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The Bayes’ Principle

The Bayes’ principle for the selection of an optimal action derives its name from the 18th century philosopher Thomas Bayes who first suggested and investigated the notion of “inverse probability”.
To make use of the Bayes Principle in statistical decision problems, the decision-maker must be able to assign probabilities to each state of nature. The probabilities assigned must conform to the requirements of the rules of probability. That is, they must all be non-negative numbers between zero and one, and the sum of the probabilities over all the possible states of nature must equal one. These probabilities represent the strength of the decision-maker’s belief, a subjective evaluation, regarding the likelihood of the occurrence of the various states of nature. The set probabilities together with the possible states of nature constitute a probability distribution, called ‘prior distribution’. We shall use the notation g (θi) = Pr [θ = θ1].

The probability distribution used at the beginning of the decision analysis is usually referred to as the prior distribution. In many cases, it is possible to develop the prior distribution from empirical information or data like historical data taken form records that are already available.

Bayes’ criterion states that the expected payoff should be computed for each act in the set of acts, and that act should be chosen for which the expected payoff is the best. “Best” would refer to the highest value if the payoffs are expressed in terms of profit or gain, and the lowest value when payoffs are expressed in terms of cost or loss. The optimum act is one with the highest EMV. It is frequently used in decision-making to find the last strategy. The steps are listed below:

1.    A payoff table listing the acts and events that are considered to be possibilities is constructed. This table considers the economics of the problem by calculating a conditional value for each act-event combination.

2.    Probabilities are assigned to the events.

3.    EMV for each act is calculated by weighing the conditional values by the assigned probabilities and add the weighted conditional values to obtain EMV of the act.

4.    That act is chosen which has the highest EMV.

After determining the prior distribution, the Bayes principle is to be used phase wise. The three phases in order of their occurrence are: (i) prior analysis, (ii) preposterior analysis, and (iii) posterior analysis. A brief description of each of these is given:

(i) Prior analysis.

Once the relevant prior distribution of various states of nature is found out, the decision-maker needs to compute the expected payoff or expected opportunity loss (EOL) for each action. The EP for action aj, denoted by B (aj), is computed by the formula:

            B (aj) = ∑pij g(θi) for all j

If in this formula payoffs are replaced by corresponding opportunity losses. We get EOL for action aj. The decision-maker chooses that action for which EP is maximum or EOL is minimum.

(ii) Preposterior Analysis.

After making prior analysis the decision-maker must decide either to collect additional information regarding the states of nature or to take action as suggested of nature. This is more so in business decision problems. However, if somehow the decision-maker finds a perfect predictor, he would prefer actions based on perfect predictor it would enable him to maximize his profits or minimize his losses. The highest expected profits resulting in the presence of perfect predictor is called the expected payoffs of perfect information (EPPI). EPPI is often called the expected value o payoff under certainty. The perfect predictor is EP of the optimal action. The difference between EPPI and EP is called the expected value of perfect information.

(iii) Posterior Analysis.

If it is decided to gather further information regarding the states of nature, the information to be gather by conducting survey or by performing an experiment or by some other means the posterior analysis is used only after fathering the information. In posterior analysis relevant information is combined with prior information in order to make the degree of belief regarding the states of nature more stronger.

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