The Decision Problem Under Uncertainty
To those whom learn Statistics in their high schools and undergraduate courses it comes across often as a rather “boring” , “pedantic”, “math-heavy” subject with no intrinsic real world applications. I feel that this maybe due to the fact that sometimes the motivation for learning the subject is not clearly presented to the students and this in turns leads to demotivation in terms of not being able to visualize why they (the students) would need it in real life. One of the most important applications of Statistics is how to make better decisions based on the underlying body of evidence provided to you (subject to certain assumptions of the data of course ! — which you can allow for) which I hope to delve in to a bit in this article.
Inference and Modelling
So when you start learning Statistics for the first time, you learn about things like random variables and probability distributions as well as in what scenarios each of these are applicable (e.g. using a binomial distribution to model the probability of getting a certain number of heads in a fixed number of coin tosses ). How this extends to the real world is that when you start doing these exercises, you get a feel of how to model the behavior of data given certain mathematical assumptions of the data (e.g. the probability of getting a head is 1/2 for a fair coin, etc.), which in turn, allows you to model quite a bit of phenomena in the natural world to a good deal of accuracy. Usually though you are required to validate the assumptions you make — which is where statistical inference comes handy. Since the data you work is usually characterized by probability distributions with their own parameters, what this boils downs to is trying to guess which parameter in the whole possible “space” of parameter values best fits the data being modelled. (e.g. the p for the probability of getting a head for the coin toss example)
What now after Inference ? — Decisions, Decisions, Decisions !
So the typical reason for making inference about data is that based on the insight that you — the decision maker, arrives at, you will take the best course of action since now you are better informed. In order to look at how we formulate it mathematically suppose that a choice has to be made between a number of alternative actions. The data, by providing information about the probability distribution from which they came, also provide guidance as to the best decision. The problem then reduces to determining a rule which, for each set of values of the data, specifies what decision should be taken. Mathematically such a rule is a function δ, which to each possible value X of the random variables assigns a decision d = δ(x), that is, a function whose whose domain is the set of values of X and whose range is the set of possible decisions.
In order to see how δ should be chosen, the decision maker needs to compare the consequences of using different rules. Suppose that the consequence of taking decision d when the distribution of X is P(θ) is a loss function, which in turn can be expressed as a nonnegative real number L(θ, d). Then the long-term average loss that would result from the use of δ in a number of repetitions of the experiment is the expectation — E[L(θ, δ(X))].
In short, what the above mathematical terminology translates to is that when making decisions under uncertainty you need to look at minimizing the average loss with regrets to all possible choices of actions available to you, given that the data you based your decisions on, satisfies the assumptions you made about it. By being able to quantitatively evaluate uncertainty and risk when you arrive at decisions as you can see, is quite an advantageous tool to have in your toolkit when facing life in general. In order to do that of course, you need to pay attention to all the basic statistics and probability theory you learn at high school and undergraduate courses and refine your knowledge and understanding, which I have come to learn is a constant work-in-progress.
