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A function P(|) defined on an arbitrary set of conditional events C is a coherent conditional probability if admits an extension on E H0, E Boolean algebra, H additive set , HE and H0 =H{F}, which is a conditional probability            Characterization    Theorem - Let be C an arbitrary family of conditional events , P a real function defined on C. The following two statements are equivalent: i) P is a coherent conditional probability on C; ii) for every finite subset FC, there exists a (unique) class Aa of sets of atoms, with Aa Aa for a > a , A0= A and a relevant class of (unconditional) probabilities {P a}, Pa: Aa [0,1] such that, for any E|H C, x = P(E|H) is solution of all the equations (*) Pa(EH)= x Pa(H) (with a such that H Aa) and there is a unique a such that, Pa(H)>0 (and so, for every EE, x=P(E|H) is the unique solution of (*)). Moreover an atom Ar belongs to Aa with a > 0 if and only if any H, with ArH, is such that Pa-1(H)=0 . z? -/,!       #   ?   !       +           Another fundamental result( $    Theorem - Let C be a family of conditional events and P a corresponding assessment; then there exists a (possibly not unique) coherent extension of P to an arbitrary family K C, if and only if P is coherent on C.p  tFFirst syntactic brick   Theorem - Let C be a family of conditional events {Ei|Hi} (iI), where card(I) is arbitrary and the events Hi's are a partition of W. Then any function f: C [0,1], such that f(Ei|Hi) = 0 if EiHi = f(Ei|Hi) = 1 if EiHi is a coherent conditional probability,2&%             %3  6  D           + V The bridge& .(first step),, +  X is a a random quantity with range CX Ax the event {X=x} j a related property Ej the event  You declare (that X is) j XZff# ff ffffff ffff (  X 0The bridge& .(first step)   So a coherent conditional probability P(Ej|Ax) is a measure of how much You, given the event Ax ={X=x}, are willing to claim the property j and it plays the role of the membership function of the fuzzy subset Ej* 'ffffff ffffff"ffffffff ffffffffffffff@'  1  z  0The bridge& .(first step)   =Is there any limitation for the values of the membership? 8=:=  0The bridge& .(first step)   Is the above definition able to represent a fuzzy subset as an actual generalization of a  crisp subset?2j?i  Second syntactic brick   0Theorem - Let C be a family of conditional events {E|Hi} (iI), where card(I) is arbitrary and the events Hi's are a partition of W and let P(|) be a coherent conditional probability such that P(E|Hi) {0,1}. Then the following two statements are equivalent (i) P(|) is the only coherent assessment on C ; (ii) Hi E = for every Hi H0 and EHi for every HiH1 where Hr={Hi : P(E| Hi)=r}, r=0,1.Z(      7 >         k  * The bridge& ...  Using this definition of fuzzy subsets, we can arrive in a very natural way to introduce the basic continuous T-norms and the relevant dual T-conorms, bound to the former by coherence. In fact, given a T-norm that in this framework singles-out the value of the conjunction P(Ej Ey |Ax Ay) then the corresponding choice of the T-conorm (which determines the value of the disjunction) is uniquely driven by the coherence of the relevant conditional probability (and the dual operation is what is actually obtained).ZZZ; <5A%4 v6     pD Fuzzy logic&       Ej* Ey* = {Ej Ey= Ejy, P(Ejy| Ax Ay)} Ej* Ey* = {Ej Ey= Ej y, P(Ej y| Ax Ay)} aZ6"  2    sE Fuzzy logic&       All the t-norms and t-conorms satisfying Frank equation S(x,y)= x + y-T(x,y) are coherent extensions of P to the new events.2:ZFZ(*  a  uGExample X =  income j =  high m(x)= P(Ej|Ax). Aggregating the Ax & .. B={X[a,b]}, (in general, B element of algebra spanned by the events Ax) f(B)= P(Ej|B) ?? Which limitations come out when looking for a coherent extension ??  ZjZ    9  G       T  Y  NCoherent Conditional Probability as uncertainty measure of conditioning eventsOON  Let E be an arbitrary event and P any coherent conditional probability on the family G= {E}{Ax}, x CX, admitting P(E|W)=1 as (coherent) extension. A (pointwise)-distribution of uncertainty measure on CX is the real function f defined by f(x)= P(E| Ax). Z$  4    V\  <   Z    An interesting information |What implies the constrain:  admitting P(E|W)=1 as (coherent) extension ? When CX is finite we necessarily have 1= P(E|W) maxx {P(E|Ax)} therefore P(E|Ax)=1 for at least one event Ax.(    ,   #               Third brick   bSo any (pointwise)-distribution of uncertainty measure on CX is also a possibility distributionfc$$4$$ $$*    O   Another brick  Theorem. Let C be a family of conditional events E|Hi (i I), where card(I) is arbitrary and the events Hi 's are a partition of W and let P(E|) an arbitrary (coherent) conditional probability on C . Then any coherent extension of P to C ={E|H :HHo}, where H is the algebra spanned by Hi , is such that, for every H, K Ho, with HK= min{P(E|H), P(E|K)} P(E|HK) max{P(E|H), P(E|K)}.% /    4         In particular&  HLet C be a family of conditional events E|Hi (i I), where card(I) is arbitrary and the events Hi's are a partition of W and let P(E|) any (coherent) conditional probability on C. Then any coherent extension of P to C ={E|H: HHo}, where H is the algebra spanned by Hi , is such that, for every H, K Ho with HK= P(E|HK) P(E|H)+P(E|K). (the equality holding only when at least one of the two terms is 0) 2Z0" Z8" ZE0" ZZ% /    +       A*a  ?  4Extend coherent conditional probability by using max55 4 &Can we choose in all situations exactly the maximum among the values? ???? For every H, K Ho, with HK= put P(E|HK) =max{P(E|H), P(E|K)} >sG    5From coherent conditional probability to possibility665 vTheorem. Let E be an arbitrary event and C be the family of conditional events E|Hi (i I), where card(I) is arbitrary and the events Hi's are a partition of W. Denote by H the algebra spanned by the Hi's and let f: C [0,1] be any function such that f(E|Hi) = 0 if EHi = f(E|Hi) = 1 if EHi Then any P extending f on K={E}Ho and such that P(E|HK) =max{P(E|H), P(E|K)} for every H, K Ho is a coherent conditional probability. Z.Z3ZZ<ZZ ( /     &        %O9   ' 0. 3  *  0Toward the main result &  VIs the function f(H)= P(E|H), with P a coherent conditional probability, a capacity? 0UV WSurprisingly & . Theorem: Let C be a family of conditional events E|Hi (i I), where card(I) is arbitrary and the events Hi 's are a partition of W and let P(E|) an arbitrary (coherent) conditional probability on C. A coherent extension of P to C ={E|H: HHo}, is monotone with respect to if and only if for every H, K Ho P(E|HK) = max{P(E|H), P(E|K)}. h@ 2 % /    4  V   <+  %Complete the first part of the bridge&&% &The above results assure (in our context) that any possibility measure can be obtained as coherent extension (unique, in the finite case) of a possibility distribution P(E|Ax). Vice versa given any possibility measure P on an algebra H, there exists an event E and a coherent conditional probability P on K={E}Ho agreeing with P, i.e. whose extension to {E}H (putting P(E|)=0) coincides with PZ    )    )Information measure: a general definition** J   Hrecall the definition of information measure as given by Kamp de Feriet and Forte: A function I from an arbitrary set of events E, and with values on R* =[0,+], is an information measure if it is antimonotone, that is the following condition holds: for every A, B on E A B I(B) I(A). TZZZU !  <9       9    Information measure&  lSo, if both and W belong to E, it follows that 0 I(W)= inf_AE I(A) sup AE I(A)= I() Kamp de Feriet claims that: the above inequality is necessary and sufficient to build up an information theory; nevertheless, only to attribute an universal value to I(W) and I() the further conditions I()= + and I(W)= 0 are given. The choice of these two values is obviously aimed at reconciling with the Wiener-Shannon theory. 2Z,ZZZoZ  33 f                 U 8          4 , Information measure&  In general, axiom A B I(B) I(A) implies only that I(A B) min {I(A), I(B)}; we can specify the rule of composition by introducing a binary operation to compute I(A B) by means of I(A) and I(B). J          ^  H -decomposable information measures(% $ 8   :Definition - An information measure defined on an additive set of events A is -decomposable if there exists a binary operation from [0, +] [0, + ] such that, for every A, B A with A B = we have I(A B)= I(A) I(B). So ``min'' is only one of the possible choices of . 47 <$            6 g-transformations Obviously, every function defined on an algebra of events with values on [0, a] and with I(W)=0, satisfying I(A B)= min {I(A), I(B)} can be regarded as an increasing transformation g of an information measure. oN M2\        C     Let F be an arbitrary event and P any coherent conditional probability on the family G= {F}{Ax}, x CX, admitting P(F|W) = 0 as (coherent) extension. A (pointwise) information measure on CX is the real function y defined by y(x)= P(E| Ax). $  $    V\  >   J    An interesting information 8   What implies the constrain:  admitting P(E|W) = 0 as (coherent) extension ? When CX is finite we necessarily have 0 = P(E|W) minx {P(E|Ax)} therefore P(E|Ax) = 0 for at least one event Ax. &'   .                  *Less& surprisingly... (   Theorem: Let C be a family of conditional events E|Hi (i I), where card(I) is arbitrary and the events Hi 's are a partition of W and let P(E|) an arbitrary (coherent) conditional probability on C. A coherent extension of P to C ={E|H: HHo}, is antimonotone with respect to if and only if for every H, K Ho P(E|HK) = min{P(E|H), P(E|K)}. y 2 % /    4   $V    @+  Extending c.c.p. by using min 8   tTheorem. Let E be an arbitrary event and C be the family of conditional events E|Hi (i I), where card(I) is arbitrary and the events Hi's are a partition of W. Denote by H the algebra spanned by the Hi's and let f: C [0,1] be any function such that f(E|Hi) = 0 if EHi = f(E|Hi) = 1 if EHi Then any P extending f on K={E}Ho and such that P(E|HK) =min{P(E|H), P(E|K)} for every H, K Ho is a coherent conditional probability. lZ.Z3ZZ<Z ( /     &        %O9   ' 0. 3  * Complete the bridge$ Theorem - Let E be an arbitrary event and C be the family of conditional events E|Hi (i I), where card(I) is arbitrary and the events Hi's are a partition of W. Denote by H the algebra spanned by the Hi's and let f: C [0,1] be any function such that f(E|Hi) = 0 if EHi = f(E|Hi) = 1 if EHi Then any P extending f on K={E}Ho is an increasing transformation of an information measure if and only it is an increasing transformation of a min-decomposable information measure. Z.ZZZZ ( /     &       ;   F9   ' /-  # g ' qCBuild the second bridge    T(KEY IDEA 0  y In the computational theory of perceptions, perceptions are dealt with through their descriptions in a natural language xz z U) CONTINUED ,  xVersion 2. Perception-based You are allowed n seconds to look at the box. n seconds is not enough to allow you to count the balls You describe your perceptions in a natural language p1: there are about 20 balls p2: most are black p3: there are several times as many black balls as white balls PT s solution?h=Z   N = V*A SERIOUS LIMITATION OF PT  W+ CONTINUED ,  gIn PT, there are no tools for dealing with semantics of natural languages What is the meaning of: There are several times as many small balls as large balls? PT is not equipped to operate on perception-based information expressed in a natural language PT has a limited capability to deal with world knowledge, since much of world knowledge is perception-basedhZh h vH Our thesis     EZadeh is right: the classical PT is not able to manage these problems2F % A ^@ Recall&      X is a a random quantity with range CX Ax the event {X=x} j a related property Ej the event  You declare (that X is) j XZff# ff ffffff ffff (  V X,A useful model  Y-Z.A useful model  [/A useful model  _0Trivial computation  `1Complicate a bit  xKThe ingredients   (Y random quantity  color CY={c1,& ,cm} Bi= {Y= cj} We have P(EL|Ai) for every si We have P(ED|Bj) for every cj We have P(Ai) , P(Bj) 5 2_  2            x1  "         wI   We can compute P(ELD)= Si P(ELD|AiBj) P(Ai Bj) = By using one of the  coherent t-norm and to compute P(Ai Bj) w 2 2$$$$$$$$$$$$$$$$$$$$9$$$$$$$L$     A   d5Unknown composition   e6semplify   f7*A  not fuzzy problem$. g8*A  not fuzzy problem$,`We need to introduce a new random variable : P =  percentage of balls with diameter d1 or d2 CP={p1 +p2} In fact we consider only the urns such that 7/10 p1 +p2 9/10H&    6  r     6   h9Unknown composition  yJ Ingredients    A new random variable P random variable of possible values of percentage of big balls (percentage of big balls is the probability of the event EL) V>;   i: Ingredients    A new  fuzzy subset Aa = {P=a} EH = You declare that (la percentage) is high. P(EH| Aa) (membership function)u, u j;#The urn again (unknown composition)$   $ k< Ingredients    The range of P CP ={a1,& , ar } (different values of P(FL| Hk) Br = { P= ar } P(EHL) =SrP(EH|Br)P(Br) = SrmHL(ar )P(Br) \}           6<    # l= Last brick   < EL|EHL P(EL|EHL) Tools: disintegration and Bayes theorem=      ! /   m> the last brick&       P(EDEL|A)= SrP(EDEL |Hk)P(Hk|A ) P(Hk|A ) by Bayes formula by using P(A| Hk ) taking into account that P(A| Hk )=P(ElD|Hk)P(EHL|Hk) Z                    #          n?" Last brick##  #  P(EL|EHL)= SrP(EL |Hk)P(Hk|EHL) P(Hk|EHL ) by Bayes using P(EHL| Hk ) L                       /IRSr {  ` ^WN̙f` ^WNff3` 999MMM` ^WNf` ff33` ff33` ^WN̙f` ^WNf3` ^WNf3>?" dd@(?zdd@ y x@" `  n?" dd@   @@``@n?" dd@  @@``PR   @ ` ` p>>  S( @     s *t" `  m9Fare clic per modificare lo stile del titolo dello schema: :-  c $"  @  uFare clic per modificare gli stili del testo dello schema Secondo livello Terzo livello Quarto livello Quinto livello: v  s *4# Tt  a* 2    s *# Tt   c*(2    s *# T@t  c*(2  j  BA paint<@.Z  B޽h))? ? ^WNf3 2Pittura contemporanea.pot  ~P(    s *  @  m9Fare clic per modificare lo stile del titolo dello schema: :  c $  @   r>Fare clic per modificare lo stile del sottotitolo dello schema? ?  s *d T  a* 2^WN    s * T   c*(2^WN    s *$ T@  c*(2^WN  j  BA paint@rZ  B޽h))? ? ^WNf3K 0 0[(  H   0d @9   [*     0 z 9  ]*   d  c $ ?Ay  ;  0$  ;  uFare clic per modificare gli stili del testo dello schema Secondo livello Terzo livello Quarto livello Quinto livello: v   6 4@m   [*     6d 4z m  ]*   H  09g ? ̙3380___PPT10.Upt @w(  r  S @    c $AT ??G8 $D 01  c $D  ,$D 0 # Giulianella Coletti Praga 2005 B$     H  0޽h ? ̙33y___PPT10Y+D=' = @B +q  /'` ( T     c $$ `<$D 0      0@ k ,$D 0 8 Perception-based probabilistic reasoning $9 2ff& 9    0D0 ,$D 0 coherent probability@ 2      0  ,$D 0 W conditional 2      0@,$D 0 UFuzzy set theory 2  l   C $  @      ` >?P I 7  N&(" 2  X   0A ?/ H   0޽h ? ^WNff3___PPT10.+ D' = @B Du' = @BA?%,( < +O%,( < +D' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ ++0+ +  mep((  ( ( c $D `<$D 0    ( 0@ ,$D 0 j&Measure of information and possibility' 2' ' $ ( 00 ,$D 0 Coherent probability@ 2   ( 0d  ,$D 0 W conditional 2    ( 0$ ,$D 0 Z Fuzzy subets 2    ( S ~0e0eA  0 <$D 0  H ( 0޽h ? ^WNff3  zD(  D D c $D `<$D 0    D 0@ ,$D 0 'Measure of information and possibility( 2(  $ D 0ģ0 ,$D 0 Coherent probability@ 2   D 0  ,$D 0 W conditional 2    D 0D ,$D 0 Z Fuzzy subets 2    D S ~0e0eA  0 <$D 0  H D 0޽h ? ^WNff3  *( w x  c $ `   r  S d  @  H  0޽h ? ^WNf3y  0(  x  c $ p   x  c $  @  H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +[  z(  ~  s * `     0$ ? `   `  c $AT ??$ g  `  c $AT ??$ g  H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +   h$(  hr h S  @   r h S   @  H h 0޽h ? ^WNf3y  00( w x  c $t `   x  c $4u  @  H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +  D<@t( @| tr t S tw `   r t S w  @0   : t 3 A 00f t 04xP D,$D 0 A fuzzy subset Ej* of CX is the pair Ej* ={Ej ,, P(Ej|Ax)} Where P(Ej|Ax) is a coherent conditional probabilityuZ (B9    (  H t 0޽h ? ^WNf3  Px$( @| xr x S y    r x S z  @  H x 0޽h ? ^WNf3  6.`|(  | | S z `<$D 0    | S 4{  @<$D 0  B | 0{P p,$D 0 NO ! By using previous result V(z 2(((  H | 0޽h ? ^WNf3  p!(    S T| `<$D 0     S |  @<$D 0    0} ,$D 0 Q Is it possible to obtain crisp subsets of CX as a specific particular case? ~/ z" z 2    $ Q  H  0޽h ? ^WNf3  $( `l r  S } `   r  S 4~  @  H  0޽h ? ^WNf3&  f(  r  S ~ p `   r  S T  @  :  3 AH  0޽h ? ^WNf3   UM(  r  S  `   r  S   @0   :  3 A 00  0P  ,$ 0 G Z     Zt >?a   By probability rules we necessarily have : P(Ejy| Ax Ay) = P(Ej| Ax Ay) + P(Ey| Ax Ay) - P(Ej y| Ax Ay)4s(" 2(+                      H  0޽h ? ^WNf3___PPT10+{GD' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+  +   +(  7 r  S  `   r  S  @   :  3 A 00  0TP  ,$ 0 G Z     Z >?  PRIn particular: max and min probabilistic sum and product L t-norm and L t-conorm S " 2 K  H  0޽h ? ^WNf3___PPT10+{GD' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+  +  $( w r  S t `   r  S   @  H  0޽h ? ^WNf3  $(  r  S  `   r  S   @  H  0޽h ? ^WNf3  $( X|\P r  S  `   r  S 4  @  H  0޽h ? ^WNf3  $( uX|\P r  S  `   r  S T  @  H  0޽h ? ^WNf3  $( lhpuX| r  S   `   r  S t  `  H  0޽h ? ^WNf3  $( D^lhpu r  S   `   r  S 4   @  H  0޽h ? ^WNf31   q(    S   `<$D 0     S    @0 <$D 0    0T  @ -,$D 0 YYES(2  H  0޽h ? ^WNf3  0$( =YD^l r  S 4 `   r  S   @  H  0޽h ? ^WNf3#  @c( |=YD^   S  `<$D 0     S   @ <$D 0  7  0Ԛ  ,$D 0  In general, NO V z 2    04 @,$D 0 dThe following theorem focuses the (surprising) condition that assures the monotonicity of P(E|). ld z 2% 1"c H  0޽h ? ^WNf3  P$( D=Y r  S  `   r  S   @  H  0޽h ? ^WNf3  `$( Dlm= r  S  `   r  S   @  H  0޽h ? ^WNf3  p$( Dlmt r  S ԝ `   r  S 4  @  H  0޽h ? ^WNf3  $( Dlmt r  S  `   r  S t  @  H  0޽h ? ^WNf3  0( Dlmt x  c $4 `   x  c $  @  H  0޽h ? ^WNf3  $( Dlmt r  S  `   r  S T  @  H  0޽h ? ^WNf3  $(  r  S  `   r  S   @  H  0޽h ? ^WNf3  $( X|\P r  S ԣ `   r  S   @  H  0޽h ? ^WNf3  0( Dlmt x  c $4 `   x  c $  @  H  0޽h ? ^WNf3  $( Dlmt r  S  `   r  S   @  H  0޽h ? ^WNf3  $( Dlmt r  S  `   r  S 4  @  H  0޽h ? ^WNf3&  f( Dlmt r  S  ``   r  S T  @  :  3 A0 H  0޽h ? ^WNf3w   5- ( 8II   c $ `<$D 0     0@ k ,$D 0 8 Perception-based probabilistic reasoning $9 2ff& 9   00 ,$D 0 coherent probability@ 2     0  ,$D 0 W conditional 2     0T @,$D 0 UFuzzy set theory 2  r  S    @     `! >?P I 7  N&(" 2  X   0A ?/ H  0޽h ? ^WNff3___PPT10.+ D' = @B Du' = @BA?%,( < +O%,( < +D' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ ++0+ +   0(  0 0  6!     0  f4"vd @ @  H 0 0޽h ? 3fffy___PPT10Y+D=' = @B +  4~( O; 4 4  0$ e    4  fvd @ @  H 4 0޽h ? ̙33y___PPT10Y+D=' = @B +   0 8e( O; 8 8  64Y m   Z 8 0Yo\  Version 4 points to a serious short coming of PT In PT there is no concept of cardinality of a fuzzy set How many large balls are in the box?4" "   j 8 B>  = 2 8 BYo |   U0.5$"    8 HZvd @om ? u% There is no underlying randomness "& " 2& & 2 8 B[o \   U0.8$"   2 8 Bt[o ;[  U0.6$"   2  8 B[o )  U0.9$"   2  8 B4\o  U0.4$"    2  8 B\o 2  U0.9$"   H 8 0޽h ? 3fffy___PPT10Y+D=' = @B +  @<( w < <  6]     < # l^vd @   H < 0޽h ? 3fffy___PPT10Y+D=' = @B +   ZRP(  r  S ` `   r  S a  @     0taT ,$D 0 *The GENERAL theory of coherent conditional probabilities, not only can, but can  in a natural way 6c 2Q   d H  0޽h ? ^WNf3___PPT10+D' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+ +   NF`( <- x  c $b `   x  c $b  @0   :  3 A 00d  0TcP D,$D 0 A fuzzy subset Ej* of CX is the pair Ej* ={Ej ,, P(Ej|Ax)} Where P(Ej|Ax) is a coherent conditional probabilityuZ (@9    (  H  0޽h ? ^WNf3___PPT10+x:D' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4+8+0+4  +   aYpH( <- H H c $d  <$D 0    H 0@  @ff  R2 H s *p@ 0X2 H 0 0 X2 H 0 @P X2 H 0P `  X2 H 0  @ X2  H 0 0 X2  H 0` ` X2  H 0` ` X2  H 0  X2  H 00  K H 0@   {Consider an experiment consisting on drawing a ball by the urn PROBLEM: which is the probability to obtain a big ball?B|C & | X2 H 0 H H 0޽h ? ^WNf31)___PPT10 .+ȇD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8+8+0+8  + LT( <- L L 0ĩT Z In this case 2 (  Z L 0$/   X is the  size of the ball CX ={s1,& ,sk} Ai the event {X= si} L (j)  big EL the event  You declare that the size is big jZ 2ff  ff ffff ffff$T        H  H L 0޽h ? ^WNf3y___PPT10Y+D=' = @B +C  PJ( <- P P S   <$D 0    P 0$@  @ff  R2 P s *p@ 0X2 P 0 0 X2 P 0 @P X2 P 0P `  X2 P 0  @ X2  P 0 0 X2  P 0` ` X2  P 0` ` X2  P 0  X2  P 00   P 0@   JjConsider an experiment consisting on drawing a ball by the urn and let be EL the event (referred to the drawn ball) EL=  you declare that (the size of the drawn ball) is big L *    X2 P 0 H P 0޽h ? ^WNf31)___PPT10 .+ȇD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+  +    T( <- T T S D  <$D 0    T 0@  @ff  R2 T s *p@ 0X2 T 0 0 X2 T 0 @P X2 T 0P `  X2 T 0  @ X2  T 0 0 X2  T 0` ` X2  T 0` ` X2  T 0  X2  T 00  X2 T 0  T 0d]6   What is the probability of EL?N 2      H T 0޽h ? ^WNf31)___PPT10 .+ȇD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+  +   meX(  Xr X S d `    X 0r  We have P(EL|Ai) for every si We have P(Ai) We can compute P(EL)= Si P(EL|Ai) P(Ai) C 2(2 2        = W X 0$C c<n,$D 0 KFor us : PROBABILITY OF AN EVENT For Zadeh: PROBABILITY OF A FUZZY EVENT 4L 2 ! &  ! H X 0޽h ? ^WNf3___PPT10+D' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+  +_  \f( <- \ \ S D  <$D 0    \ 0@  @ff  R2 \ s *p@ 02 \ 0d 0  @  R2 \ s * @P X2 \ 0P `  R2 \ s *33  @ R2  \ s *f 0 X2  \ 0` ` 2  \ 0$` `  @f  R2  \ s *f  X2  \ 00  X2 \ 0 T \ 0` L  ELD=  you declare that (the size of the drawn ball) is big and dark  :G 2@ff G  \ 0DO ELD = EL EDL 2  H \ 0޽h ? ^WNf31)___PPT10 .+ȇD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+  +m  p$( w r  S t5< c   r  S 5<  @  H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +m  $( w r  S  `   r  S D  @  H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +  PH l (  l l S   <$D 0    l 0d@  @ff  R2 l s *p@ 02 l 0 0  @  R2 l s * @P 2 l 0$fP `   <  R2 l s *33  @ R2  l s *f 0 X2  l 0` ` 2  l 0` `  @f  R2  l s *f  X2  l 00  2 l 0ff  @   l 0D  X Probability to draw a big and dark ball - 2, - X2 l 0 . X2 l 00 2 R2 l s * >s R2 l s *+ X2 l 0 Qr R2 l s *f X2 l 0_ y X2 l 0 i _ X2 l 0 4 R2 l s *33B 7 X2 l 0X2 l 0eF^=X2 l 0v h l T >?   zIn the hypothesis that most balls are big and few are dark 4= 2(" 2>= H l 0޽h ? ^WNf31)___PPT10 .+ȇD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+  +  91p ( <- p p c $  <$D 0    p 0d@  @ff  R2 p s *p@ 02 p 0 0  @  R2 p s * @P 2 p 0$fP `   <  R2 p s *33  @ R2  p s *f 0 X2  p 0` ` 2  p 0` `  @f  R2  p s *f  X2  p 00  2 p 0ff  @  B p 0D    Probability of drawing a big ball in the hypothesis that most balls are big Q 2Q Q X2 p 0 . X2 p 00 2 R2 p s * >s R2 p s *+ X2 p 0 Qr R2 p s *f X2 p 0_ y X2 p 0 i _ X2 p 0 4 R2 p s *33B 7 X2 p 0X2 p 0eF^=X2 p 0v   p T >?w    P(EL| ??)d (" 2 H p 0޽h ? ^WNf31)___PPT10 .+ȇD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0+8+0+0  +   t ( <- t t c $  <$D 0    t 0$@  @ff  R2 t s *p@ 02 t 0 0  @  R2 t s * @P 2 t 0fP `   <  R2 t s *33  @ R2  t s *f 0 X2  t 0` ` 2  t 0D` `  @f  R2  t s *f  X2  t 00  2 t 0䠤ff  @  $ t 0D    Probability to draw a ball with diameter d1 in the hypothesis that the percentage of balls with diameter d1 or d2 is between 70% and 90%  x 2,'    X2 t 0 . X2 t 00 2 R2 t s * >s R2 t s *+ X2 t 0 Qr R2 t s *f X2 t 0_ y X2 t 0 i _ X2 t 0 4 R2 t s *33B 7 X2 t 0X2 t 0eF^=X2 t 0v H t 0޽h ? ^WNf31)___PPT10 .+ȇD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,+8+0+,  +  x0( <- xx x c $ `   x x c $d  @  H x 0޽h ? ^WNf3___PPT10i.0^+D=' = @B +=   |D( <- | | c $䣤  <$D 0    | 0D@  @ff  R2 | s *g7 ' | 0`  , Most (balls) are big ( 2ff  t | 0d e  P=  percentage of balls such that you declare big  CP= {??????} dB 23   B H | 0޽h ? ^WNf31)___PPT10 .+ȇD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<+8+0+<  +m  P$( w r  S  `   r  S t  @  H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +m  $( <- r  S $ `   r  S   @  H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +k  r( <-   S   <$D 0     0d@  @ff  R2  s *g7 '"  0Ĩu  bHk (k=1,& ., r) possible compositions of the urn&2 2. -   0$   For every Hk we compute ak= P(EL| Hk ) the value of P in the hypothesis HkN 2     L     $   L  0  P(EL| Hk ) = Si P(EL|Ai Hk) P(Ai | Hk ) = Si P(EL|Ai) P(Ai | Hk ) C 2     b          H  0޽h ? ^WNf31)___PPT10 .+ȇD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+  +m  $( <- r  S D `   r  S   @  H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +   6( <- r  S $ `   r  S   @    64W ? ~ ) D 2  X  0A ?v{H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +  6.0( `@ r  S X `   r  S Y  @  :  3 A 00X  0A ?v{H  0޽h ? ^WNf3y___PPT10Y+D=' = @B +  @( <- r  S 4Z `   r  S Z d*J  X  0A ?v{H  0޽h ? ^WNf3y___PPT10Y+D=' = @B + 0 ( TP X  C Bx     S =< ;     H  09g ? ̙33 0 H*(  H^ H S Ay    H c $6< :     H H 09g ? ̙33& 0 $*( w~̃ O $^ $ S Bx    $ c $T7< ;     H $ 09g ? ̙33' 0 ,*(  ,^ , S Ay    , c $6< :     H , 09g ? ̙33C 0 *(  ^  S Bx     c $t8< ;     H  09g ? ̙33 xU=OA}|HbAa, Ƙ&Q~h boLoXi&G;+BA ʻ3;ovfxz_^0ZtL uT>icmi# Dw<?p7׭[펹͞CqZ3k*~~qW5{=<4mA% 2ť7\z*{ZVn3N]UDAZJ tl >g7PkRp)y˷t4q@j,*5K)7Wg;f7B"Mb{?߭S+P2Jg٧t7 IؑQ>?:(RPmu8[Ιh,1 qK5 xU=OA}|(JBAa, Ɛ(&Q~h boL +m,ZmVٽ=P;c.{۹g `.tSCbJn1F 41Cm3 b~͋~kVٍvv\BfO85~0W(C;w$~ELv}I͹|M$ca8뫚Խ^|16XFҌFY-bz*W+d<-nul $>g7PkyRp.|9ʷt4q@j,,5)k?7Xu%1b擝3OИchNi|K xU=OA} A"#X\a,Ƙ(&~DQNM +iJmJ [9gBaʻ3;ovfhсiвxtF,lvv 4|`=5ȢL8l_b=W@Jv罶ͳq|%BJnOY80 7~Cpg)]9ֱUT1W< }"׏8p~u'ZG>A]߃L Nu~^}jmlP+Yes238-ͭu IM]T+^a8ukyRPkQJ6UFH@k2"5RuL~q3B3G+/b٥)I1cPQSrTepc桏7 ΎZ%eXbWW6ݛO~KcԏOifCMHrhPa>HUH Nq@h{ˉ8ԓN7#ϿwcOG3GPVtX) p&s*e8 >UM]@Zfd{VpQM  x*v,5 jyb Oh+'0d px( LX x   0Measure of information through a ``likelihood''Prof. Giulianella ColettiouJC:\Programmi\Microsoft Office\Modelli\Strutture\Pittura contemporanea.potuProf. Giulianella Colettiic42fMicrosoft PowerPointett@ q@q׿@AsH@I GboM  s1'& &&#TNPP0D & TNPP &&TNPP    & "--- !- "----- !---&&J& - &Jv& &_C0(``(0  y?xB@???? 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