**2. An application to determine the best binomial distribution**

In the best population selection problem, a similar decision-making process exits. First, the decision space can be divided into several subspaces (one for each population); second, the solution of the problem is one of the subspaces (the best population). Finally, we can assign a belief to each subspace where the belief denotes the performance of the population in term of its parameter. Based upon the updated beliefs in iterations of the data gathering process, we may decide which population possesses the best parameter value.

Consider *n* independent populations *P*1, *P*2, ..., *Pn*, where for each index *i* =1, 2, ..., *n*, popu‐ lation *Pi* is characterized by the value of its parameter of interest *pi* . Let *<sup>p</sup>* <sup>1</sup> <sup>≤</sup>...<sup>≤</sup> *<sup>p</sup> n* denote the ordered value of the parameters *p*1, ..., *pn*. If we assume that the exact pairing between the ordered and the unordered parameter is unknown, then, a population *Pi* with *pi* <sup>=</sup> *<sup>p</sup> <sup>n</sup>* is called the best population.

There are many applications for the best population selection problem. As one application in supply chain environments, one needs to select the supplier among candidates that performs the best in terms of the quality of its products. As another example, in statistical analysis, we need to select a distribution among candidates that fits the collected observations the most. Selecting a production process that is in out-of-control state, selecting the stochastically optimum point of a multi-response problem, etc. are just a few of these applications.

The problem of selecting the best population was studied in papers by Bechhofer and Kulkarni [5] using the indifference zone approach and by Gupta and Panchapakesan [6] employing the best subset selection approach.
