<s>
In	O
statistical	O
decision	O
theory	O
,	O
a	O
randomised	B-General_Concept
decision	I-General_Concept
rule	I-General_Concept
or	O
mixed	B-General_Concept
decision	I-General_Concept
rule	I-General_Concept
is	O
a	O
decision	O
rule	O
that	O
associates	O
probabilities	O
with	O
deterministic	O
decision	O
rules	O
.	O
</s>
<s>
In	O
finite	O
decision	O
problems	O
,	O
randomised	B-General_Concept
decision	I-General_Concept
rules	I-General_Concept
define	O
a	O
risk	O
set	O
which	O
is	O
the	O
convex	O
hull	O
of	O
the	O
risk	O
points	O
of	O
the	O
nonrandomised	O
decision	O
rules	O
.	O
</s>
<s>
As	O
nonrandomised	O
alternatives	O
always	O
exist	O
to	O
randomised	O
Bayes	O
rules	O
,	O
randomisation	O
is	O
not	O
needed	O
in	O
Bayesian	O
statistics	O
,	O
although	O
frequentist	B-General_Concept
statistical	O
theory	O
sometimes	O
requires	O
the	O
use	O
of	O
randomised	O
rules	O
to	O
satisfy	O
optimality	O
conditions	O
such	O
as	O
minimax	B-Algorithm
,	O
most	O
notably	O
when	O
deriving	O
confidence	O
intervals	O
and	O
hypothesis	O
tests	O
about	O
discrete	O
probability	O
distributions	O
.	O
</s>
<s>
A	O
statistical	O
test	O
making	O
use	O
of	O
a	O
randomized	B-General_Concept
decision	I-General_Concept
rule	I-General_Concept
is	O
called	O
a	O
randomized	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Then	O
the	O
randomised	B-General_Concept
decision	I-General_Concept
rule	I-General_Concept
is	O
defined	O
as	O
and	O
its	O
associated	O
risk	O
function	O
is	O
.	O
</s>
<s>
Alternatively	O
,	O
a	O
randomised	B-General_Concept
decision	I-General_Concept
rule	I-General_Concept
may	O
assign	O
probabilities	O
directly	O
on	O
elements	O
of	O
the	O
actions	O
space	O
for	O
each	O
member	O
of	O
the	O
sample	O
space	O
.	O
</s>
<s>
The	O
introduction	O
of	O
randomised	B-General_Concept
decision	I-General_Concept
rules	I-General_Concept
thus	O
creates	O
a	O
larger	O
decision	O
space	O
from	O
which	O
the	O
statistician	O
may	O
choose	O
his	O
decision	O
.	O
</s>
<s>
As	O
non-randomised	O
decision	O
rules	O
are	O
a	O
special	O
case	O
of	O
randomised	B-General_Concept
decision	I-General_Concept
rules	I-General_Concept
where	O
one	O
decision	O
or	O
action	O
has	O
probability	O
1	O
,	O
the	O
original	O
decision	O
space	O
is	O
a	O
proper	O
subset	O
of	O
the	O
new	O
decision	O
space	O
.	O
</s>
<s>
As	O
with	O
nonrandomised	O
decision	O
rules	O
,	O
randomised	B-General_Concept
decision	I-General_Concept
rules	I-General_Concept
may	O
satisfy	O
favourable	O
properties	O
such	O
as	O
admissibility	O
,	O
minimaxity	O
and	O
Bayes	O
.	O
</s>
<s>
The	O
risk	O
set	O
,	O
henceforth	O
denoted	O
as	O
,	O
is	O
the	O
set	O
of	O
all	O
vectors	O
in	O
which	O
each	O
entry	O
is	O
the	O
value	O
of	O
the	O
risk	O
function	O
associated	O
with	O
a	O
randomised	B-General_Concept
decision	I-General_Concept
rule	I-General_Concept
under	O
a	O
certain	O
parameter	O
:	O
it	O
contains	O
all	O
vectors	O
of	O
the	O
form	O
.	O
</s>
<s>
Note	O
that	O
by	O
the	O
definition	O
of	O
the	O
randomised	B-General_Concept
decision	I-General_Concept
rule	I-General_Concept
,	O
the	O
risk	O
set	O
is	O
the	O
convex	O
hull	O
of	O
the	O
risks	O
.	O
</s>
<s>
A	O
minimax	B-Algorithm
Bayes	O
rule	O
is	O
one	O
that	O
minimises	O
the	O
supremum	O
risk	O
among	O
all	O
decision	O
rules	O
in	O
.	O
</s>
<s>
Sometimes	O
,	O
a	O
randomised	B-General_Concept
decision	I-General_Concept
rule	I-General_Concept
may	O
perform	O
better	O
than	O
all	O
other	O
nonrandomised	O
decision	O
rules	O
in	O
this	O
regard	O
.	O
</s>
<s>
In	O
a	O
finite	O
decision	O
problem	O
with	O
two	O
possible	O
parameters	O
,	O
the	O
minimax	B-Algorithm
rule	O
can	O
be	O
found	O
by	O
considering	O
the	O
family	O
of	O
squares	O
.	O
</s>
<s>
The	O
value	O
of	O
for	O
the	O
smallest	O
of	O
such	O
squares	O
that	O
touches	O
is	O
the	O
minimax	B-Algorithm
risk	O
and	O
the	O
corresponding	O
point	O
or	O
points	O
on	O
the	O
risk	O
set	O
is	O
the	O
minimax	B-Algorithm
rule	O
.	O
</s>
<s>
If	O
the	O
risk	O
set	O
intersects	O
the	O
line	O
,	O
then	O
the	O
admissible	O
decision	O
rule	O
lying	O
on	O
the	O
line	O
is	O
minimax	B-Algorithm
.	O
</s>
<s>
If	O
or	O
holds	O
for	O
every	O
point	O
in	O
the	O
risk	O
set	O
,	O
then	O
the	O
minimax	B-Algorithm
rule	O
can	O
either	O
be	O
an	O
extreme	O
point	O
(	O
i.e.	O
</s>
<s>
A	O
randomised	O
Bayes	O
rule	O
is	O
one	O
that	O
has	O
infimum	O
Bayes	B-General_Concept
risk	I-General_Concept
among	O
all	O
decision	O
rules	O
.	O
</s>
<s>
In	O
the	O
special	O
case	O
where	O
the	O
parameter	O
space	O
has	O
two	O
elements	O
,	O
the	O
line	O
,	O
where	O
and	O
denote	O
the	O
prior	O
probabilities	O
of	O
and	O
respectively	O
,	O
is	O
a	O
family	O
of	O
points	O
with	O
Bayes	B-General_Concept
risk	I-General_Concept
.	O
</s>
<s>
The	O
minimum	O
Bayes	B-General_Concept
risk	I-General_Concept
for	O
the	O
decision	O
problem	O
is	O
therefore	O
the	O
smallest	O
such	O
that	O
the	O
line	O
touches	O
the	O
risk	O
set	O
.	O
</s>
<s>
correspond	O
to	O
two	O
nonrandomised	O
decision	O
rules	O
and	O
randomised	B-General_Concept
decision	I-General_Concept
rules	I-General_Concept
combining	O
the	O
two	O
.	O
</s>
<s>
This	O
is	O
also	O
true	O
in	O
the	O
general	O
case	O
,	O
even	O
with	O
infinite	O
parameter	O
space	O
,	O
infinite	O
Bayes	B-General_Concept
risk	I-General_Concept
,	O
and	O
regardless	O
of	O
whether	O
the	O
infimum	O
Bayes	B-General_Concept
risk	I-General_Concept
can	O
be	O
attained	O
.	O
</s>
<s>
Randomized	B-General_Concept
tests	I-General_Concept
should	O
not	O
be	O
confused	O
with	O
permutation	B-General_Concept
tests	I-General_Concept
.	O
</s>
<s>
In	O
the	O
usual	O
formulation	O
of	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
rejected	O
whenever	O
the	O
likelihood	O
ratio	O
is	O
smaller	O
than	O
some	O
constant	O
,	O
and	O
accepted	O
otherwise	O
.	O
</s>
<s>
However	O
,	O
this	O
is	O
sometimes	O
problematic	O
when	O
is	O
discrete	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
when	O
is	O
possible	O
.	O
</s>
<s>
A	O
solution	O
is	O
to	O
define	O
a	O
test	O
function	O
,	O
whose	O
value	O
is	O
the	O
probability	O
at	O
which	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
accepted	O
:	O
</s>
<s>
This	O
can	O
be	O
interpreted	O
as	O
flipping	O
a	O
biased	O
coin	O
with	O
a	O
probability	O
of	O
returning	O
heads	O
whenever	O
and	O
rejecting	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
if	O
a	O
heads	O
turns	O
up	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
consider	O
the	O
case	O
where	O
the	O
underlying	O
distribution	O
is	O
Bernoulli	O
with	O
probability	O
,	O
and	O
we	O
would	O
like	O
to	O
test	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
against	O
the	O
alternative	O
hypothesis	O
.	O
</s>
