Introduction
Bayesian decision theory (BDT) is a mathematical framework that is used to make decisions with the aim of maximizing expected utility. It is the application of Bayesian probability to the domain of decision-making. The theory assigns probabilities to each action to be undertaken and values each action against a predetermined goal to select the strategy most likely to achieve that goal. Bayesian decision theory can be applied to a wide variety of scenarios, including marketing, game theory, classification problems, and economic decisions.
Overview
In order to make decisions, Bayesian decision theory requires three components which are the prior probabilities, posterior probabilities and expected utility. Prior probabilities refer to the probability of each possible outcome at the beginning of a decision-making process. These probabilities may be based on previous experiences or events such as a customer survey or past customer purchases. Posterior probabilities refer to the revised or updated probability of each outcome after the decision-making process has been completed. In most cases, the posterior probabilities are based on information that has been collected during the decision-making process.
The expected utility is the expected gain or reward of each action which is calculated by multiplying the probability of each outcome by its specific utility. It’s a measure of the desirability of an action or outcome. The goal of using Bayesian decision theory is to select the action or strategy that will provide the highest expected utility. This is typically done by comparing the expected utilities of various possible strategies.
Bayes Rule
Bayes Rule is the mathematical formula used in Bayesian decision theory that enables you to calculate prior and posterior probabilities. It states that the probability of an event occurring is equal to the probability of the event given a prior event multiplied by the probability of the prior event, divided by the probability of the prior event. In other words, Bayes Rule is used to update the probabilities of an event occurring based on new information.
Example
Suppose you are considering a new marketing strategy and need to decide which of three strategies to use. You decide to use Bayesian decision theory to decide which strategy is the most likely to lead to success. For each strategy you determine the prior probability, posterior probability, and the potential reward. After computing the expected utility of each strategy, you select the strategy that provides the highest expected utility.
Conclusion
Bayesian decision theory is a mathematical approach to decision-making which utilizes prior probabilities of outcomes and expected utilities of actions in order to systematically identify the most appropriate action or outcome. It is widely used in fields such as marketing, game theory, and economics. By using posterior probabilities, the Bayesian decision maker is able to take new information into account while assessing the likelihood of success associated with each action. Furthermore, Bayes Rule can be used to calculate prior and posterior probabilities in order to generate more accurate predictions.
