SMHS DecisionTheory

Scientific Methods for Health Sciences - Decision Theory

Overview

Decision theory is concerned with determining the optimal course of action when a number of alternatives, whose consequences cannot be forecasted with certainty, are present. Namely, decision theory is method to make decisions in the presence of statistical knowledge when some uncertainties are involved. In this section, we present an introduction to decision theory and illustrate its application with specific examples. Sample R codes will also be provided to help apply decision theory in the programming background.

Motivation

Suppose a drug company is deciding whether they should market a new drug. Two of the main factors to consider including the proportion of people for which the drug will prove effective $(\theta_{1})$ and the proportion of the market the drug will capture ($\theta_{2})$. Both of these two factors are generally unknown even with experiments conducted to obtain statistical information about them. This kind of problem is one of the application where decision theory in that ultimate purpose is to decide whether to market the drug and how much to market and questions like this. So, what is decision theory and how does it work?

Theory

• Decision theory: concerned with the problem of making decisions in the presence of statistical knowledge which sheds light on some of the uncertainty involved in the decision problem. In most cases, we will assume that these uncertainties can be considered to be unknown numerical quantities, and will represent them by $\theta$, which could be a vector or matrix.
• Statistics is directed towards the use of sample information in making references about $\theta$ without regard to the use to which they are to be put. Beside, we try to combine the sample information with other relevant aspects of the problem in order to make the optimal decisions. The relevant information include knowledge of the possible consequences of the decision, quantified by determining the loss that would be incurred for each possible decision and for various think in terms of losses and non-sample information that is useful to consider, which is called prior information considering about $\theta$ arising from sources other than statistical investigation. Generally speaking, prior information comes from past experience about similar situations involving similar $\theta$ and l as the set of all possible actions under consideration.
• The uncertain quantity $\theta$, which affects the decision process is commonly referred to as the state of nature. It is clearly important to consider what the possible states of nature are when making decisions. We use the symbol $\Theta$ to denote the set of all possible states of nature (parameter space) and $\theta$ (parameter). Loss function is an important element in decision theory. If a particular action $a_{1}$ is taken and $\theta_{1}$ turns out to be the true state of nature, then a loss function $L(\theta_{1},a_{1})$ is defined for all $(\theta,a) \in\Theta×\ell.$ For technical convenience, only loss function satisfying $L(\theta,a)≥-K>-\infty$ will be considered.
• With a statistical investigation, the outcome will be denoted as X, which is often referred to as a vector $X=(X_{1},X_{2},…,X_{n})$, where $X_{i}$ are independent observations from a common distribution. A particular realization of X will be denoted x and the set of possible outcomes is the sample space, which is denoted as $\mathcal {L}$, usually a subset of $R^{n}$, n-dimensional Euclidean space. The possible distribution of X depends on the unknown state of nature $\theta$. Let $P_{\theta}(A)$ or $P_{\theta}$ $(X\in A)$ denote the probability of the event $A(A\subset \mathcal {L}$ when $\theta$ is the true state of nature. For simplicity $X$ will be assumed to be either continuous or discrete random variable with density $\mathcal{f}(x|\theta)$. If $ X $ is continuous then $P_{θ} (A)=\int_{A}\mathcal {f}(x│\theta)dx$ when $X$ is discrete $P_{\theta}(A)=\sum_{X\in A} \mathcal {f}(x│\theta)$.

References

• Chapter 3 Decision Theory / York University
• Decision Theory Wikipedia
• Statistical Decision Theory and Bayesian Analysis / James O. Berger

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• SOCR Home page: http://www.socr.umich.edu

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