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Bayesian Fuzzy Hypothesis Testing with Imprecise Prior Distribution

This paper considers the classical (normal) Bayesian method for testing fuzzy hypotheses. For this purpose, using a notion of prior distribution with interval-valued or fuzzy-valued parameters, a concept of posterior probability of a fuzzy hypothesis is proposed and its main properties are also verified. The feasibility and effectiveness of the proposed methods are also clarified by some numerical examples.


Introduction
Hypothesis testing, also called significance testing, is the the most frequent statistical method widely used by scientists to guard against making claims unjustified by data. However, there are a variety of problems, difficulties, and inconsistencies associated with the most frequent methods of testing hypotheses that are overcome by using Bayesian analysis. However, the key ingredients to a Bayesian analysis are the likelihood function, which reflects information about the parameters contained in the data, and the prior distribution, which quantifies what is known about the parameters before observing data. The prior distribution and likelihood are then combined to the posterior distribution, which represents total knowledge about the parameters after the data have been observed.
Notably, a prior is a subjective assessment for the distribution of a parameter from an experienced expert. It is the probability distribution that would express expert's beliefs about a parameter(s) before some evidence is taken into account. However, when the experts agree about the family of prior distribution, the main concern is to assign some proper values for parameters which would satisfy all experts. For such cases, it is difficult to set exact values for parameters of the prior distribution due to uncertainty in estimations. In these cases, it is better that an expert estimates the value of a parameter θ based on the smallest possible value (or "pessimistic") and the highest possible value (or "optimistic") or some fuzzy sets instead of crisp ones. For instance, let X be a random variable denoting the expected time (per hour) to do a certain task. It is known that X is uniformly distributed in an interval (0, θ), θ > 0. However, due to uncertainty related to the exact estimation, it is hard to say that our belief is that θ is distributed according to exponential distribution with an exact mean of 10 hours. For such a case, it is better to say that θ has an exponential distribution with a mean of about, for instance, 10 hours which may be modelled by an interval-valued [8,12] or a triangular fuzzy number of about, for instance, 10 hours.
On the other hand, the classical hypotheses testing may fail in many environmental practices and we may be forced to use fuzzy hypotheses testing as an alternative approach for demonstrating its utility in real applications Arnold (1996); Ortega et al. (2008); Parchami et al. (2011). Therefore, in order to achieve suitable Bayesian testing statistical methods dealing with imprecise information, there is a need to model the imprecise information and extend the classical methods of probability theory to imprecise environments.
The topic of probability theory with imprecise information has been studied by some authors (Baldwin et al. , 1996;Heilpern , 1993;Klement , 1982;Plasecki , 1985;Singpurwalla and Booker , 2004;Smets , 1982;Stein , 1985;Toth , 1992;Yager , 1984, see for example ). Especially, introducing a loss function, Meeden (2008) suggested a fuzzy Bayes estimator for any given (crisp) prior density. He also investigated some conditions to transform the uncertainty expressed in a fuzzy decision into the (crisp) Bayesian paradigm. Moreover, there are several articles relevant to the Bayesian approach combining probability and/or fuzzy sets theory that can be found in literature Chib and Jacobi (2015); Chou and Yuan (1993); Gholizadeh et al. (2012); Jeet and Dhir (2012); Leu et al. (2015); Marttinen et al. (2009);Osoba et al. (2011Osoba et al. ( , 2012; Viertl (2011);Xiao et al. (2013). In the sequel, there is a review of some studies relevant to testing statistical hypotheses based on Bayesian approaches. Casals (1993) considered the problem of Bayesian fuzzy hypotheses with fuzzy observations, as a special case of the fuzzy decision problem. Casals and Gil (1994) also extended the Bayesian sequential problem of testing fuzzy hypotheses when the outcomes of which provide fuzzy information. Taheri and Behboodian Taheri and Behboodian (2001) considered the problem of Bayesian approach of testing hypotheses when the data (observations) are ordinary (crisp) and the hypotheses are fuzzy sets. Ortega et al. (2008) applied Taheri and Behboodian's method to evaluate whether the mother's serostatus and children's seroconversion to measles vaccine could be considered as high or low in a setting of fuzzy basedrules.
The purpose of this study is extending an approach to (Normal) Bayes test with unknown imprecise parameters given the crisp observations. For this purpose, after choosing a probability model for the (crisp) data, first a concept of prior distribution for the unknown fuzzy-valued or interval-valued parameters is introduced. Inspired by Zadeh's idea of probability of a fuzzy set, a posterior probability of the fuzzy hypotheses is introduced by averaging on the combination of the likelihood and prior distribution which quantifies the uncertainty in the values of the unknown imprecise parameters after the data are observed. The main properties of the proposed posterior probability are also put into investigation. For practical reasons, we will illustrate the proposed method by some numerical examples.
This paper is organized as follows. In the next section, we briefly review some definitions of fuzzy numbers and some results from possibility theory. In addition, the classical normal Bayes test is also reviewed briefly in this section. In Section 2, based on a crisp random sample of observations, first we propose a new concept of prior distribution with imprecise parameters which are reported by closed intervals or fuzzy numbers. Then, a notion of posterior probability of the underlying fuzzy hypotheses are introduced and discussed. Some numerical examples are provided in Section 4 to clarify the discussions made in this paper and to show the applicability of the proposed method in the fuzzy environment. Finally, a brief conclusion is provided in Section 5.

Preliminaries
A fuzzy set of X is a mapping A : X → [0, 1], which assigns to each In what follows, a fuzzy number is called continuous if its membership function is continuous over R. The set of all continuous fuzzy numbers will be denoted by F c (R). The imprecision or vagueness can be treated by means of special fuzzy numbers, i.e. the LR-fuzzy numbers, since they can be characterized using three real numbers: the center, the left spread, and the right spread. A LR-fuzzy number A = (a l , a c1 , a c2 , a r ) T , a l , a c1 , a c2 , a r ≥ 0, is defined as where a l and a r are the left-hand and the right-hand spreads, respectively. L and R are continuous and strictly decreasing functions with L(0) = R(0) = 1 and A special type of LR-fuzzy numbers is the so-called trapezoidal fuzzy numbers with the shape functions The membership function of a trapezoidal fuzzy number A = (a, a l , a c1 , a c2 , a r ) T is given by Remark 1. (Hesamian and Shams , 2016) Let A ∈ F c (R) and A α be defined for each α ∈ [0, 1] as (2.1) Examples 2.1. For a given LR-fuzzy number A = (a l , a, a r ) LR , it is easily seen that Remark 2. The following operators on fuzzy numbers A and B (Dubois and Prade , 1998) are used in this paper:

The classical Bayesian testing hypothesis: The normal case
Let X = (X 1 , X 2 , . . . , X n ) T be a random sample from a parametric density function f θ , θ ∈ Θ ⊆ R, where θ denotes a population parameter. The observed random sample is denoted by x = (x 1 , x 2 , . . . , x n ) T . The general format of the null and alternative hypotheses is as H 0 : θ ∈ Θ 0 and H 1 : θ ∈ Θ c 0 , where Θ 0 is a subset of the parameter space and Θ c 0 is its complement. However, hypothesis testing problems may be formulated in a Bayesian framework. A Bayesian framework includes both the sampling distribution f θ (x) and the prior distribution π(θ), with the prior distribution reflecting the experimenter's opinion about the parameter θ prior to sampling. The Bayesian paradigm prescribes that the sample information should be combined with the prior information using Bayes's Theorem to obtain the posterior distribution π(θ|x) = Therefore, all inferences about θ are investigated based on the posterior distribution. In the Bayesian testing hypothesis problem, the posterior distribution may be used to calculate the probabilities that H 0 and H 1 are true. One way a Bayesian hypothesis tester may choose to use the posterior distribution is to consider H 0 as true if P(θ ∈ Θ 0 |x) ≥ P(θ ∈ Θ c 0 |x) and to reject H 0 otherwise. It is mentioned that the prior distribution for testing the two-sided hypothesis H 0 : θ = θ 0 v.s. H 1 : θ θ 0 is given by where π 0 + π 1 = 1 (Berger , 1985;Bolstad , 2007, see for more).

Bayesian testing for fuzzy hypothesis with imprecise prior distribution
This section proposes a procedure for Bayesian testing statistical fuzzy hypothesis in case where the unknown parameters of the prior density are given by closed intervals or fuzzy numbers.

Fuzzy hypothesis
Here, some models for extending one-sided or two-sided ordinary (crisp) hypotheses to fuzzy ones are introduced.

Any hypothesis of the form
is called a simple hypothesis. For the above hypotheses, we suppose that

Any hypothesis of the form
where 0 < ϵ, ϵ 1 ≤ θ 0 and ⊖ denotes the subtraction operator between fuzzy numbers. A typical form of fuzzy hypotheses H S , H TS , H LOS and H ROS are shown in Fig. 1. In this paper, we are going to test the following hypotheses: (1)

Posterior probability of a fuzzy hypothesis
Suppose that the prior distribution π(θ) has a vector of imprecise parameters. In this regards, two sources of uncertainty are considered in this paper: 1) a vector of interval parameters γ = (γ 1 , γ 2 , . . . , γ k ) ∈ (C(Θ)) k , where C(Θ) is the class of nonempty compact intervals on real numbers R and 2) a vector of fuzzy parameters γ = ( γ 1 , γ 2 , . . . , γ k ) ∈ (F c (R)) k . Now, let X be a random sample from a parametric density function f θ , θ ∈ Θ ⊆ R. It is worth noting that Zadeh (1965) suggested to evaluate the probability of a fuzzy event A by P( dxdα (for more discussions on Zaheh's criterion see Singpurwalla and Booker (2004)). Here, inspired by Zadeh's idea, we introduce a posterior probability of the null and alternative fuzzy hypotheses as follows.
From the operations on fuzzy numbers introduced in Remark 2, we have the following results.
Lemma 3.1. Let A and B be two fuzzy numbers. Then, Proof. Let X = (X 1 , X 2 , . . . , X n ) T be a random sample from a parametric density function f θ , θ ∈ Θ ⊆ R, and {π γ : γ = ( γ 1 , γ 2 , . . . , γ k ) ∈ (F c (R)) p } be a family of prior distribution with fuzzy-valued parameters. Therefore, the assertions 1)-3) are immediately followed. To prove the assertion 4), note that Let y = 1 − β. Since g γ β (θ|x) = g γ 1−β (θ|x) for all β ∈ [0, 1], it is easily concluded that which completes the proof. The proof for the case that the family of prior distributions is considered as {π γ : γ = (γ 1 , γ 2 , . . . , γ k ) ∈ (C(R)) p } with intervalvalued parameters is similar. □ Remark 4. It is pointed out that if the vector of interval-valued parameter γ or fuzzy-valued γ reduces to the crisp one, i.e. a vector of the real-valued parameter γ, then it is easy to observe that the posterior probability of the fuzzy hypothesis H is reduced as which is the same result as Taheri and Behbodian's method (Taheri and Behboodian , 2001).

Numerical examples
In this section, we apply two numerical examples to demonstrate the proposed method.
In this example, since P( H 1 |x = 120) ≫ P( H 0 |x = 120), the data strongly supports the alternative hypothesis that Martin's IQ is essentially greater than the standard intelligence 100. Now, assume that our prior beliefs are normal with the mean of about 100 and variance of about 225 represented by triangular fuzzy numbers µ = (95, 100, 105) T and σ 2 = (220, 225, 230) T , respectively. In this case, the posterior probability of H 0 and H 1 are evaluated as where µ β = 105 − 10β and τ 2 β = 230 − 10β. So, with respect to the observed IQ value of 125, the null hypothesis H 1 is also strongly accepted in this case.

Conclusion
There are many situations in applied statistics where we deal with imprecise information. In this paper, the subject of Bayesian test based on fuzzy hypothesis is introduced and discussed. Unlike the classical Bayesian method that usually applies a two stage hierarchical model for priors, the proposed method assumes that the unknown parameters of the prior distribution in the first stage are closed intervals or fuzzy numbers instead of putting a second prior distribution on the prior. Averaging over the interval-valued or fuzzy-valued parameters, we then construct a notion of posterior probability of the fuzzy hypotheses. Numerical results show that we invoked successfully the induced prior distribution with imprecise parameters to overcome such uncertainties which is often arisen in many real applications of Bayesian hypothesis testing. Notably, the proposed method is general which can be applied for other kinds of Bayesian testing methods of fuzzy hypothesis such as Bayes factor. Extending the proposed method with imprecise observations and hypotheses is also a potential subject for future studies.