SMHS DecisionTheory - Revision history

Dinov: /* Theory */

2014-09-04T17:29:22Z

‎Theory

Dinov: /* Theory */

2014-09-04T17:28:01Z

‎Theory

Dinov: /* Theory */

2014-09-04T17:12:10Z

‎Theory

Dinov: /* Theory */

2014-09-04T17:08:10Z

‎Theory

Dinov: /* Decision principles */

2014-09-04T17:00:21Z

‎Decision principles

Dinov: /* Applications */

2014-09-04T16:53:26Z

‎Applications

Dinov: /* Decision principles */

2014-09-04T16:51:43Z

‎Decision principles

Dinov: /* Decision principles */

2014-09-04T16:50:24Z

‎Decision principles

Dinov: /* Decision principles */

2014-09-04T16:48:53Z

‎Decision principles

Dinov: /* Decision principles */

2014-09-04T16:34:29Z

‎Decision principles

@@ Line 12: / Line 12: @@
 *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 $A\subset \mathcal {L}$ 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 $P_{\theta}(A)=\sum_{X\in A} \mathcal {f}(x│\theta)$.
-*Example: In the drug example above, assume it is desired to estimate $\theta_{2}$, which is a proportion and $\Theta={\theta_{2}:0≤\theta_{2}≤1}=[0,1]$. The action is the process of estimating $\theta_{2}$, hence $\ell=[0,1]$ as for estimation actions the action and parameter spaces coincide ($\ell \identity \Theta$). The company might determine the loss function (in units of utility) to be
+*Example: In the drug example above, assume it is desired to estimate $\theta_{2}$, which is a proportion and $\Theta={\theta_{2}:0≤\theta_{2}≤1}=[0,1]$. The action is the process of estimating $\theta_{2}$, hence $\ell=[0,1]$ as for estimation actions the action and parameter spaces coincide ($\ell \equiv \Theta$). The company might determine the loss function (in units of utility) to be
 $$
 L(\theta_{2},a) =

@@ Line 12: / Line 12: @@
 *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 $A\subset \mathcal {L}$ 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 $P_{\theta}(A)=\sum_{X\in A} \mathcal {f}(x│\theta)$.
-*Example: In the drug example above, assume it is desired to estimate $\theta_{2}$, which is a proportion and $\Theta={\theta_{2}:0≤\theta_{2}≤1}=[0,1]$. Action need to estimate a set of $\theta_{2}$. Hence $\ell=[0,1]$. The company might determine the loss function (in units of utility) to be
+*Example: In the drug example above, assume it is desired to estimate $\theta_{2}$, which is a proportion and $\Theta={\theta_{2}:0≤\theta_{2}≤1}=[0,1]$. The action is the process of estimating $\theta_{2}$, hence $\ell=[0,1]$ as for estimation actions the action and parameter spaces coincide ($\ell \identity \Theta$). The company might determine the loss function (in units of utility) to be
 $$
 L(\theta_{2},a) =
 \begin{cases}
 \theta_{2}-a, & \text{if }\theta_{2}-1\ge 0 \\
 (a-\theta_{2}), & \text{if } \theta_{2}-a\le 0
 \end{cases}.
 $$

@@ Line 21: / Line 21: @@
 $$
-: Note that an overestimate of demand is considered twice as costly as an underestimate of demand, and that otherwise the loss is linear in the error. We could perform a sample survey to get reasonable information about $\theta_{2}$. For example, assume n people are interviewed and the number $X$ who would buy the drug is observed. It might be reasonable to assume that $X$ is $B(n,\theta_{2})$, which has density function $f(x│θ_{2})={n\choose x} \theta_{2}^{x} (1-\theta_{2} )^{n-x}$. There could well be considerable prior information about $\theta_{2}$ arising from previous introductions of new similar drugs into the market. Recall in the past, new drugs tended to capture between $\frac{1}{10}$ and $\frac{1}{5}$ of the market with all values between $\frac{1} {10}$ and $\frac{1}{5}$ being equally likely. This prior information could be modeled by giving $\theta_{2}$ a $U(0.1,0.2)$ prior density, i.e. letting $\pi(\theta_{2})=10I_{(0.1,0.2)}\theta_{2}$. The above estimates of $L$, $f$, and $\pi$ are quite crude and usually much more detailed constructions are required to obtain satisfactory results. The techniques for doing this will be developed as we proceed.
+: Note that an overestimate of demand is considered twice as costly as an underestimate of demand, and that otherwise the loss is linear in the error. We could perform a sample survey to get reasonable information about $\theta_{2}$. For example, assume n people are interviewed and the number $X$ who would buy the drug is observed. It might be reasonable to assume that $X$ is $B(n,\theta_{2})$, which has density function $f(x│θ_{2})={n\choose x} \theta_{2}^{x} (1-\theta_{2} )^{n-x}$. There could well be considerable prior information about $\theta_{2}$ arising from previous introductions of new similar drugs into the market. Assume new drugs tend to capture between $\frac{1}{10}$ and $\frac{1}{5}$ of the market with all values between $\frac{1} {10}$ and $\frac{1}{5}$ being equally likely. This prior information could be modeled by giving $\theta_{2}$ a $U(0.1,0.2)$ prior density, i.e. letting $\pi(\theta_{2})=10I_{(0.1,0.2)}\theta_{2}$. The above estimates of $L$, $f$, and $\pi$ are quite crude and usually much more detailed constructions are required to obtain satisfactory results. The techniques for doing this will be developed as we proceed.
 *The main point to remember that a well-defined loss function and explicit prior information are needed in decision theory. Many statisticians chose statistical inference as a shield to ward off consideration of losses and prior information, which is a mistake for the following reasons: (1) it reports from statistical inference should be constructed so that they can be easily utilized in individual decision-making; (2) investigator may very well possess such information like losses and prior information; (3) the choice of an inference (beyond mere data summarization) can be viewed as a decision problem where the action space is the set of all possible inference statements and a loss function reflecting the success in conveying knowledge is used.

@@ Line 11: / Line 11: @@
 *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 $P_{\theta}(A)=\sum_{X\in A} \mathcal {f}(x│\theta)$.
+*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 $A\subset \mathcal {L}$ 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 $P_{\theta}(A)=\sum_{X\in A} \mathcal {f}(x│\theta)$.
 *Example: In the drug example above, assume it is desired to estimate $\theta_{2}$, which is a proportion and $\Theta={\theta_{2}:0≤\theta_{2}≤1}=[0,1]$. Action need to estimate a set of $\theta_{2}$. Hence $\ell=[0,1]$. The company might determine the loss function (in units of utility) to be
 $$

← Older revision		Revision as of 17:00, 4 September 2014
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	plot(quadraticF, 0.1, 0.2)		plot(quadraticF, 0.1, 0.2)
		+
		+	<center>[[Image:SMHS_EBook_DecisionTheory_Fig1.png\|500px]]</center>

	====Frequentist Decision Principles====		====Frequentist Decision Principles====

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 ===Applications===
-[http://www.jstor.org/discover/10.2307/2946694?uid=3739728&uid=2&uid=4&uid=3739256&sid=21104006412381 This article] suggests that decision-making under uncertainty is at least partly, case-based. It proposed a model in which cases are primitive, and provided a simple approximation of a decision rule that chooses a “best” act based on its past performance in similar cases. Each act is evaluated by the sum of the utility levels that resulted from using this act in past cases, each weighted by the similarity of that past case to the problem at hand. The formal model of case-based decision theory naturally rises to the notions of satisficing decisions and aspiration levels.
+* [http://www.jstor.org/discover/10.2307/2946694?uid=3739728&uid=2&uid=4&uid=3739256&sid=21104006412381 This article] suggests that decision-making under uncertainty is at least partly, case-based. It proposed a model in which cases are primitive, and provided a simple approximation of a decision rule that chooses a “best” act based on its past performance in similar cases. Each act is evaluated by the sum of the utility levels that resulted from using this act in past cases, each weighted by the similarity of that past case to the problem at hand. The formal model of case-based decision theory naturally rises to the notions of satisficing decisions and aspiration levels.
-[http://psycnet.apa.org/psycinfo/1993-97634-000 This article] was written to describe naturalistic decision-making. This is our attempt to understand how human decision makers actually make decisions in complex real-world settings and to learn how to support those processes.  Section A introduces the main themes of naturalistic decision-making, describes classical decision theory in order to discuss some of its limitations, and presents examples of the types of decisions that need to be explained.  Section B presents examples of naturalistic research paradigms that have emerged within the last few years.  Section C examines a range of issues concerning our need to develop methodology to conduct research in naturalistic settings.  Section D examines applications and extensions of naturalistic decision-making.  Section E attempts to evaluate the issues raised by this book. This article]
+* [http://psycnet.apa.org/psycinfo/1993-97634-000 This article] was written to describe naturalistic decision-making. This is our attempt to understand how human decision makers actually make decisions in complex real-world settings and to learn how to support those processes.  Section A introduces the main themes of naturalistic decision-making, describes classical decision theory in order to discuss some of its limitations, and presents examples of the types of decisions that need to be explained.  Section B presents examples of naturalistic research paradigms that have emerged within the last few years.  Section C examines a range of issues concerning our need to develop methodology to conduct research in naturalistic settings.  Section D examines applications and extensions of naturalistic decision-making.  Section E attempts to evaluate the issues raised by this book.
 ===Problems===

@@ Line 61: / Line 61: @@
 \end{cases}
 $$
-: is minimized ($ρ(π,a^*)=\frac{1}{30}$) for $a^*=2/15$. This would be the estimate for $θ_2$, the market share of the new drug assuming no data was available.
+: is minimized ($ρ(π,a^*)=\frac{1}{30}=0.03333333$) for $a^*=2/15=0.1333333$. This would be the estimate for $θ_2$, the market share of the new drug assuming no data was available.
 : You can try the following R code:

@@ Line 58: / Line 58: @@
 .15-1, & \text{if } a\le 0.1 \\
 a^{2}-4a+0.3, & \text{if } 0.1\le a\le 0.2 \\
-a-0.3, & \text{if } a\le 0.2
+a-0.3, & \text{if } a\ge 0.2
 \end{cases}
 $$
@@ Line 65: / Line 65: @@
 You can try the following R code:
   quadraticF <- function (x){
-      result = 15*x*x -4*x +0.3
+      result <- rep(0, length(x))
       return(result)}
   quadraticF(2/15)
   optim(0.1, quadraticF, method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B",
-  "SANN","Brent"), lower = -10, upper = 10)
+  "SANN","Brent"), lower = 0.1, upper = 0.2)
   $\$ $par
@@ Line 76: / Line 80: @@
   $\$ $value
   [1] 0.03333333
 ====Frequentist Decision Principles====