Decision Theory

Indroduction

In general, the theory which deals process of identifying and selecting the course of action to solve a spefic problem is known as decision theory.Everybody always faces problems of decision making and s/he must decide before starting any work or job or bussiness etc. so that s/he can get a best solution or an optimal solution to the problem. His/her decisions may be in terms of money, time, cost of items etc. as the case may be. The optimal solution to the decision problem will be obtained only when the decision problems are solved by making the correct decision by selecting a best startegy or alternatives among the various alternatives available to the decision maker. The successfilness of a person or decision maker depends upon the capability of taking the correct decision from amongst different alternatives or actions. Therefore, a decision has direct and termendous impact on everybody's bussiness, economic condition and the real practical life. Thus, the word "decision" plays a important and vital role in our daily life.

Just for example, (i) a person must decide whether to do work (job) or not to do it before starting it, (ii) a physicisn must decide whether a patient has a particular ddiseases or not before starting the treatment, (iii) a manager has to make a decision whether to introduce a new product or not, (iv) a businessman or business orginazation must decide whether to expand his plant capacity this year or next year, (v) a student must decide whether to be a doctor or an engineer or lowyers or manager or businessman or any, etc. In first four cases there are two options or alternatives where as in the last case there are many (multiples) alternatives. Out of these available alternatives, the decision maker has to select one alternatives such that the solution obtained from this selection is best or optimum.

In order to make the correct or right decision we need to study an important approach to Statistical Inference what is called decision theory.

Definition

Decision theory is defined as an important approach to decision making by choosing a best action or alternative from a set of different possible actions which optimizes the objective of decision maker.

Specifically, the optimization of the objective of the decision maker is to maximize excepted profits, to minimizes the excepted loss, to minimizes excepted costs, to maximized the excepted sales and so on. In making decision theory, since the probability of occurance of events on which the outcomes of each actitoon depend, are not precisely known, we make the decision under the condition of uncertanity and risk. Therefore, the probability theory plays and important role in the statistical decision theory, which was first explained by Abraham Wald. The problems of estimation, testing of hypothesis and decision of experiments may be regarded as the particular cases of the decision problems. Hence, the statistical decision theory has tremendous impact on the development of statistics, economics, business, industry and operation research where we make decisions in the presence of uncertanity.

Components of decision theory

suppose x₁, x₂, ..., x_n is a random sample of size n drawn from a population with probability density / mass functionƒ_θ (x), whereθ is the parameter of the distribution. Then the decision maker or statistician makes a decision on the basis of sample data (x₁. x₂, ...,x_n) depending on unknown parameterθ. The set of all possible values ofθ is called parameter space denoated byΘ. In decision problems, the probability density function or probability mass function of the random variable X is conditioned on the true value ofθ. Thereforeƒ_θ (x) can be written asƒ(x/θ) for all x∈ s where S = { x₁, x₂, ..., x_n}, the set of all possible sample points is called sample space.

In order to put a decision problem in to a single logical framework, we give definations of basic components or elements of the decision theory as follows:

1. State of nature

The different possible future events which are uncertain are said to be states of nature. The state of nature are not under the control of decision maker. These are important components of statistical decision theory. There are many states of nature in decision problems. A set of staes of nature in a decision problem is denoated byΘ = {θ₁,θ₂, ...θ_k} whereθ_i denotes the i^th value of the parameterθ; i = 1, 2, 3, ..., k; k≥ 2. For example, the result of an election such as a party A will get majority or minotary of votes, consumer demand for a product such as high, medium or low demand, etc.

The parametric value ofθ or the state of nature affeact the selection of the actions or decision making

2. Actions or Alternative

The differenttypes of choices or opinion available to a decision maker to make decision in a decision problem are known as action or alternatives. The action or alternatives are under the controll of decision maker. The actions are, in fact, the decision which are open to decision maker or statistician in decision making problems.

Science there are several cources of action in a decision making problem, it is needed to considre only some actions which are relevent to the objectives of decision maker. A set of actions including a best action is called action space and it is denoated by A. Thus an action space is A = {a₁, a_2, ..., a_m} where a_i denoates the i^th action in the decision problem; i = { 1, 2, ..., m; m≥ 2 }.

Decision function

If a random sample (x₁, x₂, ..., x_n) is drawn from a population having probability mass function (p.d.f) or probability density function (p.d.f)ƒ (x /θ ), then the sample space S = { x₁, x₂, ..., x_n} may be divided into a finate or countable infinite number of subsets depending on the number of possible actions available. So, in decision theoretic problems, we define mapping if all possible sample points or outcomes uniquely to some actions and this mapping or function is called decision function.

Definition

A decision function is defined as a rule or function which maps S into A and it is denoated by d(x).In other words, a decision function is a rule which assigns a unique action to every possible outcomes x.

3. Loss function.

A loss function is another important component of decision theory. In any decision thepretic problems, there is always uncertainty and therefore there can be some loss to the decision maker or stastician while taking an action for a certain value ofθ. So, a quantity of result of the decision made is called "loss". The loss depends upon the action a_i and the unknown parameterθ. The loss may be either positive or negative and the negative loss represents the profit or gain to the stastician. This gain is called utility obtained by stastician.

Definitio

A loss function is defined as a real valued function which measures the loss incurred if true state of nature (i.e. the true value of the parameter) isθ and if we take an action a. The loss function is denoted by L(θ,a) for all (θ,a)∈Θ× A. If we use the decision function d(x), the loss function is denoated by L (θ, d(x)).