On the Bayes formula for fuzzy probability measures

Fuzzy Sets and Systems 18 (1986) 183-185 North-Holland

183

SHORT COMMUNICATION

ON THE B A Y E S F O R M U L A F O R F U Z Z Y PROBABILITY MEASURES Krzysztof P I A S E C K I

Department of Mathematics, Academy of Economy, ul. Marchlewskiego 146/150, 60-967 Pozna~i, Poland Received January 1985 Revised April 1985 The fuzzy P-measure is presented here as the unique useful fuzzy probability measure satisfying the Bayes formula.

Keywords: Probability of fuzzy events, Bayes formula.

1. Preliminary notions L e t tr = {ix :/2 ~ [0, 1]} be a fuzzy g - a l g e b r a . If it does not contain the m a p p i n g !~j]a :/2 ~ {!2} then it is called a soft fuzzy g - a l g e b r a . K l e m e n t , L o w e n and Schwychla [1] define a fuzzy probability m e a s u r e on tr as the continuous f r o m below m a p p i n g m : tr ~ [0, 1] fulfilling the following p r o p e r ties: m (0,~) = 0, (1.1) m ( l a ) = 1,

(1.2)

m(Izvv)+m(p, Av)=m(tz)+m(v)

for each pair (/~, v) e tr 2.

(1.3)

T h e m a p p i n g c(. Iv, m):o"--> [0, 1] defined, for any fuzzy probability m e a s u r e m and f o r each v ~ o- such that re(v)~ 0, by the identity m (iI~A It)

m)=-toO,)

is called a conditional fuzzy probability given v. M o r e o v e r , by m e a n s of the fuzzy probability m e a s u r e m we can define a B a y e s fuzzy partition of O as the s e q u e n c e {t~m}~ tr r~ of fuzzy subsets satisfying the following conditions: • (R1) the fuzzy subsets t~, are pairwise W - s e p a r a t e d , i.e. ~k ~< 1 - ~ for e v e r y pair (k, l) of positive integers such that k ~ 1; (R2) re(sup, {/z,})= 1; 0165-0114/86/$3.50 t~) 1986, Elsevier Science Publishers B.V. (North-Holland)

184

K. Piasecki

(R3) m(~,,)>0 for each natural number n. Another approach to the probability of a fuzzy event is described in [2]. It is based on the notion of fuzzy P-measure on the soft fuzzy g-algebra 6. defined as the mapping p:6. ~ R+tA {0} fulfilling the following properties: (P1) for any tz e 6. we have p(t~ v ( 1 - ~)) = 1;

(P2) for any sequence {~.}e 6.N satisfying the condition (R1) we have p(sup{/xn}) = ~ P(I~). I1

Any fuzzy P-measure is a fuzzy probability measure as defined by Klement et al. Bayes fuzzy partitions containing uncrisp subsets exist for each fuzzy Pmeasure [2]. Furthermore, any fuzzy P-measure satisfies the Bayes formula expressed as follows: If {vq} and {/zr} are Bayes fuzzy partitions defined by the fuzzy P-measure p, then

c(vk I ~,, p) =

p(vD- c(~,l vk, p) Z. p(vq) • c(~, I ~q, p)

for any pair (~t, Uk) ~ 6 .2 [2]. On the other hand the Zadeh probability measure of fuzzy events [4] is a fuzzy probability measure that is not a fuzzy P-measure. All facts presented above induce the following problem. Can the Bayes formula be generalized for any fuzzy probability measure? The answer to this question is presented in the next section.

2. The main thesis

According to the identity (1.2) the one-element sequence {la} is a Bayes fuzzy partition for any fuzzy probability measure. In general, if we have m(/~)= 1 for the fuzzy probability measure m then {/z} is a Bayes fuzzy partition generated by the measure m. As we know, one-element Bayes partitions are not useful for the Bayes method of inference. We observe that the sequence {la} is the unique Bayes fuzzy partition defined by the Zadeh probability measure of fuzzy events for the fuzzy o--algebra o'c = ~cla :O---> {c}; c e[0, 1]}. So, a fuzzy probability measure generating a one-element Bayes fuzzy partition exists. Therefore, the following qualification is proposed. Definition :2.1. If a fuzzy probability measure can generate a two-element or more numerous Bayes fuzzy partition then it is called a useful fuzzy probability measure.

Bayes formula for fuzzy probability measures

185

It is easy to check that if a fuzzy probability measure is not useful then it satisfies the Bayes formula whereas for a useful fuzzy probability measure we have the ,following theorem.

Theorem 2.1. A useful fuzzy probability measure satisfies the Bayes formula iff it is a fuzzy P-measure. Proof. Let m be a useful fuzzy probability measure. IS it does not satisfy the Bayes formula then there exist Bayes partitions {vq}, {t~r} and a pair (k, l) of positive integers such that

,.

m(~)-c(t~ I_~,_m_)

This together with the definition of conditional fuzzy probability implies

m(~) ~ ~ mCm ^ ~,q). q

If {Vq} is a one-element sequence than the last condition does not hold. We have m ( ~ ^ v l ) = m(~) for each (~, vl)e o-2 such that m(vl)= 1 [3]. Therefore, from (R2) we get

q

q

~

q

We see that the fuzzy probability measure m does not fulfil the condition (P2) for the sequence {/xz^vq} satisfying (R1). So, the mapping m is not a fuzzy Pmeasure. This fact along with the Bayes formula expressed for any fuzzy Pmeasure puts an end to this proof. [] We have seen above that the fuzzy P-measure is the unique fuzzy probability measure useful for the Bayes method of inference.

Acknowledgments This paper is the result of work at the Seminar on Fuzzy and Interval Mathematics directed by Prof. Jerzy Albrycht.

References [1] E.P. Klemcnt, R. Lowen and W. Schwychla, Fuzzy probabilitymeasures, Fuzzy Sets and Systems 5 (1981) 21-30.

[2] K. Piasecki, Probabilityof fuzzyevents defined as denumerable addidvitymeasure, Fuzzy Sets and Systems 17 (1985) 271-284. [3] K. Piasecki, Probability space defined by means of the fuzzy relation 'lessthan', Fuzzy Sets and Systems, to appear. [4] L.A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl. 23 (1968) 421--427.