A draft analysis on the economy of a probabilistic expert system for risk avoidance

 

 

Carlo Pellacani

June 2001

Consider a world W where S or ØS is the case. In this world there is a system M with some input units and an output that can be A orØA.

So the situation can be represented as follow:

Alarm status	A	ØA
World Satatus
S	A,S	ØA,S
ØS	A, ØS	ØA, ØS

The table represents all the possible sates of the system (W,M).

S, being a status of W, will have an associated probability P(S)

P(ØS)=1-P(S).

If M is in A, it take an action, with a cost L.

Such action’s effect is avoiding a loss D

M’s competence is defined by:

P(A/S)

P(ØA/S)

P(A/ØS)

P(ØA/ØS)

As W is always = S or  ØS and M is always = A or ØA, the elements of the table are not independent.

We can evaluate the average performance of M in W.

 (D-L) P(A,S) – D P(ØA,S) – L P(A,ØS) + L P(ØA,ØS)

that can be rewritten as

(D-L) P(A/S) P(S) – D P(ØA/S) P(S) – L P(A/ØS) P(ØS) + L P(ØA/ØS) P(ØS)

As P(ØS) = 1-P(S)

The above equation becomes:

(D-L) P(A/S) P(S) – D P(ØA/S) P(S) - L P(A/ØS) (1- P(S)) + L P(ØA/ØS) (1- P(S))

[(D-L) P(A/S)  – D P(ØA/S)] P(S) +L P(A/ØS) - L P(ØA/ØS) +[ L P(A/ØS) - L P(ØA/ØS)] P(S)

[(D-L) P(A/S)  – D P(ØA/S) + L P(A/ØS) - L P(ØA/ØS)] P(S) + L P(A/ØS) - L P(ØA/ØS)

P(ØA/S) = 1– P(A/S)

P(A/ØS) = 1 – P(ØA/ØS)

[(D-L) P(A/S)  -D (1 – P(A/S)) + L (1 – P(ØA/ØS)) - L P(ØA/ØS)] P(S) + L P(A/ØS) – L P(ØA/ØS)

[(D-L) P(A/S) – D + D P(A/S) +L – L P(ØA/ØS)  - L P(ØA/ØS)] P(S)+ L P(A/ØS) – L P(ØA/ØS)

[ (2D-L) P(S/A) –2L P(ØA/ØS) –D + L]P(S) + L P(A/ØS) – L P(ØA/ØS)

[2D-L- 2L-D+L]P(S)= [D-L] P(S)–L

for a perfect knowledge.

To have a positive effect:

L<( P(S)/(1+P(S)) ) D

For low value of P(S), L must be very small, while for P(S)=1

L<D/2

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