<s>
In	O
statistics	O
,	O
a	O
probit	B-Architecture
model	I-Architecture
is	O
a	O
type	O
of	O
regression	O
where	O
the	O
dependent	O
variable	O
can	O
take	O
only	O
two	O
values	O
,	O
for	O
example	O
married	O
or	O
not	O
married	O
.	O
</s>
<s>
The	O
purpose	O
of	O
the	O
model	O
is	O
to	O
estimate	O
the	O
probability	O
that	O
an	O
observation	O
with	O
particular	O
characteristics	O
will	O
fall	O
into	O
a	O
specific	O
one	O
of	O
the	O
categories	O
;	O
moreover	O
,	O
classifying	O
observations	O
based	O
on	O
their	O
predicted	O
probabilities	O
is	O
a	O
type	O
of	O
binary	B-General_Concept
classification	I-General_Concept
model	O
.	O
</s>
<s>
A	O
probit	B-Architecture
model	I-Architecture
is	O
a	O
popular	O
specification	O
for	O
a	O
binary	O
response	O
model	O
.	O
</s>
<s>
When	O
viewed	O
in	O
the	O
generalized	O
linear	O
model	O
framework	O
,	O
the	O
probit	B-Architecture
model	I-Architecture
employs	O
a	O
probit	O
link	O
function	O
.	O
</s>
<s>
It	O
is	O
most	O
often	O
estimated	O
using	O
the	O
maximum	O
likelihood	O
procedure	O
,	O
such	O
an	O
estimation	O
being	O
called	O
a	O
probit	B-Architecture
regression	I-Architecture
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
motivate	O
the	O
probit	B-Architecture
model	I-Architecture
as	O
a	O
latent	O
variable	O
model	O
.	O
</s>
<s>
Gibbs	B-Algorithm
sampling	I-Algorithm
of	O
a	O
probit	B-Architecture
model	I-Architecture
is	O
possible	O
because	O
regression	O
models	O
typically	O
use	O
normal	O
prior	O
distributions	O
over	O
the	O
weights	O
,	O
and	O
this	O
distribution	O
is	O
conjugate	O
with	O
the	O
normal	O
distribution	O
of	O
the	O
errors	O
(	O
and	O
hence	O
of	O
the	O
latent	O
variables	O
Y*	O
)	O
.	O
</s>
<s>
The	O
result	O
for	O
β	O
 '	O
is	O
given	O
in	O
the	O
article	O
on	O
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
,	O
although	O
specified	O
with	O
different	O
notation	O
.	O
</s>
<s>
If	O
a	O
large	O
fraction	O
of	O
the	O
original	O
mass	O
remains	O
,	O
sampling	O
can	O
be	O
easily	O
done	O
with	O
rejection	B-Algorithm
sampling	I-Algorithm
—	O
simply	O
sample	O
a	O
number	O
from	O
the	O
non-truncated	O
distribution	O
,	O
and	O
reject	O
it	O
if	O
it	O
falls	O
outside	O
the	O
restriction	O
imposed	O
by	O
the	O
truncation	O
.	O
</s>
<s>
General	O
sampling	O
from	O
the	O
truncated	O
normal	O
can	O
be	O
achieved	O
using	O
approximations	O
to	O
the	O
normal	O
CDF	O
and	O
the	O
probit	O
function	O
,	O
and	O
R	B-Language
has	O
a	O
function	O
rtnorm( )	O
for	O
generating	O
truncated-normal	O
samples	O
.	O
</s>
<s>
Consider	O
the	O
latent	O
variable	O
model	O
formulation	O
of	O
the	O
probit	B-Architecture
model	I-Architecture
.	O
</s>
<s>
When	O
the	O
variance	O
of	O
conditional	O
on	O
is	O
not	O
constant	O
but	O
dependent	O
on	O
,	O
then	O
the	O
heteroscedasticity	B-General_Concept
issue	O
arises	O
.	O
</s>
<s>
Under	O
heteroskedasticity	B-General_Concept
,	O
the	O
probit	O
estimator	O
for	O
is	O
usually	O
inconsistent	O
,	O
and	O
most	O
of	O
the	O
tests	O
about	O
the	O
coefficients	O
are	O
invalid	O
.	O
</s>
<s>
To	O
deal	O
with	O
this	O
problem	O
,	O
the	O
original	O
model	O
needs	O
to	O
be	O
transformed	O
to	O
be	O
homoskedastic	B-General_Concept
.	O
</s>
<s>
When	O
the	O
assumption	O
that	O
is	O
normally	O
distributed	O
fails	O
to	O
hold	O
,	O
then	O
a	O
functional	O
form	O
misspecification	O
issue	O
arises	O
:	O
if	O
the	O
model	O
is	O
still	O
estimated	O
as	O
a	O
probit	B-Architecture
model	I-Architecture
,	O
the	O
estimators	O
of	O
the	O
coefficients	O
are	O
inconsistent	O
.	O
</s>
<s>
The	O
probit	B-Architecture
model	I-Architecture
is	O
usually	O
credited	O
to	O
Chester	O
Bliss	O
,	O
who	O
coined	O
the	O
term	O
"	O
probit	O
"	O
in	O
1934	O
,	O
and	O
to	O
John	O
Gaddum	O
(	O
1933	O
)	O
,	O
who	O
systematized	O
earlier	O
work	O
.	O
</s>
<s>
A	O
fast	O
method	O
for	O
computing	O
maximum	O
likelihood	O
estimates	O
for	O
the	O
probit	B-Architecture
model	I-Architecture
was	O
proposed	O
by	O
Ronald	O
Fisher	O
as	O
an	O
appendix	O
to	O
Bliss	O
 '	O
work	O
in	O
1935	O
.	O
</s>
