Information Technology Reference
In-Depth Information
KR
bin
the input of the
system which, as mentioned earlier, under our probabilistic interpretation takes the
form of a collection of conditional probabilistic statements
Consider now the theory
T
=(
Φ
,
R
)
, with
R
=
and
Φ
κ
1
,
q
1
,
η
1
, ...,
κ
n
,
q
n
,
η
n
,
for some
q
1
, ...,
q
n
basic medical entities,
η
1
, ...,
η
n
∈
[
0
,
1
]
and
Ω
=
{
κ
1
, ...,
κ
n
}⊂
SL
the initial evidence about the patient. The diagnose
η
would be given as an output in a run of the inference process of CADIAG2 on in-
put
φ
along with the value
only if there exists a maximal proof (defined for
CadL
∗
essentially as for
CadL
)of
Φ
Ω
,
φ
,
η
in
CadL
∗
. In our probabilistic interpretation, a run-
time inconsistency in CADIAG2 can be manifested by the existence of maximal
proofs of
from
T
Ω
,
φ
,
Ω
,
φ
,
T
0
and
1
from
, for some medical entity
φ
,orbythe
Ω
,
φ
,
η
non-existence of a proof of
together with the existence of a proof of a
(due to the fact that
max
∗
is not defined for
statement of the form
κ
,
φ
,
ζ
from
T
(
0
,
1
)
) -for more details on all these issues see [19]-.
CadL
∗
and Probabilistic Soundness
Among the manipulation rules in
CadL
∗
, probabilistic soundness of
ME
∗
is clear
(i.e., that any probability function on
L
that satisfies
κ
,
θ
,
1
and
θ
,
φ
,
0
also sat-
). So is soundness of
AO
∗
.However,
C
∗
is certainly not sound with
respect to probabilistic semantics. Among the two new additional rules in
CadL
∗
introduced to provide a probabilistic interpretation of the inference,
MAX
is clearly
not sound and
EX
assumes some probabilistic independence among entities that
may not actually be independent. Overall,
CadL
∗
does not score well within prob-
ability theory. This is no surprise. The computation of conditional probabilistic
statements in a compositional way, as done by CADIAG2 primarily by means of
the
min
and
max
∗
operators, is clearly bound to be probabilistically unsound. One
may wonder though what could be done in order to improve the inference on prob-
abilistic grounds from a knowledge base like
KR
bin
. The answer seems to be 'not
much'. Certainly a
KR
bin
-like knowledge base (i.e., a knowledge base given by
some binary probabilistic conditional statements) is not the most convenient for in-
ferential purposes in probability theory for medical applications like CADIAG2. As
is well known, there are other knowledge-base structures better suited for that pur-
pose, Bayesian networks being the most celebrated among them -see for example
[5] or [18]-.
It is worth noting that
CadL
∗
satisfies what we can call
weak consistency
-called
weak soundness
in [11]-, defined as follows: if there is a maximal proof in
CadL
∗
of a statement of the form
isfies
κ
,
φ
,
0
Δ
,
φ
,
Δ
,
φ
,
1
(or
0
) from some theory
T
, with
SL
then, if there is a maximal proof in
CadL
∗
of a statement of
φ
∈
SL
and
Δ
⊂
Δ
∗
,
φ
,
η
Δ
⊂
Δ
∗
,then
the form
0 respectively). That is to
say, if
CadL
∗
concludes certainty about the occurrence of some event or about the
truth or falsity of some sentence then adding new evidence does not alter this cer-
tainty. Weak consistency is provided in
CadL
∗
and so in the inference mechanism
of CADIAG2 by the operator max
∗
defined over the ordering
, with
η
=
1(or
η
=
.
