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Rajendra Akerkar




           R. Akerkar   1
Personalisation - user p
                       profile and p     yp
                                   prototypes
to provide: What is received is what is required
     EXAMPLES
     Web page retrieval to satisfy customer
     Computer Interface
     News Services Advertising
           Services,
     Regulation of messages -SMS. E-Mail
     Personalised Data Mining

        AI Machine Learning, Inference
         and Linguistics provides the
        intelligent agents for this
        personalisation service
           R. Akerkar                              2
message



  P1       P2                P3   P4    P5    prototypes



  s1       s2                s3   s4    s5   supports for interest


            Personalised Fusion
Final Support      s
                               send ?         to user
                R. Akerkar                                       3
y y
  Fuzzy Bayesian Nets

  Fuzzy Decision Trees
                                   Can we construct a theory
  Fril programs                    of prototypes ?

  Evidential Logic Rules
                                     Can be have a theory of
  Neural nets                        fusion ?

  Fuzzy Conceptual graphs

Cluster over persons to get clusters for prototypes
                R. Akerkar                                     4
Data Base

                                    WEB


 answer            fuzzy decision
                   tree
user query                      search agent



    R. Akerkar                                 5
Fril Evidential logic rule
               x1 is g1 with weight w1 through
               x2 is g w we g w
                   s g2 with weight w2 filt
class is f IFF
 l i                                     filter
               .       .      .      .     h
               xn is gn with weight wn

               For input {xi = fi}         f, gi, fi, h are fuzzy sets
               Let i = Pr(gi | fi)
                          (g      )
     class is f with probability 
     where
      =  h (w11 + w22 + … + wnn)

                R. Akerkar                                         6
Type             company                  Random Variables
Offer           LoanRate   size  company          with conditional
        Rate
                                   age            independencies

            Rate                        Medical
Offer                           Comp
            Su t
            Suit                         Req
Suit                             Suit
                                 S it
                                            Company Size :
Loan           Interest                     {large, medium, small}
                                            {l        di       ll}
Time
         Offer : {mortgage, personal loan, car loan, credit card
         car insurance, holiday insurance, payment protection,
         charge card, home insurance}
                   R. Akerkar                                      7
Offer : {mortgage, personal loan, car loan, credit card
        {mortgage            loan       loan
car insurance, holiday insurance, payment protection,
charge card, home insurance}
Type Rate : {fixed, variable}
LoanRate : {good, fair, bad}
Company Size : { g , medium, small}
     p y          {large,        ,       }
Company Age : {old, middle, young}
Loan Period : {long, medium, short}
Rate Suitable : {very, average, little}
                 {very average
Company Suitable : {very, average, little}
Medical req : {yes, no}
Offer Suit : {good, av, bad}
Interest : {high, medium, low}
            R. Akerkar                                    8
X1           X2                   X3

                                         We update this prior
                               X5        joint distribution with
       X4
                                         respect to any given
                                                tt        i
                                         variable instantiations

               X6
                         n
P (X1 , ..., Xn )   P (X i | P
Pr(X                  Pr(X Parents(Xi ))
                                 t (X            Prior
                                                   i
                         i1

            R. Akerkar                                             9
C0 = {Offer*, LoanPeriod*, OfferSuit*}
       C0    C1 = {OfferSuit,Offer, RateSuit}
             C2 = {RateSuit*, LoanRate*, Offer, TypeRate*}
             C3 = {CompSuit, MedReq,* OfferSuit, RateSuit
                  {CompSuit MedReq * OfferSuit RateSuit,
                   Interest*}
        C1   C4 = {CompSize*, AgeComp*, CompSuit*}


C2            C3
                           C4


     Compile Clique tree : equivalent to forming p
           p      q          q                  g prior j
                                                        joint
     distribution using an efficient message passing algorithm
                   R. Akerkar                                    10
Message :
        4% fixed rate long term mortgages available
        from 40 year old fairly large Company            FRIL

                  conceptual graph
                                              graph instantiations

            Offer = mortgage, Type rate = fixed, loan time = long
            CompSize = dist, CompAge = dist.

    Use message passing algorithm again to update clique distributions
    with graph instantiations.
Result will be distribution over interest variable - defuzzify to
obtain degree of interest
                       R. Akerkar                                    11
w1       w2   w3   w4     w5    w6     w7   X replaced with {wi}
1

                                                     We can also
                                                     fuzzify discrete
0                                                    sets
                        X
Mass Assignment Theory provides:
{Pr(wi | x)} where x is point value, interval or fuzzy set
                              value
i.e we translate value on X to distribution over {wi}

    Mass Assignment Theory provides:
    Defuzzification: if {i} is distribution over {wi} then
    x   i  i where i is expected value of wi
                 R. Akerkar                                         12
          i
medium
            small                                             large


        0           1       2   3    4                        5            6
                            DICE SCORE
Random Set of Voters            x       ≤3           4                5            6
% accept x as large             %          0         20               80           100
                    large = 4 / 0.2 + 5 / 0.8 + 6 / 1
   1        2           3   4          5         6       7        8            9   10
                                                                                                    {6} : 0.2,
    6        6          6   6           6        6        6       6            6       6   MAlarge ={5, 6} : 0.6,
    5        5          5   5           5        5        5       5                                 {4, 5, 6} : 0.2
    4        4

constant threshold : if voter accepts x and y>x then he must accept y
                                    R. Akerkar                                                                   13
weighted dice is small where small = 1 / 1 + 2 / 0.7 + 3 / 0.3
                                                         07       03
         prior for dice : 1:0.1, 2:0.1, 3:0.5, 4 :0.1, 5:0.1, 6:0.1
              MA      = {1} : 0.3, {1, 2} : 0.4, {1, 2, 3} : 0.3
                small
 Pr(dice is x | small) = probability that a randomly chosen voter
 chooses x as value of dice after being told dice value is small
  h               l    f di  f b i         ld di    l i       ll
   Least Prejudiced Probability Distribution for dice value =
   1 : 0 3 + 0.2 + 0.3(1/7) = 0.5429
       0.3 0 2 0 3(1/7) 0 5429
   2 : 0.2 + 0.3(1/7) = 0.2429        Pr(x | small)
   3 : 0.3(5/7) = 0.2143
For continuous fuzzy set we can derive a least prejudiced density
function whose expected value can be used for defuzzification
                   R. Akerkar                                       14
What is Pr(dice is about_2 | dice is small) where
                 (                              )
      about 2 = 1 / 0.5 + 2 / 1 + 3 / 0.5
                                  small
               0.3             0.4
                               04       0.3
                                        03
               {1}            {1, 2} {1, 2, 3}
   about_2                                Prior
                     1/2(0.2) 1/7(0.15)   1 : 0.1
                 0
     0 5 {2}
     0.5              =01
                        0.1     = 0.0214
                                  0 0214
                                          2 : 0.1
                                          3 : 0.5
                0.15   0.2       0.15
0.5 { , ,
    {1, 2, 3}                             4 : 0.1
                                          5 : 0.1
             Pr(about_2 | small) = 0.6214 6 : 0.1

     Point Semantic Unification used to determine Pr(f | g)
     where f is fuzzy set, g is point, interval or fuzzy set
                 R. Akerkar                                    15
prediction
output_sm                output_me          output_la   Means of
                                                        output_sm,
                                                        output_me,
                                                        output me
                                                        output_la
                                                        are
        mutually exclusive fuzzy sets on OUTPUT
          t ll      l i f          t                    1, 2, 3

input                                                    output_sm : 1
                      output {distribution over words}   output me : 2
                                                         output_me
            model
                                                         output_la : 3



               Prediction = 11 + 22 + 33           Defuzzification
                    R. Akerkar                                            16
Variable X - value x
                  V i bl           l
                  x is point or fuzzy set defined on R

               Bayes Node
Prior            { }
                 {fi}             Instantiate :       Updated
Distribution                      {fi : P (fi | x)}
                                        Pr(fi )}      Distribution
               NET


          Classical probability theory does not allow
          for this distribution update.
                                update

                     R. Akerkar                                      17
P’

                                         Updated
                                        Distribution
                                             o
                                             on
          Distribution Constraint      X = (X1…Xn)
          on Xk - D                                       Minimum Relative Entropy
P
                                            MIN
                                                                      (             )
                                                                           Pr’(X)
    Prior                                                    Pr’(X)
 Distribution                               P'(X)            Ln            Pr(X)
      on                                    D satisfied
X = (X1…Xn)

                   var 1 distribution     var 2 distribution
                Prior               Update               Update

                                   until convergence
                      R. Akerkar                                                     18
Bayes Net Compiled to give prior probability distributions.

         Update Compiled Net with
         input variables instantiated
         to probability distributions using
         relative entropy updating

    Bayes Net message passing algorithms for
    Clique tree updating modified to
    be equivalent to this relative entropy updating

            R. Akerkar                                        19
Fuzzify :
   A             B             C                                A : same as A above
                                                                B : {small, medium, large}
                                   A : {a1, a2, a3}             C : {low, av, high}
                                   B : [1, 10]
                                       [1                       D : same as above
                                   C : {1, 2, 3, 4, 5, 6}
                 D                 D : {d1, d2, d3}             small       medium       large

        A         B        C         D
        a1        x        y        {d1, d2}                      1                        10
 = Pr(medium | x)  = Pr(large | x)
                                                            low = 1 / 1 + 2 / 0.8 + 3 / 0.4
 = Pr(low | y)  = Pr(av | y)
                                                            av = 2 / 0.2 + 3 / 0.6 + 4 / 1 + 5 / 0.2
Pr(d1) = 0.5 Pr(d2) = 0.5
                                                            high = 5 / 0.8 + 6 / 1
                                                                       08
Pr(small | x) = Pr(high | y) = 0
Note
x and y can be point values, intervals or fuzzy sets
                           R. Akerkar                                                           20
Reduced Data Base
Red ced                                     Joint
                                            Probability
    A             B         C       D       Distribution

    a1       medium         low     d1      .5
                                             5
    a1       medium         low     d2      .5
    a1       large          low     d1      .5
    a1       large          low     d2      .5
    a1       medium         av      d1      .5
    a1       medium         av      d2      .5
    a1       large          av      d1      .5
    a1       large          av      d2      .5

         Repeat for all lines of database
           Calculate Pr(D | A, B, C)

            R. Akerkar                                     21
1. Search and Scoring based algorithms
1

2. Dependency Analysis algorithms            Complexity
                                             (a) n2
A. Node Ordering                             (b) n4
B.
B Without node ordering


      Querying
         Markov Cover :
         Parents of query node + children of query
         node + parents of these children
           R. Akerkar                                     22
((prototype p1 (offer Mortgage) ) (evlog disjunctive (
      (AgeOfCompany old_age) 0.1
      (CompanySize large_size) 0.15
      (LoanRate small rate) 0.4
                 small_rate)
      (TypeRate quite_fixed) 0.1
      (LoanPeriod long_period) 0.2
      (MedicalRequired bias_to_yes) 0.05))) : ((1 1) (0 0))

 ((prototype p1 (offer PersonalLoan) ) (evlog disjunctive (   Rules can also
    (AgeOfCompany old_age) 0.1                                be used for
    (
    (CompanySize large_size) 0.15
          p y         g      )                                fusion
    (LoanRate medium_rate) 0.4
    (TypeRate quite_variable) 0.1
    (LoanPeriod short_period) 0.2
    (MedicalRequired bi t
    (M di lR      i d bias_to_no) 0.05))) : ((1 1) (0 0))
                                  ) 0 05)))
ETC - rules for other offers and other prototypes
                  R. Akerkar                                             23

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Personalisation and Fuzzy Bayesian Nets

  • 1. Rajendra Akerkar R. Akerkar 1
  • 2. Personalisation - user p profile and p yp prototypes to provide: What is received is what is required EXAMPLES Web page retrieval to satisfy customer Computer Interface News Services Advertising Services, Regulation of messages -SMS. E-Mail Personalised Data Mining AI Machine Learning, Inference and Linguistics provides the intelligent agents for this personalisation service R. Akerkar 2
  • 3. message P1 P2 P3 P4 P5 prototypes s1 s2 s3 s4 s5 supports for interest Personalised Fusion Final Support s send ? to user R. Akerkar 3
  • 4. y y Fuzzy Bayesian Nets Fuzzy Decision Trees Can we construct a theory Fril programs of prototypes ? Evidential Logic Rules Can be have a theory of Neural nets fusion ? Fuzzy Conceptual graphs Cluster over persons to get clusters for prototypes R. Akerkar 4
  • 5. Data Base WEB answer fuzzy decision tree user query search agent R. Akerkar 5
  • 6. Fril Evidential logic rule x1 is g1 with weight w1 through x2 is g w we g w s g2 with weight w2 filt class is f IFF l i filter . . . . h xn is gn with weight wn For input {xi = fi} f, gi, fi, h are fuzzy sets Let i = Pr(gi | fi) (g ) class is f with probability  where  =  h (w11 + w22 + … + wnn) R. Akerkar 6
  • 7. Type company Random Variables Offer LoanRate size company with conditional Rate age independencies Rate Medical Offer Comp Su t Suit Req Suit Suit S it Company Size : Loan Interest {large, medium, small} {l di ll} Time Offer : {mortgage, personal loan, car loan, credit card car insurance, holiday insurance, payment protection, charge card, home insurance} R. Akerkar 7
  • 8. Offer : {mortgage, personal loan, car loan, credit card {mortgage loan loan car insurance, holiday insurance, payment protection, charge card, home insurance} Type Rate : {fixed, variable} LoanRate : {good, fair, bad} Company Size : { g , medium, small} p y {large, , } Company Age : {old, middle, young} Loan Period : {long, medium, short} Rate Suitable : {very, average, little} {very average Company Suitable : {very, average, little} Medical req : {yes, no} Offer Suit : {good, av, bad} Interest : {high, medium, low} R. Akerkar 8
  • 9. X1 X2 X3 We update this prior X5 joint distribution with X4 respect to any given tt i variable instantiations X6 n P (X1 , ..., Xn )   P (X i | P Pr(X Pr(X Parents(Xi )) t (X Prior i i1 R. Akerkar 9
  • 10. C0 = {Offer*, LoanPeriod*, OfferSuit*} C0 C1 = {OfferSuit,Offer, RateSuit} C2 = {RateSuit*, LoanRate*, Offer, TypeRate*} C3 = {CompSuit, MedReq,* OfferSuit, RateSuit {CompSuit MedReq * OfferSuit RateSuit, Interest*} C1 C4 = {CompSize*, AgeComp*, CompSuit*} C2 C3 C4 Compile Clique tree : equivalent to forming p p q q g prior j joint distribution using an efficient message passing algorithm R. Akerkar 10
  • 11. Message : 4% fixed rate long term mortgages available from 40 year old fairly large Company FRIL conceptual graph graph instantiations Offer = mortgage, Type rate = fixed, loan time = long CompSize = dist, CompAge = dist. Use message passing algorithm again to update clique distributions with graph instantiations. Result will be distribution over interest variable - defuzzify to obtain degree of interest R. Akerkar 11
  • 12. w1 w2 w3 w4 w5 w6 w7 X replaced with {wi} 1 We can also fuzzify discrete 0 sets X Mass Assignment Theory provides: {Pr(wi | x)} where x is point value, interval or fuzzy set value i.e we translate value on X to distribution over {wi} Mass Assignment Theory provides: Defuzzification: if {i} is distribution over {wi} then x   i  i where i is expected value of wi R. Akerkar 12 i
  • 13. medium small large 0 1 2 3 4 5 6 DICE SCORE Random Set of Voters x ≤3 4 5 6 % accept x as large % 0 20 80 100 large = 4 / 0.2 + 5 / 0.8 + 6 / 1 1 2 3 4 5 6 7 8 9 10 {6} : 0.2, 6 6 6 6 6 6 6 6 6 6 MAlarge ={5, 6} : 0.6, 5 5 5 5 5 5 5 5 {4, 5, 6} : 0.2 4 4 constant threshold : if voter accepts x and y>x then he must accept y R. Akerkar 13
  • 14. weighted dice is small where small = 1 / 1 + 2 / 0.7 + 3 / 0.3 07 03 prior for dice : 1:0.1, 2:0.1, 3:0.5, 4 :0.1, 5:0.1, 6:0.1 MA = {1} : 0.3, {1, 2} : 0.4, {1, 2, 3} : 0.3 small Pr(dice is x | small) = probability that a randomly chosen voter chooses x as value of dice after being told dice value is small h l f di f b i ld di l i ll Least Prejudiced Probability Distribution for dice value = 1 : 0 3 + 0.2 + 0.3(1/7) = 0.5429 0.3 0 2 0 3(1/7) 0 5429 2 : 0.2 + 0.3(1/7) = 0.2429 Pr(x | small) 3 : 0.3(5/7) = 0.2143 For continuous fuzzy set we can derive a least prejudiced density function whose expected value can be used for defuzzification R. Akerkar 14
  • 15. What is Pr(dice is about_2 | dice is small) where ( ) about 2 = 1 / 0.5 + 2 / 1 + 3 / 0.5 small 0.3 0.4 04 0.3 03 {1} {1, 2} {1, 2, 3} about_2 Prior 1/2(0.2) 1/7(0.15) 1 : 0.1 0 0 5 {2} 0.5 =01 0.1 = 0.0214 0 0214 2 : 0.1 3 : 0.5 0.15 0.2 0.15 0.5 { , , {1, 2, 3} 4 : 0.1 5 : 0.1 Pr(about_2 | small) = 0.6214 6 : 0.1 Point Semantic Unification used to determine Pr(f | g) where f is fuzzy set, g is point, interval or fuzzy set R. Akerkar 15
  • 16. prediction output_sm output_me output_la Means of output_sm, output_me, output me output_la are mutually exclusive fuzzy sets on OUTPUT t ll l i f t 1, 2, 3 input output_sm : 1 output {distribution over words} output me : 2 output_me model output_la : 3 Prediction = 11 + 22 + 33 Defuzzification R. Akerkar 16
  • 17. Variable X - value x V i bl l x is point or fuzzy set defined on R Bayes Node Prior { } {fi} Instantiate : Updated Distribution {fi : P (fi | x)} Pr(fi )} Distribution NET Classical probability theory does not allow for this distribution update. update R. Akerkar 17
  • 18. P’ Updated Distribution o on Distribution Constraint X = (X1…Xn) on Xk - D Minimum Relative Entropy P MIN  ( ) Pr’(X) Prior Pr’(X) Distribution P'(X) Ln Pr(X) on D satisfied X = (X1…Xn) var 1 distribution var 2 distribution Prior Update Update until convergence R. Akerkar 18
  • 19. Bayes Net Compiled to give prior probability distributions. Update Compiled Net with input variables instantiated to probability distributions using relative entropy updating Bayes Net message passing algorithms for Clique tree updating modified to be equivalent to this relative entropy updating R. Akerkar 19
  • 20. Fuzzify : A B C A : same as A above B : {small, medium, large} A : {a1, a2, a3} C : {low, av, high} B : [1, 10] [1 D : same as above C : {1, 2, 3, 4, 5, 6} D D : {d1, d2, d3} small medium large A B C D a1 x y {d1, d2} 1 10  = Pr(medium | x)  = Pr(large | x) low = 1 / 1 + 2 / 0.8 + 3 / 0.4  = Pr(low | y)  = Pr(av | y) av = 2 / 0.2 + 3 / 0.6 + 4 / 1 + 5 / 0.2 Pr(d1) = 0.5 Pr(d2) = 0.5 high = 5 / 0.8 + 6 / 1 08 Pr(small | x) = Pr(high | y) = 0 Note x and y can be point values, intervals or fuzzy sets R. Akerkar 20
  • 21. Reduced Data Base Red ced Joint Probability A B C D Distribution a1 medium low d1 .5 5 a1 medium low d2 .5 a1 large low d1 .5 a1 large low d2 .5 a1 medium av d1 .5 a1 medium av d2 .5 a1 large av d1 .5 a1 large av d2 .5 Repeat for all lines of database Calculate Pr(D | A, B, C) R. Akerkar 21
  • 22. 1. Search and Scoring based algorithms 1 2. Dependency Analysis algorithms Complexity (a) n2 A. Node Ordering (b) n4 B. B Without node ordering Querying Markov Cover : Parents of query node + children of query node + parents of these children R. Akerkar 22
  • 23. ((prototype p1 (offer Mortgage) ) (evlog disjunctive ( (AgeOfCompany old_age) 0.1 (CompanySize large_size) 0.15 (LoanRate small rate) 0.4 small_rate) (TypeRate quite_fixed) 0.1 (LoanPeriod long_period) 0.2 (MedicalRequired bias_to_yes) 0.05))) : ((1 1) (0 0)) ((prototype p1 (offer PersonalLoan) ) (evlog disjunctive ( Rules can also (AgeOfCompany old_age) 0.1 be used for ( (CompanySize large_size) 0.15 p y g ) fusion (LoanRate medium_rate) 0.4 (TypeRate quite_variable) 0.1 (LoanPeriod short_period) 0.2 (MedicalRequired bi t (M di lR i d bias_to_no) 0.05))) : ((1 1) (0 0)) ) 0 05))) ETC - rules for other offers and other prototypes R. Akerkar 23