Personalisation and Fuzzy Bayesian Nets

Rajendra Akerkar R. Akerkar 1

Personalisation - user p profile and p yp prototypesto 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 FusionFinal 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 graphsCluster over persons to get clusters for prototypes R. Akerkar 4

Data Base WEB answer fuzzy decision treeuser 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 filtclass 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 VariablesOffer LoanRate size company with conditional Rate age independencies Rate MedicalOffer Comp Su t Suit ReqSuit 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 loancar 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 averageCompany 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 nP (X1 , ..., Xn )   P (X i | PPr(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 toobtain degree of interest R. Akerkar 11

w1 w2 w3 w4 w5 w6 w7 X replaced with {wi}1 We can also fuzzify discrete0 sets XMass Assignment Theory provides:{Pr(wi | x)} where x is point value, interval or fuzzy set valuei.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 SCORERandom 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 4constant 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) = 0.2143For continuous fuzzy set we can derive a least prejudiced densityfunction 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 0.150.5 { , , {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

predictionoutput_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, 3input 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 NodePrior { } {fi} Instantiate : UpdatedDistribution {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 EntropyP MIN  ( ) Pr’(X) Prior Pr’(X) Distribution P(X) Ln Pr(X) on D satisfiedX = (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 / 0.2Pr(d1) = 0.5 Pr(d2) = 0.5 high = 5 / 0.8 + 6 / 1 08Pr(small | x) = Pr(high | y) = 0Notex and y can be point values, intervals or fuzzy sets R. Akerkar 20

Reduced Data BaseRed 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 algorithms12. Dependency Analysis algorithms Complexity (a) n2A. Node Ordering (b) n4B.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

Personalisation and Fuzzy Bayesian Nets

by Rajendra Akerkar

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