The Fuzzy-Bayes Decision Rule

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1 Academic Web Journal of Business Management Volume 1 issue 1 pp December, Accepted 18 th November, 2016 Research paper The Fuzzy-Bayes Decision Rule Houju Hori Jr. and Yukio Matsumoto The chief in Nara Community, Tsubakikishi Shine The Visiting Research Fellow, the Institute of Mathematical Statistics. 655, Yuzaki, Kawanishi-cho, Shiki-gun, , Nara, Japan. *Corresponding author. uemura0742@yahoo.co.jp. ABSTRACT We first showed that in the no-data problem, if the subjective distribution is OR-connective or ANDconnective, the fuzzy mathematics extension principle is then equivalent to the OR-connective operation or the AND-connective operation of utility function theory (loss function theory) and fuzzy mathematics is applicable in utility function theory (loss function theory). Next, for cases involving observed information (data), we identified the fuzzy events in the observed information as a normal type-2 fuzzy set by the fuzzy polynomial regression model and further assumed that the utility function (loss function) is identified as a type-2 fuzzy set under subjectivity of decision-maker in an infinite lottery for risk tolerance and risk aversion. We formulated (fuzzy) two-dimensional liner problem in risk tolerance and risk aversion having two fuzzy goals by obtaining vagueness on a fuzzy event, introducing the α-level cut concept. Lastly, in our decision rule, we selected the max-product of the possibility measure and inevitability measure operations and derived the utility function (loss function) for the fuzzy events by multiplication with the subjective possibility distribution. The method may therefore be considered a variational method that incorporates the utility function (loss function) instead of individually varying the subjective distribution or the utility function (loss function) itself. Key words: Extension principle, fuzzy event, max-product, type-2 fuzzy set, utility function (loss function), variational method. INTRODUCTION Tanaka et al. (1979) specified fuzzy environments in states of nature by fuzzy events and applied fuzzy event probability as defined by Zadeh (1968) to propose the calculation of utility function and fuzzy-event membership function expected values as fuzzy utility function values, multiplication of fuzzy event probability and fuzzy utility value to calculate fuzzy expected utility and the selection of the action that maximizes the fuzzy expected utility as the optimum action. Uemura (1991) noted that fuzzy events are events that emerge from the mapping and transforming of a state of nature by the fuzzy-event membership function and formulated the decision rule for fuzzy events which applies only to no-data problems in which the subjective distribution is OR-connective. In the present paper, however, we extended the discussion on the decision problem to cases where the fuzzy-bayes decision rule and the subjective distribution under observed information are AND-connective. Uemura (1995) also showed that if the subjective distribution and the fuzzy-event membership function are both bell-shaped distributions, then application of the max-product method proposed by Tanaka et al. (1979) when the decision-maker is risk-neutral results in a reversion from the decision rule to the normal distribution theory of Bayesian statistics and thus showed that the

2 Academic Web Journal of Business Management; Hori and Matsumoto; 002 formulation of Tanaka et al. (1979) is effective and valid for the case of a risk-neutral decision-maker. Uemura (2001) then extended the discussion to multi-dimensional states, which conflicts with the actual state of nature and constructed a new game theory. Uemura (2011) went on to represent the transition of the multi-dimensionalized natural state in a fuzzy transition matrix and expressed the revival-inversion-annihilation cycle in a Markov Monte Carlo simulation model of fuzzy events. This was developed into the type-2 fuzzy transition matrix by Hori and Tsubaki (2014), who introduced the α-level cut concept and following interval-type inference formulation used representative curves at α-level 1 to construct the Monte Carlo simulation for fuzzy events in the riskneutral case. In the present study, we identified the fuzzy events in observed data as a type-2 fuzzy set and applied fuzzy polynomial regression analysis as proposed by Tanaka et al. (1982). We then showed that taking the two fuzzy event possibility measures obtained at the maximum and minimum of the α-level cut representative sections, with the maximum representing the minimum risk-tolerant measure and the minimum representing the maximum riskaverse measure ultimately leads us from the biobjective planning problem to a fuzzy-bayes decision rule for observed information. In this treatment, the two objective functions are nonlinear, but with the fuzziness of the fuzzy events at the maximum and the minimum on the platform when the α-level is 0 given as fuzzy goals and applying the fuzzy planning method of Sakawa (1985), the result is an interactive compromise solution based on the bi-objective function trade-off ratio, in which the decision-maker can select the action. On the α- level 0 platform, the risk-tolerant measure and the risk-averse measure may mutually conflict, but a more risk-neutral decision can be obtained in a second stage, which is application of the fuzzyevent Monte Carlo simulation formulated by Uemura (2011) using a risk-neutral evaluation function and a transition matrix. infinitesimal sections (Equation 2). Bayesian possibility distribution: f 1/σ (1) Fuzzy possibility distribution: or {fi 1/σi }( i = 1,..., ) (2) A scale parameter is applied to the prior information for simplicity. The possibility distribution or utility function is identified by certainty equivalence obtained by lottery in the infinitesimal sections. Piecewise linear approximation is therefore possible in the infinitesimal sections whether they are apparently upward- or downward-convex and the utility function itself must therefore also be fuzzy in nature, as shown in Equation 2. The fuzzy events Fi are mapped and transformed by the membership function ( ) in correspondence with the state of natures. Then, by the mapping extension principle and using the notation of Zadeh (1997), we can derive the utility function for the fuzzy events Fi as: (3) Where U(s d) is the utility function given decision d for state of natures and U-1(x d) is the inverse function. Using utility function theory, we can also take the fuzzy events as OR-connective on the utility function for the infinitesimal sections and in this way define Equation 3 as: (4) Equations 3 and 4 are thus completely identical and as shown by utility function theory, Zadeh s extension principle is applicable and the same utility function for fuzzy events can be derived from both utility function theory and fuzzy theory. If we apply the max-product operation, the possibility measure μ( ) for fuzzy events Fi, is then: FUZZY BAYES DECISION RULE FOR NO-DATA PROBLEM WITH OR-CONNECTIVE SUBJECTIVE DISTRIBUTION The no-data problem can be regarded statistically as inference from prior information. Fuzzy theory can then be characterized as the introduction of fuzziness into the Bayesian subjective prior distribution as shown here by Equation (1) together with division of the possibility distribution into Where π is the subjective (possibility) distribution. For each action dj and fuzzy event Fi, the possibility measures μ( ) for the fuzzy events are summed as in Equation 6 and the dj*yielding the largest value is taken as the optimum action given as: (5) (6)

3 Academic Web Journal of Business Management; Hori and Matsumoto; 003 If the decision-maker is risk-averse, it is preferable to use the expected utility optimization operation, which closely agrees with the Bayesian decision, rather than the maxproduct operator (Max-Max decision), which emphasizes variance. FUZZY-BAYES DECISION RULE WITH INDIFFERENT EVENTS With the fuzzy-event membership function, there is no need to take a direct sum. In cases where the sum is less than 1, it can be freely specified in accordance with the subjectivity of the decision-maker and new indifferent events can be automatically added to obtain a sum of 1. This addition gives rise to a new decision risk and it is therefore desirable to add a reserved judgment action so that the sum is also 1 for the utility function based on fuzzy theory. In many cases, moreover, the height of the fuzzy events is irregular and the height of the fuzzy-event membership function does not reach 1. We therefore propose that the fuzzy event possibility measure be weighted by the fuzzy entropy, calculated as: μ( )logμ( ) (7) [ ( ), ( )] And the utility function as: [ ( ), ( )]. We note that the minimum is thus risk-averse and the maximum is thus risk-tolerant. The fuzzy event possibility measures for the risk-averse and risk-tolerant cases are then: (9) (10) If 0<α<1, the risk-tolerant and risk-averse contexts mutually conflict and Equations 9 and 10 are then objective functions and the bi-objective planning problem (Equation 11) can be formulated to enable maximization of the risk-averse measure and minimization of the risk-tolerant measure by varying α, as: (11a) We sum the fuzzy event possibility measures weighted by Equation 7 for each fuzzy event as: 0 α 1 (11b) (11c) (8) And take the maximum decision dj as the optimum decision. FUZZY-BAYES DECISION RULE WITH OBSERVED INFORMATION In this paper, for cases in which observed information (data) is present, we assumed that the fuzzy events for the observed information constitute a normal type-2 fuzzy set and that the utility function for the infinitesimal sections is identified as type-2 by risk-averse and risk-tolerant lottery. With this assumption of normal distribution, the fuzzy polynomial regression model can then be identified (Uemura, 1995). Moreover, the central curves in fuzzy regression models do not generally take the form of regression lines, but the technique was shown to be effective if the origin is fixed, as for example, in the enterprise efficiency evaluation problem (Uemura, 1999). We now introduce the α-level cut technique. Maximum fuzziness occurs at an α-level of 0 and the risk-averse representative curve is obtained at an α-level of 1. At a given α-level, we denote the type-2 membership function of fuzzy events Fi as: Fuzzy goals are necessary for the solution and from the event fuzziness we then have: (12a) (12b) (13a) (13b) We calculate and sum this (fuzzy) bi-objective planning problem over all fuzzy events and use the largest value to obtain the optimum action. We note that for a risk-neutral decision-maker, the α-level is 1 and the representative curve is calculated on this basis enabling derivation of the optimum action from Equations 5 and 6 without solving the bi-objective linear planning problem. TWO-STAGE FUZZY-BAYES DECISION RULE If α=0, the risk-tolerant and risk-averse contexts mutually conflict and as such we first applied a discriminant-analysis

4 Academic Web Journal of Business Management; Hori and Matsumoto; 004 approach that renders the boundaries fuzzy and specifies the type-2 fuzzy set of fuzzy events rather than a fuzzy-event meta-fuzzy set from the data. We then propose the use of a risk-neutral evaluation measure and a risk-neutral transition matrix to derive a single compromise optimum action by Monte Carlo simulation for the fuzzy events. Identification of fuzzy-event normal possibility distribution and possibility measure It is natural to regard data (xi, yi) (i=1,, n) as existing and fuzzy events Fj (j=1,, J) as having normal possibility distributions f1j and f2j that enclose these data on the upper (risk-tolerant) and lower (risk-averse) sides, respectively. Therefore, we may formulate this as the linear planning problem given as: min f1j(x) s.t. f1j(xi) yi (14) max f2j(x) s.t. f2j(xi) yi (15) Using the max-product operation, we obtain risk-tolerant fuzzy event possibility measures (Equations 16 and 17) for f1j and f2j, respectively, as: max z f 1j(U -1 (z)) π 1(U -1 (z)), (16) max z f 2j(U -1 (z)) π 2(U -1 (z)), (17) Where U(s) is the utility function for state s and πk (k=1, 2) is the normal subjective possibility distribution. The actions yielding the maximum possibility measures in Equations 16 and 17 are then optimum actions. Two-stage fuzzy-bayes decision rule If the optimum actions with respect to Equations 16 and 17 are the same, then, the decision is uniquely determined and we end it there. If they are different, then, we construct a fuzzy transition matrix with risk-neutral fuzzy probability as shown in Figure 1 and apply the two-dimensional riskneutral fuzzy event possibility measure Equations 18 and by 19 obtain the second-stage fuzzy-event fuzzy measure. We take the action yielding the maximum by this fuzzy measure as the optimum action. Note that it is possible to apply the early formulation and integral operation of Tanaka et al. (1979) since the decision-maker is riskneutral and the fuzzy events have a normal possibility distribution given as: M t(s 1,s 2) := f(s 1,s 2) U(s 1,s 2) π(s 1,s 2) ={f 1j(s 1) f 2j(s 2)} {U(s 1) U(s 2)} π 1(s 1)π 2(s 2) = f 2j(s 2) {U(s 1) U(s 2)} π 1(s 1)π 2(s 2), (18) (19) FUZZY-BAYES DECISION RULE FOR NO-DATA PROBLEM WITH AND-CONNECTIVE SUBJECTIVE DISTRIBUTION Extension principle for AND-connectives For subjective distributions with AND-connectives, we apply the loss function L( ) rather than the utility function U( )used in decision problems involving subjective distribution with OR-connectives. The extension principle for AND-connectives is then defined as follows. We note here that the minimum of the mapped and transformed fuzzy-event membership function is used in this definition of the loss function in contrast to the use of its maximum in the utility function given as: (20) On the other hand, when considered from the perspective of utility function theory, the AND-connective operation of the loss function is defined by Equation 21 and yields the same value as Equation 20. This clearly shows that the fuzzy mathematics extension principle can be applied for AND-connectives as well as for OR-connectives and that the application of fuzzy mathematics is therefore effective and valid in decision problems dealing with the utility function (loss function) (Equation 21) given as: Fuzzy-event necessity measure (21) The subjective necessity distribution can similarly be regarded as a loss function obtained by mapping and transforming the fuzzy-event membership function. Applying the extension principle to this map, we derive the subjective necessity distribution as: (22) In the case of AND-connectives, this can be regarded as dealing with necessity and in accordance with the necessity measure definition by Zadeh, (1997) we may then define the fuzzy-event necessity measure as: (23) In this way, the fuzzy-bayes decision rule for necessity ultimately becomes a matter of selecting the action that minimizes the sum of the fuzzy-event necessity measures as the optimum action.

5 p dnorm(seq(-3, 3, 0.1), 0, 0.5) seq(-5, 5, 0.1) risk-neutral Academic Web Journal of Business Management; Hori and Matsumoto; 005 and the fuzzy events have a normal possibility distribut f 1j (s 1 ) n t S1 f 2j (s 2 ) Figure 1: Risk-neutral fuzzy transition matrix. AND-connective subjective distribution with observed information If observed information (data) is present, we assume that the fuzzy events constitute a normal type-2 fuzzy set as in the case of OR-connective subjective distribution. In this case, we may note that the fuzzy-event membership function has a type-2 membership function with the same values as in the OR-connective case. The loss function is given subjectively by the decision-maker as a type-2 fuzzy set based on risk-tolerant and risk-averse lottery in infinitesimal sections. We now introduce the α-level cut technique. The fuzziness is largest when the α-level is 0, and the riskaverse representative curve is obtained when the eα-level is 1. We denote the type-2 membership function for the fuzzy events Fi at the α-level as: [ ( ), ( )] and the loss function as: [ ( ), ( )]. Let us note that the minimum is risk-averse and the maximum is risktolerant. The fuzzy-event necessity measures for riskaverse and risk-tolerant are expressed, respectively, as: S Fig. 1. Risk-neutral fuzzy transi (26b) 0 α 1 (26c) Fuzzy goals are necessary for solution of this problem and can be specified on the basis of event fuzziness as: (27a) (27b) max {{1{ x (28a) (28a) (28b) ((24) (25) Equations 24 and 25 are thus objective functions and the bi-objective planning problem can then be formulated to provide the risk-averse measure as the maximum and the risk-tolerant measure as the minimum by varying α, as: (26a) We calculate and sum of this (fuzzy) bi-objective planning problem over all fuzzy events and take the lowest value to represent the optimum action. We note here that if the decision-maker is risk-neutral, then we can take the α-level as 1, derive the representative curve and obtain the optimum action from Equation 23 without solving thebiobjective linear planning problem. We note also that if α=0, then, we use the risk-neutral evaluation function of Equation 29 whereby the utility function of Equation 18 is converted to a loss function together with the risk-neutral transition matrix to obtain the minimum expected loss in the second stage as the optimum action given as:

6 Academic Web Journal of Business Management; Hori and Matsumoto; 006 REFERENCES (29) (30) Where nt is the risk-neutral fuzzy transition matrix (Figure 1). CONCLUSION In this paper, we first showed that for cases in which the subjective distribution is OR-or AND-connective, fuzzy mathematics can be applied to utility function theory (loss function theory). We next showed that if the decisionmaker is risk-neutral, then, selection of the fuzzy-event membership function for fuzzy events identified as type-2 together with the representative curve of the utility function leads to resolution by the early formulation of Tanaka and Ishibuchi (1992). Lastly, we introduced the α- level cut technique and formulated the risk-tolerant and risk-averse possibility measures in a bi-objective planning problem and showed that a more risk-neutral compromise solution can then be obtained through interaction of the decision-maker in a trade-off between the fuzziness of the fuzzy events in the two objectives of the bi-objective planning problem. In summary, the decision theory itself may be regarded as a variational problem comprising construction of an objective function with the variables corresponding to the probability distributions. In Bayesian decision theory, this is simply a variational problem of subjective distributions (functions) in the state of nature (a type of functions) for separate utility functions, but as shown here, variational problems with incorporated fuzzy utility functions can be solved by introducing fuzzy events. Further study will be necessary to extend this to cases involving subjective distributions having a mixture of AND-and OR-connectives. Zadeh LA (1968). Probability Measures of Fuzzy Events, J. Math. Anal. Appl. 22: Zadeh LA (1997). Fuzzy Sets as Basis for a Theory of Possibility, Fuzzy Sets and Systems, 1: Tanaka H, Okuda T. Asai K (1979). Fuzzy Information and Decision in Statistical Model, Advance in Fuzzy Sets Theory and Application, pp , North-Holland. Tanaka H, Uejima S, Asai K (1982). Linear Regression Analysis with Fuzzy Model, IEEE Transaction on Systems, Man, and Cybernetics, 12: Tanaka H, Ishibuchi M (1992). Evidence Theory based on Normal Possibility Theory, System Control and Information, 5: , (in Japanese). Uemura Y (1991). Decision Rule on Fuzzy Events, Fuzzy Sets and Theory, 3: (in Japanese). Uemura YA (1995). Normal Possibility Decision Rule, Control and Cybernetics,24: Uemura YA (1999). Comparative Study in Evaluation on f the Efficiency for DMU: Fuzzy Log linear Model and DEA, Control and Cybernetics, 28 : Uemura Y (2001). Application of Normal Possibility Theory to Silence, Control and Cybernetics, 30: Uemura Y (2011). Application of Fuzzy Decision Making to Smoking Problem. J. Fuzzy Math. 18: Hori HJ, Tsubaki H (2014). Fuzzy Decision Making Rule under no-data Problem, Applied Math. (to appear). Sakawa M, Yano H (1985). An Interactive Fuzzy Satisfying Method Using Augmented Minima Problems and Its Applications to Environmental Systems, IEEE Trans. On Systems, Man, and Cybernetics, 15: