**2.1. Belief and the approach of its improvement**

Assume that there are *n* available *Binomial* populations and we intend to select the one with the highest probability of success. Furthermore, in each stage of the data gathering process and for each population, we take an independent sample of size *m*. Let us define *αi*,*<sup>t</sup>* ' and *βi*,*<sup>t</sup>* ' to be the observed number of successes and failures of the *i* th *Binomial* population in the *t* th stage (sample) and *αi*,*k* and *βi*,*<sup>k</sup>* to be the cumulative observed number of successes and failures of the *i* th *Binomial* population up to the *k*th stage (sample) respectively. In other words, *<sup>α</sup>i*,*<sup>k</sup>* <sup>=</sup>∑ *t*=1 *k αi*,*<sup>t</sup>* ' and *<sup>β</sup>i*,*<sup>k</sup>* <sup>=</sup>∑ *t*=1 *k βi*,*t* ' . Then, in the *k*th stage defining *p*¯ *<sup>i</sup>*,*k* to be the estimated probability of success of the *i* th population obtained by *αi*,*<sup>k</sup> km* , referring to Jeffrey's prior (Nair et al.[7]), for *p*¯ *<sup>i</sup>*,*<sup>k</sup>* , we take a *Beta* prior distribution with parameters αi,0=0.5 and βi,0=0.5. Then, using Bayesian inference, we can easily show that the posterior probability density function of *<sup>p</sup>*¯ *<sup>i</sup>*,*k* is

Since the idea behind the sequential analysis modeling is completely similar to the decisionmaking process of a human being in his life, it may perform better than available methods in decision-making problems. In these problems, when we want to make a decision, first we divide all of the probable solution space into smaller subspaces (the solution is one of the subspaces). Then based on our experiences, we assign a probability measure (belief) to each

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

Consider *n* independent populations *P*1, *P*2, ..., *Pn*, where for each index *i* =1, 2, ..., *n*, popu‐

the ordered value of the parameters *p*1, ..., *pn*. If we assume that the exact pairing between the

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

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

Assume that there are *n* available *Binomial* populations and we intend to select the one with the highest probability of success. Furthermore, in each stage of the data gathering process

(sample) and *αi*,*k* and *βi*,*<sup>k</sup>* to be the cumulative observed number of successes and failures of

th *Binomial* population up to the *k*th stage (sample) respectively. In other words,

and for each population, we take an independent sample of size *m*. Let us define *αi*,*<sup>t</sup>*

. Then, in the *k*th stage defining *p*¯

optimum point of a multi-response problem, etc. are just a few of these applications.

. Let *<sup>p</sup>* <sup>1</sup>

th *Binomial* population in the *t*

*<sup>i</sup>*,*k* to be the estimated probability

<sup>≤</sup>...<sup>≤</sup> *<sup>p</sup> n* denote

' and *βi*,*<sup>t</sup>* '

th stage

with *pi* <sup>=</sup> *<sup>p</sup> <sup>n</sup>* is called

subspace, and finally we update the beliefs and make the decision.

126 Dynamic Programming and Bayesian Inference, Concepts and Applications

may decide which population possesses the best parameter value.

lation *Pi* is characterized by the value of its parameter of interest *pi*

the best population.

best subset selection approach.

and *<sup>β</sup>i*,*<sup>k</sup>* <sup>=</sup>∑

*t*=1 *k βi*,*t* '

the *i*

*<sup>α</sup>i*,*<sup>k</sup>* <sup>=</sup>∑ *t*=1 *k αi*,*<sup>t</sup>* '

**2.1. Belief and the approach of its improvement**

to be the observed number of successes and failures of the *i*

ordered and the unordered parameter is unknown, then, a population *Pi*

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

$$f(\overline{p\_{i,k}}) = \frac{\Gamma(\alpha\_{i,k} + \beta\_{i,k} + 1)}{\Gamma(\alpha\_{i,k} + 0.5)\Gamma(\beta\_{i,k} + 0.5)} \overline{p\_{i,k}}^{\alpha\_{i,k} - 0.5} (1 - \overline{p\_{i,k}})^{\beta\_{i,k} - 0.5} \tag{1}$$

At stage *k* of the data gathering process, after taking a sample and observing the numbers of failures and successes, we update the probability distribution function of *<sup>p</sup>*¯ *<sup>i</sup>*,*<sup>k</sup>* for each popu‐ lation. To do this, define *B*(*αi*,*<sup>k</sup>* , *βi*,*<sup>k</sup>* ) as a probability measure (called belief) of the *i* th population to be the best one given *αi*,*k* and *βi*,*k* as

$$B\left(\alpha\_{i,k'}\beta\_{i,k}\right) = \Pr\left|i^{\text{th}}\text{population is the best}\right|a\_{i,k'}\beta\_{i,k}\Big|\tag{2}$$

We then update the beliefs based on the values of (*αi*,*<sup>k</sup>* , *βi*,*<sup>k</sup>* ) for each population in iteration *k*. If we define *B*(*αi*,*<sup>k</sup>* <sup>−</sup>1, *βi*,*<sup>k</sup>* <sup>−</sup>1) as the prior belief for each population, in order to update the posterior belief *B*(*αi*,*<sup>k</sup>* , *βi*,*<sup>k</sup>* ), since we may assume that the data are taken independently in each stage, we will have

$$\begin{aligned} &\operatorname{B}\left(\boldsymbol{\alpha}\_{i,k},\boldsymbol{\beta}\_{i,k}\right) = \\ &\quad \frac{\Pr\Big{(}\boldsymbol{i}^{\text{th}}\text{ Population is the best}\Big{|}\big{(}\boldsymbol{\alpha}\_{i,k-1},\boldsymbol{\beta}\_{i,k-1}\big{)}\big{)}\Pr\Big{\Big{(}\alpha\_{i,k},\beta\_{i,k}\big{)}\Big{|}\big{\boldsymbol{i}^{\text{th}}\text{ Population is the best}\Big{)}}}{\sum\_{j=1}^{n}\left[\Pr\Big{(}\boldsymbol{i}^{\text{th}}\text{ Population is the best}\Big{|}\big{(}\boldsymbol{\alpha}\_{j,k-1},\boldsymbol{\beta}\_{j,k-1}\big{)}\big{)}\Pr\Big{\Big{(}\alpha\_{j,k},\beta\_{j,k}\big{)}\Big{|}\big{)}^{\text{th}}\text{ Population is the best}\Big{)}\right]}} \\ &= \frac{\operatorname{B}\left(\boldsymbol{\alpha}\_{i,k-1},\boldsymbol{\beta}\_{i,k-1}\right)\Pr\Big{\Big{(}\alpha\_{i,k},\beta\_{i,k}\big{)}\Big{|}\big{)}^{\text{th}}\text{ Population is the best}\Big{)}}{\sum\_{j=1}^{n}\left[\operatorname{B}\left(\boldsymbol{\alpha}\_{j,k-1},\beta\_{j,k-1}\right)\Pr\Big{\Big{(}\alpha\_{j,k},\beta\_{j,k}\big{)}\Big{|}\big{)}^{\text{th}}\text{ Population is the best}\Big{]}\right]}} \end{aligned} \tag{3}$$

From equation (3) we see that to update the beliefs, we need to evaluate Pr{(*αi*,*<sup>k</sup>* , *<sup>β</sup>i*,*<sup>k</sup>* )|*<sup>i</sup> th* Population is the best} ; *<sup>i</sup>* =1, 2, ..., *<sup>n</sup>* in each decision-making stage. One way to do this is to use

$$\Pr\left\{ \left( \alpha\_{i,k'} \beta\_{i,k} \right) \middle| i^{th} \text{ Population is the best} \right\} = \frac{\overline{p\_{i,k}}}{\sum\_{j=1}^{n} \overline{p\_{j,k}}} \tag{4}$$

Then, the probability given in equation (3) will increase when a better population is selected. In the next theorem, we will prove that when the number of decision-making stages goes to infinity this probability converges to one for the best population.
