Towards an Interpretation of the Medical Expert System CADIAG 2 - Fuzziness and Medicine - page 317

CADIAG2. We can identify the graded statement ( q , η ) with a probabilistic con-

κ , , η

κ ∈

ditional statement of the form q ,where SL is the evidence that supports

the presence of q in the patient.
Let us assume that the input of the system consists of

κ 1 , q 1 , η 1 , ..., κ n , q n , η n ,

for some q 1 , ..., q n basic medical entities and η 1 , ..., η n ∈ [ 0 , 1 ] . Under this view, the

set SL constitutes the initial evidence about the patient, which is

then propagated along the inference process by the application of the rules in KR bin .
Within our probabilistic interpretation the reflexivity and manipulation rules in
CadL adopt the following form, for input

Ω = { κ 1 , ..., κ n }⊂

Φ in T now formally given by the above

conditional statements:

κ , φ , η ∈ Φ
T κ , φ , η

( REF* )

R c θ , φ , η ∈ T κ , θ , ζ

( C* ) for ζ > 0

T κ , φ , min ( η , ζ )

R me θ , φ , 0 ∈ T κ , θ , 1

( ME* )

T κ , φ , 0

R ao

θ , φ , 1 ∈ T κ , φ , 0

( AO* )

T κ , θ , 0

Within this frame, final outputs of the form ( φ , η ) produced by the inference engine

Ω , φ , η

shall be interpreted as conditionals of the form (i.e., as the probability of

φ given all the medical evidence available about the patient). In order to make such

interpretation operative and formalize it we need to extend CadL by introducing
two new inference rules (the extended system will be denoted by CadL ∗ ). The first
of these rules formalizes the maximization process done by the system in order to
yield as output the set of medical entities (diagnoses) along with the maximal value
generated by it, with respect to the ordering :

T Δ 1 , φ , η T Δ 2 , φ , ζ
T ( Δ 1 ∪ Δ 2 ) , φ ,

( MAX )

max ∗ ( η , ζ )

for Δ 1 , Δ 2 ⊆ Ω .

An additional rule is necessary to produce the desired outcome:

T Δ , φ , η T κ , φ , ζ for all ζ ∈ [ 0 , 1 ]

( EX )

T κ ∧ Δ , φ , η

for .

This last rule, which we call EX as abbreviation of ' exhaustive ', simply states
that, if

Δ ⊂ Ω and κ ∈ Ω

κ is a piece of evidence that says nothing about the presence of φ in the

patient (i.e., that κ and φ are independent ) then the probability of φ given Δ should

stay the same if in addition we consider the piece of evidence κ (i.e., Δ ∪{ κ } ).