<s>
Fiducial	B-Error_Name
inference	I-Error_Name
is	O
one	O
of	O
a	O
number	O
of	O
different	O
types	O
of	O
statistical	O
inference	O
.	O
</s>
<s>
In	O
modern	O
statistical	O
practice	O
,	O
attempts	O
to	O
work	O
with	O
fiducial	B-Error_Name
inference	I-Error_Name
have	O
fallen	O
out	O
of	O
fashion	O
in	O
favour	O
of	O
frequentist	B-General_Concept
inference	I-General_Concept
,	O
Bayesian	O
inference	O
and	O
decision	O
theory	O
.	O
</s>
<s>
However	O
,	O
fiducial	B-Error_Name
inference	I-Error_Name
is	O
important	O
in	O
the	O
history	O
of	O
statistics	O
since	O
its	O
development	O
led	O
to	O
the	O
parallel	O
development	O
of	O
concepts	O
and	O
tools	O
in	O
theoretical	O
statistics	O
that	O
are	O
widely	O
used	O
.	O
</s>
<s>
Some	O
current	O
research	O
in	O
statistical	O
methodology	O
is	O
either	O
explicitly	O
linked	O
to	O
fiducial	B-Error_Name
inference	I-Error_Name
or	O
is	O
closely	O
connected	O
to	O
it	O
.	O
</s>
<s>
The	O
general	O
approach	O
of	O
fiducial	B-Error_Name
inference	I-Error_Name
was	O
proposed	O
by	O
Ronald	O
Fisher	O
.	O
</s>
<s>
Fiducial	B-Error_Name
inference	I-Error_Name
can	O
be	O
interpreted	O
as	O
an	O
attempt	O
to	O
perform	O
inverse	O
probability	O
without	O
calling	O
on	O
prior	O
probability	O
distributions	O
.	O
</s>
<s>
Fiducial	B-Error_Name
inference	I-Error_Name
quickly	O
attracted	O
controversy	O
and	O
was	O
never	O
widely	O
accepted	O
.	O
</s>
<s>
Indeed	O
,	O
counter-examples	O
to	O
the	O
claims	O
of	O
Fisher	O
for	O
fiducial	B-Error_Name
inference	I-Error_Name
were	O
soon	O
published	O
.	O
</s>
<s>
These	O
counter-examples	O
cast	O
doubt	O
on	O
the	O
coherence	O
of	O
"	O
fiducial	B-Error_Name
inference	I-Error_Name
"	O
as	O
a	O
system	O
of	O
statistical	O
inference	O
or	O
inductive	O
logic	O
.	O
</s>
<s>
Other	O
studies	O
showed	O
that	O
,	O
where	O
the	O
steps	O
of	O
fiducial	B-Error_Name
inference	I-Error_Name
are	O
said	O
to	O
lead	O
to	O
"	O
fiducial	B-Error_Name
probabilities	I-Error_Name
"	O
(	O
or	O
"	O
fiducial	B-Error_Name
distributions	I-Error_Name
"	O
)	O
,	O
these	O
probabilities	O
lack	O
the	O
property	O
of	O
additivity	O
,	O
and	O
so	O
cannot	O
constitute	O
a	O
probability	O
measure	O
.	O
</s>
<s>
The	O
concept	O
of	O
fiducial	B-Error_Name
inference	I-Error_Name
can	O
be	O
outlined	O
by	O
comparing	O
its	O
treatment	O
of	O
the	O
problem	O
of	O
interval	O
estimation	O
in	O
relation	O
to	O
other	O
modes	O
of	O
statistical	O
inference	O
.	O
</s>
<s>
A	O
confidence	O
interval	O
,	O
in	O
frequentist	B-General_Concept
inference	I-General_Concept
,	O
with	O
coverage	O
probability	O
γ	O
has	O
the	O
interpretation	O
that	O
among	O
all	O
confidence	O
intervals	O
computed	O
by	O
the	O
same	O
method	O
,	O
a	O
proportion	O
γ	O
will	O
contain	O
the	O
true	O
value	O
that	O
needs	O
to	O
be	O
estimated	O
.	O
</s>
<s>
This	O
has	O
either	O
a	O
repeated	O
sampling	O
(	O
or	O
frequentist	B-General_Concept
)	O
interpretation	O
,	O
or	O
is	O
the	O
probability	O
that	O
an	O
interval	O
calculated	O
from	O
yet-to-be-sampled	O
data	O
will	O
cover	O
the	O
true	O
value	O
.	O
</s>
<s>
Credible	B-General_Concept
intervals	I-General_Concept
,	O
in	O
Bayesian	O
inference	O
,	O
do	O
allow	O
a	O
probability	O
to	O
be	O
given	O
for	O
the	O
event	O
that	O
an	O
interval	O
,	O
once	O
it	O
has	O
been	O
calculated	O
,	O
does	O
include	O
the	O
true	O
value	O
,	O
since	O
it	O
proceeds	O
on	O
the	O
basis	O
that	O
a	O
probability	O
distribution	O
can	O
be	O
associated	O
with	O
the	O
state	O
of	O
knowledge	O
about	O
the	O
true	O
value	O
,	O
both	O
before	O
and	O
after	O
the	O
sample	O
of	O
data	O
has	O
been	O
obtained	O
.	O
</s>
<s>
Fisher	O
designed	O
the	O
fiducial	O
method	O
to	O
meet	O
perceived	O
problems	O
with	O
the	O
Bayesian	O
approach	O
,	O
at	O
a	O
time	O
when	O
the	O
frequentist	B-General_Concept
approach	O
had	O
yet	O
to	O
be	O
fully	O
developed	O
.	O
</s>
<s>
The	O
method	O
proceeds	O
by	O
attempting	O
to	O
derive	O
a	O
"	O
fiducial	B-Error_Name
distribution	I-Error_Name
"	O
,	O
which	O
is	O
a	O
measure	O
of	O
the	O
degree	O
of	O
faith	O
that	O
can	O
be	O
put	O
on	O
any	O
given	O
value	O
of	O
the	O
unknown	O
parameter	O
and	O
is	O
faithful	O
to	O
the	O
data	O
in	O
the	O
sense	O
that	O
the	O
method	O
uses	O
all	O
available	O
information	O
.	O
</s>
<s>
A	O
relatively	O
complete	O
presentation	O
of	O
the	O
fiducial	O
approach	O
to	O
inference	O
is	O
given	O
by	O
Quenouille	O
(	O
1958	O
)	O
,	O
while	O
Williams	O
(	O
1959	O
)	O
describes	O
the	O
application	O
of	O
fiducial	O
analysis	O
to	O
the	O
calibration	B-General_Concept
problem	O
(	O
also	O
known	O
as	O
"	O
inverse	O
regression	O
"	O
)	O
in	O
regression	O
analysis	O
.	O
</s>
<s>
Further	O
discussion	O
of	O
fiducial	B-Error_Name
inference	I-Error_Name
is	O
given	O
by	O
Kendall	O
&	O
Stuart	O
(	O
1973	O
)	O
.	O
</s>
<s>
This	O
distribution	O
of	O
ω	O
is	O
the	O
fiducial	B-Error_Name
distribution	I-Error_Name
which	O
may	O
be	O
used	O
to	O
form	O
fiducial	B-Error_Name
intervals	I-Error_Name
that	O
represent	O
degrees	O
of	O
belief	O
.	O
</s>
<s>
In	O
fact	O
older	O
books	O
use	O
the	O
terms	O
confidence	O
interval	O
and	O
fiducial	B-Error_Name
interval	I-Error_Name
interchangeably	O
.	O
</s>
<s>
Notice	O
that	O
the	O
fiducial	B-Error_Name
distribution	I-Error_Name
is	O
uniquely	O
defined	O
when	O
a	O
single	O
sufficient	O
statistic	O
exists	O
.	O
</s>
<s>
A	O
fiducial	B-Error_Name
interval	I-Error_Name
could	O
be	O
taken	O
to	O
be	O
just	O
a	O
different	O
name	O
for	O
a	O
confidence	O
interval	O
and	O
give	O
it	O
the	O
fiducial	O
interpretation	O
.	O
</s>
<s>
Fisher	O
would	O
have	O
denied	O
that	O
this	O
interpretation	O
is	O
correct	O
:	O
for	O
him	O
,	O
the	O
fiducial	B-Error_Name
distribution	I-Error_Name
had	O
to	O
be	O
defined	O
uniquely	O
and	O
it	O
had	O
to	O
use	O
all	O
the	O
information	O
in	O
the	O
sample	O
.	O
</s>
<s>
Fisher	O
admitted	O
that	O
"	O
fiducial	B-Error_Name
inference	I-Error_Name
"	O
had	O
problems	O
.	O
</s>
<s>
Fisher	O
wrote	O
to	O
George	O
A	O
.	O
Barnard	O
that	O
he	O
was	O
"	O
not	O
clear	O
in	O
the	O
head	O
"	O
about	O
one	O
problem	O
on	O
fiducial	B-Error_Name
inference	I-Error_Name
,	O
and	O
,	O
also	O
writing	O
to	O
Barnard	O
,	O
Fisher	O
complained	O
that	O
his	O
theory	O
seemed	O
to	O
have	O
only	O
"	O
an	O
asymptotic	O
approach	O
to	O
intelligibility	O
"	O
.	O
</s>
<s>
Later	O
Fisher	O
confessed	O
that	O
"	O
I	O
do	O
n't	O
understand	O
yet	O
what	O
fiducial	B-Error_Name
probability	I-Error_Name
does	O
.	O
</s>
<s>
Lindley	O
showed	O
that	O
fiducial	B-Error_Name
probability	I-Error_Name
lacked	O
additivity	O
,	O
and	O
so	O
was	O
not	O
a	O
probability	O
measure	O
.	O
</s>
<s>
Cox	O
points	O
out	O
that	O
the	O
same	O
argument	O
applies	O
to	O
the	O
so-called	O
"	O
confidence	B-General_Concept
distribution	I-General_Concept
"	O
associated	O
with	O
confidence	O
intervals	O
,	O
so	O
the	O
conclusion	O
to	O
be	O
drawn	O
from	O
this	O
is	O
moot	O
.	O
</s>
<s>
Fisher	O
sketched	O
"	O
proofs	O
"	O
of	O
results	O
using	O
fiducial	B-Error_Name
probability	I-Error_Name
.	O
</s>
<s>
However	O
,	O
fiducial	B-Error_Name
inference	I-Error_Name
is	O
still	O
being	O
studied	O
and	O
its	O
principles	O
appear	O
valuable	O
for	O
some	O
scientific	O
applications	O
.	O
</s>
<s>
In	O
the	O
mid-2010s	O
,	O
the	O
psychometrician	O
Yang	O
Liu	O
developed	O
generalized	O
fiducial	B-Error_Name
inference	I-Error_Name
for	O
models	O
in	O
item	O
response	O
theory	O
and	O
demonstrated	O
favorable	O
results	O
compared	O
to	O
frequentist	B-General_Concept
and	O
Bayesian	O
approaches	O
.	O
</s>
<s>
Other	O
current	O
work	O
in	O
fiducial	B-Error_Name
inference	I-Error_Name
is	O
ongoing	O
under	O
the	O
name	O
of	O
confidence	B-General_Concept
distributions	I-General_Concept
.	O
</s>
