<s>
Thompson	B-Algorithm
sampling	I-Algorithm
,	O
named	O
after	O
William	O
R	O
.	O
Thompson	O
,	O
is	O
a	O
heuristic	B-Algorithm
for	O
choosing	O
actions	O
that	O
addresses	O
the	O
exploration-exploitation	O
dilemma	O
in	O
the	O
multi-armed	O
bandit	O
problem	O
.	O
</s>
<s>
The	O
elements	O
of	O
Thompson	B-Algorithm
sampling	I-Algorithm
are	O
as	O
follows	O
:	O
</s>
<s>
As	O
such	O
,	O
Thompson	B-Algorithm
sampling	I-Algorithm
is	O
often	O
used	O
in	O
conjunction	O
with	O
approximate	O
sampling	O
techniques	O
.	O
</s>
<s>
Thompson	B-Algorithm
sampling	I-Algorithm
was	O
originally	O
described	O
by	O
Thompson	O
in	O
1933	O
.	O
</s>
<s>
In	O
2010	O
it	O
was	O
also	O
shown	O
that	O
Thompson	B-Algorithm
sampling	I-Algorithm
is	O
instantaneously	O
self-correcting	O
.	O
</s>
<s>
Thompson	B-Algorithm
Sampling	I-Algorithm
has	O
been	O
widely	O
used	O
in	O
many	O
online	O
learning	O
problems	O
including	O
A/B	O
testing	O
in	O
website	O
design	O
and	O
online	O
advertising	O
,	O
and	O
accelerated	O
learning	O
in	O
decentralized	O
decision	O
making	O
.	O
</s>
<s>
A	O
Double	O
Thompson	B-Algorithm
Sampling	I-Algorithm
(	O
D-TS	O
)	O
algorithm	O
has	O
been	O
proposed	O
for	O
dueling	O
bandits	O
,	O
a	O
variant	O
of	O
traditional	O
MAB	O
,	O
where	O
feedback	O
comes	O
in	O
the	O
form	O
of	O
pairwise	O
comparison	O
.	O
</s>
<s>
A	O
generalization	O
of	O
Thompson	B-Algorithm
sampling	I-Algorithm
to	O
arbitrary	O
dynamical	O
environments	O
and	O
causal	O
structures	O
,	O
known	O
as	O
Bayesian	O
control	O
rule	O
,	O
has	O
been	O
shown	O
to	O
be	O
the	O
optimal	O
solution	O
to	O
the	O
adaptive	O
coding	O
problem	O
with	O
actions	O
and	O
observations	O
.	O
</s>
<s>
where	O
the	O
"	O
hat	O
"	O
-notation	O
denotes	O
the	O
fact	O
that	O
is	O
a	O
causal	O
intervention	O
(	O
see	O
Causality	B-Application
)	O
,	O
and	O
not	O
an	O
ordinary	O
observation	O
.	O
</s>
<s>
Thompson	B-Algorithm
sampling	I-Algorithm
and	O
upper-confidence	O
bound	O
algorithms	O
share	O
a	O
fundamental	O
property	O
that	O
underlies	O
many	O
of	O
their	O
theoretical	O
guarantees	O
.	O
</s>
<s>
Leveraging	O
this	O
property	O
,	O
one	O
can	O
translate	O
regret	O
bounds	O
established	O
for	O
UCB	O
algorithms	O
to	O
Bayesian	O
regret	O
bounds	O
for	O
Thompson	B-Algorithm
sampling	I-Algorithm
or	O
unify	O
regret	O
analysis	O
across	O
both	O
these	O
algorithms	O
and	O
many	O
classes	O
of	O
problems	O
.	O
</s>
