The Method Of Fuzzy Bayesian Decision

The Germany mathematicians originate the set theory has provided the feasible way for information processing. However the classical set theory either this or that's disaggregated model while has established precise mathematics leading position, also fell into own contradictory region inevitably, has had in the understanding and the logical paradox, the fuzzy mathematics (set) broke this extreme understanding, the fuzzy set theory acknowledged and has solved understanding fuzzy uncertainty this objective reality well, caused it tends in a higher stratification plane subjective and the objective reality. Regarding uncertainty question, may also divide into the stochastic uncertainty and the fuzzy uncertainty two kinds. So-called stochastic uncertainty Chang Zhiyou the law of causality has the uncertainty which broken lacks creates. But the fuzzy uncertainty is refers to one kind the uncertainty which causes because of compound state's existence, is attributed to the law of excluded middle existence flaw.The Bayesian Decision is in the tradition decision theory one utility very strong decision method. Its characteristic is the information which utilizes all may obtain makes the most reasonable decision. These information include: Apriori information, sample information and consequence information. And the apriori information is the people through the experience and the historical data subjective determination. Therefore obtains the prior probability calls the subjective probability. After giving a priori distributionπ(θ)of the unknown quantityθ, the hypothesis takes a valueθ, obtains the sample distribution p ( xθ) again from the conditional distribution. Then by Bayesian formula Obtainsθto examines the distribution, after hereafter examines the distribution applies Baye, examines in the risk criterion if it has this kind of decision functionδ^*( x) to causes Theδ^*( x) is called this statistical decision-making question after examines under the risk criterion the most superior decision function, after or is called (, examines) the decision function.This article introduces the fuzzy method in the Baye decision-making, divided four to discuss the fuzzy Baye decision-making issue.First, a priori distribution fuzzy. Supposes the state spaceΘ= {θ₁ , ,θ_m} (parameter space), the elementθiprior probability p (θ_i), presently gives the state space by the fuzzy set's form ,θ_i to jthe degree of membership is , namely the fuzzy event's probability isSample information fuzzy. After tradition examines in the decision-making, the sample X = ( X₁,…, X_n) production is the establishment, in has under an observation value premise from a priori distributionπ(θ), namely the probability of X i materially is the conditional probability, is p ( X_k|θ_i). But because on a fuzzyθ, therefore the sample information also had the fuzziness, therefore also gives the sample X = ( X₁,…, X_n) by the fuzzy set's form , the degree of membership is , namelytherefore, , in which, therefore ,we obtain the sample fuzzy set probability.Then, we obtains the fuzzy condition set probability and the fuzzy sample set probability by above, according to the conditional probability formula, obtains the fuzzy probabilityaposteriori:Solves the key lies in the computation . may know by the theorem, Then And command . Therefore ( )in which , in the synthesis we obtain fuzzy probabilityaposterioriFinally, we utilize the fuzzy probabilityaposteriori to carry on the Bayesian decision-making, issues an order for the space (decision space)Δ= {a₁ , a₂ , a_r},L (θ_i , a_l)is the loss for makes the behavior al under the natural stateθ_i, makes the behavior al under the fuzzy condition the loss is: by the theorem, the loss of the behavior a_l in the probabilityaposteriori expectation is considers the sample the fuzziness, then the behavior behavior al expected finally the loss is if has the behavior a * to cause the behavior a * is after the fuzziness examines under the risk criterion the most superior decision-making.Moreover gives back to an example to explain this method the feasibility.