
is	O
either	O
a	O
local	O
maximum	O
or	O
a	O
saddle	O
point	O
.	O
</s>
<s>
The	O
EM	B-Algorithm
algorithm	I-Algorithm
seeks	O
to	O
find	O
the	O
MLE	O
of	O
the	O
marginal	O
likelihood	O
by	O
iteratively	O
applying	O
these	O
two	O
steps	O
:	O
</s>
<s>
If	O
the	O
value	O
of	O
the	O
parameters	O
is	O
known	O
,	O
usually	O
the	O
value	O
of	O
the	O
latent	O
variables	O
can	O
be	O
found	O
by	O
maximizing	O
the	O
log-likelihood	O
over	O
all	O
possible	O
values	O
of	O
,	O
either	O
simply	O
by	O
iterating	O
over	O
or	O
through	O
an	O
algorithm	O
such	O
as	O
the	O
Viterbi	B-Algorithm
algorithm	I-Algorithm
for	O
hidden	O
Markov	O
models	O
.	O
</s>
<s>
This	O
suggests	O
an	O
iterative	B-Algorithm
algorithm	I-Algorithm
,	O
in	O
the	O
case	O
where	O
both	O
and	O
are	O
unknown	O
:	O
</s>
<s>
For	O
multimodal	O
distributions	O
,	O
this	O
means	O
that	O
an	O
EM	B-Algorithm
algorithm	I-Algorithm
may	O
converge	O
to	O
a	O
local	O
maximum	O
of	O
the	O
observed	O
data	O
likelihood	O
function	O
,	O
depending	O
on	O
starting	O
values	O
.	O
</s>
<s>
A	O
variety	O
of	O
heuristic	O
or	O
metaheuristic	B-Algorithm
approaches	O
exist	O
to	O
escape	O
a	O
local	O
maximum	O
,	O
such	O
as	O
random-restart	B-Algorithm
hill	I-Algorithm
climbing	I-Algorithm
(	O
starting	O
with	O
several	O
different	O
random	O
initial	O
estimates	O
)	O
,	O
or	O
applying	O
simulated	B-Algorithm
annealing	I-Algorithm
methods	O
.	O
</s>
<s>
The	O
EM	O
method	O
was	O
modified	O
to	O
compute	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
(	O
MAP	O
)	O
estimates	O
for	O
Bayesian	O
inference	O
in	O
the	O
original	O
paper	O
by	O
Dempster	O
,	O
Laird	O
,	O
and	O
Rubin	O
.	O
</s>
<s>
Other	O
methods	O
exist	O
to	O
find	O
maximum	O
likelihood	O
estimates	O
,	O
such	O
as	O
gradient	B-Algorithm
descent	I-Algorithm
,	O
conjugate	B-Algorithm
gradient	I-Algorithm
,	O
or	O
variants	O
of	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Unlike	O
EM	O
,	O
such	O
methods	O
typically	O
require	O
the	O
evaluation	O
of	O
first	O
and/or	O
second	O
derivatives	B-Algorithm
of	O
the	O
likelihood	O
function	O
.	O
</s>
<s>
Expectation-Maximization	B-Algorithm
works	O
to	O
improve	O
rather	O
than	O
directly	O
improving	O
.	O
</s>
<s>
The	O
EM	B-Algorithm
algorithm	I-Algorithm
can	O
be	O
viewed	O
as	O
two	O
alternating	O
maximization	O
steps	O
,	O
that	O
is	O
,	O
as	O
an	O
example	O
of	O
coordinate	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
Then	O
the	O
steps	O
in	O
the	O
EM	B-Algorithm
algorithm	I-Algorithm
may	O
be	O
viewed	O
as	O
:	O
</s>
<s>
EM	O
is	O
frequently	O
used	O
for	O
parameter	O
estimation	O
of	O
mixed	B-General_Concept
models	I-General_Concept
,	O
notably	O
in	O
quantitative	O
genetics	O
.	O
</s>
<s>
The	O
EM	B-Algorithm
algorithm	I-Algorithm
(	O
and	O
its	O
faster	O
variant	O
ordered	B-Algorithm
subset	I-Algorithm
expectation	I-Algorithm
maximization	I-Algorithm
)	O
is	O
also	O
widely	O
used	O
in	O
medical	B-Application
image	I-Application
reconstruction	O
,	O
especially	O
in	O
positron	B-Application
emission	I-Application
tomography	I-Application
,	O
single-photon	O
emission	O
computed	O
tomography	O
,	O
and	O
x-ray	O
computed	O
tomography	O
.	O
</s>
<s>
In	O
structural	O
engineering	O
,	O
the	O
Structural	O
Identification	O
using	O
Expectation	B-Algorithm
Maximization	I-Algorithm
(	O
STRIDE	O
)	O
algorithm	O
is	O
an	O
output-only	O
method	O
for	O
identifying	O
natural	O
vibration	O
properties	O
of	O
a	O
structural	O
system	O
using	O
sensor	O
data	O
(	O
see	O
Operational	O
Modal	O
Analysis	O
)	O
.	O
</s>
<s>
EM	O
is	O
also	O
used	O
for	O
data	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
In	O
natural	B-Language
language	I-Language
processing	I-Language
,	O
two	O
prominent	O
instances	O
of	O
the	O
algorithm	O
are	O
the	O
Baum	O
–	O
Welch	O
algorithm	O
for	O
hidden	O
Markov	O
models	O
,	O
and	O
the	O
inside-outside	B-Application
algorithm	I-Application
for	O
unsupervised	O
induction	O
of	O
probabilistic	B-General_Concept
context-free	I-General_Concept
grammars	I-General_Concept
.	O
</s>
<s>
the	O
time	O
between	O
subsequent	O
trades	O
in	O
shares	O
of	O
stock	O
at	O
a	O
stock	O
exchange	O
the	O
EM	B-Algorithm
algorithm	I-Algorithm
has	O
proved	O
to	O
be	O
very	O
useful	O
.	O
</s>
<s>
EM	B-Algorithm
algorithms	I-Algorithm
can	O
be	O
used	O
for	O
solving	O
joint	O
state	O
and	O
parameter	O
estimation	O
problems	O
.	O
</s>
<s>
Filtering	O
and	O
smoothing	O
EM	B-Algorithm
algorithms	I-Algorithm
arise	O
by	O
repeating	O
this	O
two-step	O
procedure	O
:	O
</s>
<s>
A	O
number	O
of	O
methods	O
have	O
been	O
proposed	O
to	O
accelerate	O
the	O
sometimes	O
slow	O
convergence	O
of	O
the	O
EM	B-Algorithm
algorithm	I-Algorithm
,	O
such	O
as	O
those	O
using	O
conjugate	B-Algorithm
gradient	I-Algorithm
and	O
modified	O
Newton	O
's	O
methods	O
(	O
Newton	O
–	O
Raphson	O
)	O
.	O
</s>
<s>
Parameter-expanded	O
expectation	B-Algorithm
maximization	I-Algorithm
(	O
PX-EM	O
)	O
algorithm	O
often	O
provides	O
speed	O
up	O
by	O
"us[ing]	O
a	O
`covariance	O
adjustment	O
 '	O
to	O
correct	O
the	O
analysis	O
of	O
the	O
M	O
step	O
,	O
capitalising	O
on	O
extra	O
information	O
captured	O
in	O
the	O
imputed	O
complete	O
data	O
"	O
.	O
</s>
<s>
This	O
idea	O
is	O
further	O
extended	O
in	O
generalized	O
expectation	B-Algorithm
maximization	I-Algorithm
(	O
GEM	O
)	O
algorithm	O
,	O
in	O
which	O
is	O
sought	O
only	O
an	O
increase	O
in	O
the	O
objective	O
function	O
F	O
for	O
both	O
the	O
E	O
step	O
and	O
M	O
step	O
as	O
described	O
in	O
the	O
As	O
a	O
maximization	O
–	O
maximization	O
procedure	O
section	O
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
consider	O
the	O
EM	B-Algorithm
algorithm	I-Algorithm
as	O
a	O
subclass	O
of	O
the	O
MM	B-Algorithm
(	O
Majorize/Minimize	O
or	O
Minorize/Maximize	O
,	O
depending	O
on	O
context	O
)	O
algorithm	O
,	O
and	O
therefore	O
use	O
any	O
machinery	O
developed	O
in	O
the	O
more	O
general	O
case	O
.	O
</s>
<s>
The	O
Q-function	O
used	O
in	O
the	O
EM	B-Algorithm
algorithm	I-Algorithm
is	O
based	O
on	O
the	O
log	O
likelihood	O
.	O
</s>
<s>
Its	O
final	O
result	O
gives	O
a	O
probability	O
distribution	O
over	O
the	O
latent	O
variables	O
(	O
in	O
the	O
Bayesian	O
style	O
)	O
together	O
with	O
a	O
point	O
estimate	O
for	O
θ	O
(	O
either	O
a	O
maximum	O
likelihood	O
estimate	O
or	O
a	O
posterior	B-General_Concept
mode	I-General_Concept
)	O
.	O
</s>
<s>
For	O
graphical	B-Algorithm
models	I-Algorithm
this	O
is	O
easy	O
to	O
do	O
as	O
each	O
variable	O
's	O
new	O
Q	O
depends	O
only	O
on	O
its	O
Markov	O
blanket	O
,	O
so	O
local	O
message	O
passing	O
can	O
be	O
used	O
for	O
efficient	O
inference	O
.	O
</s>
<s>
The	O
EM	B-Algorithm
algorithm	I-Algorithm
has	O
been	O
implemented	O
in	O
the	O
case	O
where	O
an	O
underlying	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
exists	O
explaining	O
the	O
variation	O
of	O
some	O
quantity	O
,	O
but	O
where	O
the	O
values	O
actually	O
observed	O
are	O
censored	O
or	O
truncated	O
versions	O
of	O
those	O
represented	O
in	O
the	O
model	O
.	O
</s>
