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A Comprehensive Study of Statistical Decision Theory

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Decision theory is an interdisciplinary area of study that concerns mathematicians, statisticians, economists, philosophers, managers, politicians, psychologists and anyone else interested in analysis of decisions and their consequences. Decision theory in philosophy, mathematics and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision. A vast amount of effort has been devoted to explore this subject. Psychologists have studied how decision-makers work under different conditions. From a philosophical perspective, even the existence of such a thing as a “good decision” has been questioned. The logical approach has contributed to the understanding of the decision-making process. Mathematics, including Statistics, has played a major role in providing a formal structure for the process and defining criteria for optimality. The paper contains detailed study of Statistical Decision Theory .

INTRODUCTION.

One of the most common activities of human beings is that of decision-making. Every
person is constantly deciding on a wide variety of different subjects. There are easy as well as difficult decisions; there are important and irrelevant decisions; one must face personal and professional decisions. In the end, we all know, from our particular experience, that there are good and bad decisions, so a natural question arises. Are there any rules or procedures for decision-making which “guarantee” that the final result is a “good” decision? A vast amount of effort has been devoted to explore this subject. Psychologists have studied how decision-makers work under different conditions. From a philosophical perspective, even the existence of such a thing as a “good decision” has been questioned. The logical approach has contributed to the understanding of the decision-making process. Mathematics, including Statistics, has played a major role in providing a formal structure for the process and defining criteria for optimality. Under the heading of Decision Theory, the literature offers an account of the ways people actually make decisions and a discussion on the mechanisms underlying this behavior. This is called a “descriptive” decision theory. On the other hand, we can also find discussions about the principles to consider to make rational decisions. In this case, we have a “normative” decision theory.


What is Decision Making?
•    Decision making is the study of identifying and choosing alternatives based on the values and preferences of the decision maker. Making a decision implies that there are alternative choices to be considered, and in such a case we want not only to identify as many of these alternatives as possible but to choose the one that (1) has the highest probability of success or effectiveness and (2) best fits with our goals, desires, lifestyle, values, and so on.
•    Decision making is the process of sufficiently reducing uncertainty and doubt about alternatives to allow a reasonable choice to be made from among them. This definition stresses the information-gathering function of decision making. It should be noted here that uncertainty is reduced rather than eliminated. Very few decisions are made with absolute certainty because complete knowledge about all the alternatives is seldom possible. Thus, every decision involves a certain amount of risk. If there is no uncertainty, you do not have a decision; you have an algorithm--a set of steps or a recipe that is followed to bring about a fixed result.
Kinds of Decisions
There are several basic kinds of decisions.
•    Decisions whether. This is the yes/no, either/or decision that must be made before we proceed with the selection of an alternative. Should I buy a new TV? Should I travel this summer? Decisions whether are made by weighing reasons pro and con.  It is important to be aware of having made a decision whether, since too often we assume that decision making begins with the identification of alternatives, assuming that the decision to choose one has already been made.
•    Decisions which. These decisions involve a choice of one or more alternatives from among a set of possibilities, the choice being based on how well each alternative measures up to a set of predefined criteria.
•    Contingent decisions. These are decisions that have been made but put on hold until some condition is met. 
For example, I have decided to buy that car if I can get it for the right price; I have decided to write that article if I can work the necessary time for it into my schedule. 
Most people carry around a set of already made, contingent decisions, just waiting for the right conditions or opportunity to arise. Time, energy, price, availability, opportunity, encouragement--all these factors can figure into the necessary conditions that need to be met before we can act on our decision. Some contingent decisions are unstated or even exist below the awareness of the decision maker. These are the type that occur when we seize opportunity. We don't walk around thinking, "If I see a new laser printer for $38, I'll buy it," but if we happen upon a deal like that and we have been contemplating getting a new printer, the decision is made quickly. Decisions made in sports and warfare are like this. The best contingent and opportunistic decisions are made by the prepared mind--one that has thought about criteria and alternatives in the past.
The Components of Decision Making
The Decision Environment
Every decision is made within a decision environment, which is defined as the collection of information, alternatives, values, and preferences available at the time of the decision. An ideal decision environment would include all possible information, all of it accurate, and every possible alternative. However, both information and alternatives are constrained because the time and effort to gain information or identify alternatives are limited. The time constraint simply means that a decision must be made by a certain time. The effort constraint reflects the limits of manpower, money, and priorities. (You wouldn't want to spend three hours and half a tank of gas trying to find the very best parking place at the mall.) Since decisions must be made within this constrained environment, we can say that the major challenge of decision making is uncertainty, and a major goal of decision analysis is to reduce uncertainty. We can almost never have all information needed to make a decision with certainty, so most decisions involve an undeniable amount of risk.
The fact that decisions must be made within a limiting decision environment suggests two things. First, it explains why hindsight is so much more accurate and better at making decisions that foresight. As time passes, the decision environment continues to grow and expand. New information and new alternatives appear--even after the decision must be made. Armed with new information after the fact, the hindsighters can many times look back and make a much better decision than the original maker, because the decision environment has continued to expand.
The second thing suggested by the decision-within-an-environment idea follows from the above point. Since the decision environment continues to expand as time passes, it is often advisable to put off making a decision until close to the deadline. Information and alternatives continue to grow as time passes, so to have access to the most information and to the best alternatives, do not make the decision too soon. Now, since we are dealing with real life, it is obvious that some alternatives might no longer be available if too much time passes; that is a tension we have to work with, a tension that helps to shape the cutoff date for the decision.
•    Delaying a decision as long as reasonably possible, then, provides three benefits:
•    The decision environment will be larger, providing more information. There is also time for more thoughtful and extended analysis.
•    New alternatives might be recognized or created.
•    The decision maker's preferences might change. With further thought, wisdom, and maturity, you may decide not to buy car X and instead to buy car Y.
And delaying a decision involves several risks:
1. As the decision environment continues to grow, the decision maker might become overwhelmed with too much information and either make a poorer decision or else face decision paralysis.
2. Some alternatives might become unavailable because of events occurring during the delay. In a few cases, where the decision was between two alternatives (attack the pass or circle around behind the large rock), both alternatives might become unavailable, leaving the decision maker with nothing. And we have all had the experience of seeing some amazing bargain only to hesitate and find that when we go back to buy the item, it is sold out.
3. In a competitive environment, a faster rival might make the decision and gain advantage. Another manufacturer might bring a similar product to market before you (because that company didn't delay the decision) or the opposing army might have seized the pass while the other army was "letting the decision environment grow."
The Effects of Quantity on Decision Making
Many decision makers have a tendency to seek more information than required to make a good decision. When too much information is sought and obtained, one or more of several problems can arise. (1) A delay in the decision occurs because of the time required to obtain and process the extra information. This delay could impair the effectiveness of the decision or solution. (2) Information overload will occur. In this state, so much information is available that decision-making ability actually declines because the information in its entirety can no longer be managed or assessed appropriately. A major problem caused by information overload is forgetfulness. When too much information is taken into memory, especially in a short period of time, some of the information (often that received early on) will be pushed out.
The example is sometimes given of the man who spent the day at an information-heavy seminar. At the end of the day, he was not only unable to remember the first half of the seminar but he had also forgotten where he parked his car that morning.
(3) Selective use of the information will occur. That is, the decision maker will choose from among all the information available only those facts which support a preconceived solution or position. (4) Mental fatigue occurs, which results in slower work or poor quality work. (5) Decision fatigue occurs where the decision maker tires of making decisions. Often the result is fast, careless decisions or even decision paralysis--no decisions are made at all.
The quantity of information that can be processed by the human mind is limited. Unless information is consciously selected, processing will be biased toward the first part of the information received. After that, the mind tires and begins to ignore subsequent information or forget earlier information. (Have you ever gone shopping for something where you looked at many alternatives--cars, knives, phones, TVs--only to decide that you liked the first one best?)



Approaches to Decision Making
There are two major approaches to decision making in an organization, the authoritarian method in which an executive figure makes a decision for the group and the group method in which the group decides what to do.
1. Authoritarian. The manager makes the decision based on the knowledge he can gather. He then must explain the decision to the group and gain their acceptance of it. In some studies, the time breakdown for a typical operating decision is something like this:
make decision, 5 min.; explain decision, 30 min.; gain acceptance, 30 min.
2. Group. The group shares ideas and analyses, and agrees upon a decision to implement. Studies show that the group often has values, feelings, and reactions quite different from those the manager supposes they have. No one knows the group and its tastes and preferences as well as the group itself. And, interestingly, the time breakdown is something like this:
group makes decision, 30 min.; explain decision, 0 min.; gain acceptance, 0 min.
Clearly, just from an efficiency standpoint, group decision making is better. More than this, it has been shown many times that people prefer to implement the ideas they themselves think of. They will work harder and more energetically to implement their own idea than they would to implement an idea imposed on them by others. We all have a love for our own ideas and solutions, and we will always work harder on a solution supported by our own vision and our own ego than we will on a solution we have little creative involvement with.
There are two types of group decision making sessions. First is free discussion in which the problem is simply put on the table for the group to talk about. For example, Joe has been offered a job change from shift supervisor to maintenance foreman. Should he take the job?
The other kind of group decision making is developmental discussion or structured discussion. Here the problem is broken down into steps, smaller parts with specific goals. For example, instead of asking generally whether Joe should take the job, the group works on sub questions: What are Joe's skills? What skills does the new job require? How does Joe rate on each of the skills required? Notice that these questions seek specific information rather than more general impressionistic opinions.
Developmental discussion (1) insures systematic coverage of a topic and (2) insures that all members of the group are talking about the same aspect of the problem at the same time.


Criterion of Decision Making Under Uncertainty
 There are four types of criteria that we will look at.
Expected Value (Realist)
Compute the expected value under each action and then pick the action with the largest expected value. This is the only method of the four that incorporates the probabilities of the states of nature. The expected value criterion is also called the Bayesian principle.

Maximax (Optimist)
The maximax looks at the best that could happen under each action and then chooses the action with the largest value. They assume that they will get the most possible and then they take the action with the best best case scenario. The maximum of the maximums or the "best of the best". This is the lotto player; they see large payoffs and ignore the probabilities.

Maximin (Pessimist)
The maximin person looks at the worst that could happen under each action and then choose the action with the largest payoff. They assume that the worst that can happen will, and then they take the action with the best worst case scenario. The maximum of the minimums or the "best of the worst". This is the person who puts their money into a savings account because they could lose money at the stock market.

Minimax (Opportunist)
Minimax decision making is based on opportunistic loss. They are the kind that look back after the state of nature has occurred and say "Now that I know what happened, if I had only picked this other action instead of the one I actually did, I could have done better". So, to make their decision (before the event occurs), they create an opportunistic loss (or regret) table. Then they take the minimum of the maximum. That sounds backwards, but remember, this is a loss table. This similar to the maximin principle in theory; they want the best of the worst losses.

Hurwitz Criterion
           Hurwitz Criterion is weighted ratio of minimax and maximax criteria.
Hurwitz suggested a compromise between optimistic and pessimistic. He gave    the concept of coefficient of optimism=0 for pessimistic and C=1 for optimistic.
             The decision maker has to decide the value of C between 0 and 1. He will select the alternative for which Hurwitz value is maximum. Hurwitz value is given by
Hurwitz value = Maximum payoff for alternative x C+ Minimum payoff for alternative x (1-C)


CONCLUSIONS

•    Decision theory provides a useful framework to explore alternatives.
•    It forces us to recognize that deciding not to take action is just as much a decision
      as deciding which action to take.
•    It forces us to recognize that we may err either by taking an unnecessary action
      or by failing to take a necessary action.
•    It helps us formalize and categorize our thinking to make sure that we have considered all relevant possibilities.
•    Quantitative analyses must be viewed as explorations of possibilities, not hard predictions, but the process of quantization may help us to clarify our thinking.


REFERENCES

•    L. A. Maguire. Risk analysis for conservation biologists. Conservation Biology, 5:123{125, 1991.

•    Lynn A. Maguire, Ulysses S. Seal, and Peter F. Brussard. Manging critically endangered species: the sumatran rhino as a case study. In Michael E. Soul, editor, Viable Populations for Conservation, pages 141{158. Cambridge Univ. Press, Cambridge, 1987. 11