Chapter 1. Lecture notes on statistical decision theory Econ 2110, fall 2013 Maximilian Kasy March 10, 2014 These lecture notes are roughly based on Robert, C. (2007). 3.2. Contents 1. f^\star(t) = \sum_{i=1}^n f\left(\frac{i}{n}\right) 1\left(\frac{i-1}{n}\le t<\frac{i}{n}\right), \qquad t\in [0,1]. In later lectures I will also show a non-asymptotic result between these two models. Statistical Decision Theory. We repeat the iteration for \log_2 \sqrt{n} times (assuming \sqrt{n} is a power of 2), so that finally we arrive at a vector of length m/\sqrt{n} = n^{1/2-\varepsilon} consisting of sums. This book is truly a classic for the introduction to Bayesian analysis and Decision Theory. assumed, and from . Bayesian Decision Theory is a wonderfully useful tool that provides a formalism for decision making under uncertainty. The decisions of routine […] Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Another widely-used model in nonparametric statistics is the density estimation model, where samples X_1,\cdots,X_n are i.i.d. The main result in this section is that, when s>1/2, these models are asymptotically equivalent. 1 , ω. He is semi-retired and continues to teach biostatistics and clinical trial design online to Georgetown University students. is . H^2(\mathsf{K}P_{\mathbf{Y}^{(1)}}, P_{\mathbf{Z}^{(1)}}) = O\left( \frac{m}{\sqrt{n}}\cdot \frac{1}{n^\varepsilon \cdot n^{1/2}} \right) = O(n^{-2\varepsilon}) \rightarrow 0. Bayesian inference is an important technique in statistics, and especially in mathematical statistics.Bayesian updating is particularly important in the dynamic analysis of a sequence of data. It is very closely related to the field of game theory. \ \ \ \ \ (4). In the field of statistical decision theory Professors Raiffa and Schlaifer have sought to develop new analytical tech niques by which the modern theory of utility and subjective probability can actu ally be applied … \end{array}, H^2(\mathsf{K}P_{\mathbf{Y}^{(2)}}, P_{\mathbf{Z}^{(2)}}) = o(1), Lecture 3: Statistical Decision Theory: Model Distance and Equivalence, Lecture 4: Local Asymptotic Normality and Asymptotic Theorems. Statistical theory is based on mathematical statistics. (F3) A decision theory is strict ly falsified as a norma tive theory if a decision problem can be f ound in which an agent w ho performs in accordance with the theory cannot be a rational ag ent. For example, let. Logical Decision Framework 4. It provides a practical and straightforward way for people to understand the potential choices of decision-making and the range of possible outcomes based on a series of problems. Examples of effects include the following: The average value of something may be different in one group compared to another. Decision tree example Stay comfortable and dry Bear unnecessary trouble of carrying umbrella Get wet and uncomfortable Remain dry and comfortable 24. Lemma 9 Let D_{\text{\rm KL}}(P\|Q) = \int dP\log \frac{dP}{dQ} and \chi^2(P,Q) = \int \frac{(dP-dQ)^2}{dQ} be the KL divergence and \chi^2-divergence, respectively. Otherwise, we generate i.i.d. Introduction ADVERTISEMENTS: 2. Randomization Section 1.6. • Bet on Jeb. List the payoff or profit or reward 4. ⇒ Decision theory! For k\ge 0, let F_k be the CDF of \text{Binomial}(k, 1/2), and \Phi be the CDF of \mathcal{N}(0,1). Then by Lemma 9 and Jensen’s inequality, which goes to zero uniformly in P as n\rightarrow\infty, as desired. Consider a discrete probability vector P=(p_1,\cdots,p_k) with p_i\ge 0, \sum_{i=1}^k p_i=1. Lawrence D. Brown, Andrew V. Carter, Mark G. Low, and Cun-Hui Zhang. The equivalence between nonparametric regression and Gaussian white noise models (Theorem 10) was established in Brown and Low (1996), where both the fixed and random designs were studied. Bayesian inference is an important technique in statistics, and especially in mathematical statistics.Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Introduction ADVERTISEMENTS: 2. Decision Types 3. Decision space D = fB;C;T;Hg of possible actions. Decision Rule Example. Consequently, Since s'>1/2, we may choose \varepsilon to be sufficiently small (i.e., 2s'(1-2\varepsilon)>1) to make H^2(\mathsf{K}P_{\mathbf{Y}^{(2)}}, P_{\mathbf{Z}^{(2)}}) = o(1). with s=m+\alpha, m\in {\mathbb N}, \alpha\in (0,1] denotes the smoothness parameter. \sup_{\theta\in\Theta} \|Q_\theta - \mathsf{K}P_\theta \|_{\text{\rm TV}} \le \varepsilon. ADVERTISEMENTS: Read this article to learn about the decision types, decision framework and decision criteria of statistical decision theory! Equivalence between Multinomial and Poissonized Models. Here the parameter set \Theta of the unknown f is the infinite-dimensional space of all possible 1-Lipschitz functions on [0,1], and we call this model non-parametric. Motivated by this fact, we represent \mathbf{Y} and \mathbf{Z} in the following bijective way (assume that m is even): Note that (Y_1+Y_2,Y_3+Y_4,\cdots,Y_{m-1}+Y_m) is again an independent Poisson vector, we may repeat the above transformation for this new vector. The primary emphasis of decision theory may be found in the theory of testing hypotheses, originated by Neyman and Pearsonl The extension of their principle to all statistical problems was proposed by Wald2 in J. Neyman and E. S. Pearson, The testing of statistical hypothesis in relation to probability a priori. C = Take the car. August 31, 2017 1 / 20 2. In partic-ular, the aim is to give a uni ed account of algorithms and theory for sequential decision making problems, including reinforcement learning. Statistical theory is the basis for the techniques in study design and data analysis. The proof of Lemma 9 will be given in later lectures when we talk about joint ranges of divergences. A decision tree is a diagram used by decision-makers to determine the action process or display statistical probability. This site uses Akismet to reduce spam. The Bayesian choice: from decision-theoretic foundations to computational implementation. THE PROCEDURE The most obvious place to begin our investigation of statistical decision theory is with some definitions. Springer Ver-lag, chapter 2. Statistical Learning Theory and Applications Class Times: Monday and Wednesday 10:30-12:00 Units: 3-0-9 H,G Location: 46-5193 Instructors: Tomaso Poggio (TP), Lorenzo Rosasco (LR), Charlie Frogner (CF), Guille D. Canas (GJ) Ofﬁce Hours: Friday 1-2 pm in 46-5156, CBCL lounge Email Contact : 9.520@mit.edu 9.520 in 2012 Saturday, February 4, 2012. We can view statistical decision theory and statistical learning theory as di erent ways of incorporating knowledge into a problem in order to ensure generalization. The purpose of this workbook is to show, via an illustrative example, how statistical decision theory can be applied to agribusiness management. The asymptotic equivalence between nonparametric models has been studied by a series of papers since 1990s. The randomization procedure is as follows: based on the observations X_1,\cdots,X_n under the multinomial model, let P_n=(\hat{p}_1,\cdots,\hat{p}_k) be the vector of empirical frequencies. The purpose of this workbook is to show, via an illustrative example, how statistical decision theory can be applied to agribusiness management. The pioneering of statistical discrimination theory is attributed to American economists Kenneth Arrow and Edmund Phelps but has been further researched and expounded upon since its inception. Proc. Theorem 12 Sticking to the specific examples of Y_1 and Y_1 + Y_2, let P_1, P_2 be the respective distributions of the RHS in (12) and (13), and Q_1, Q_2 be the respective distributions of Z_1 + Z_2 and Z_1 - Z_2, we have, \begin{array}{rcl} H^2(P_1, Q_1) & \le & \frac{C}{n^\varepsilon (f(t_1) + f(t_2))}, \\ H^2(P_2, Q_2) & \le & C\left(\frac{f(t_1)-f(t_2)}{f(t_1)+f(t_2)} \right)^2 + Cn^\varepsilon \left(\frac{f(t_1)-f(t_2)}{f(t_1)+f(t_2)} \right)^4. H = Stay home. Contents 1. (2004). with x_i \sim P_X and y_i|x_i\sim \mathcal{N}(x_i^\top \theta, \sigma^2). Statistical decision theory. List the possible alternatives (actions/decisions) 2. \mathop{\mathbb E}_{X^n}\chi^2(P_n,P ) = \sum_{i=1}^k \frac{\mathop{\mathbb E}_{X^n} (\hat{p}_i-p_i)^2 }{p_i} = \sum_{i=1}^k \frac{p_i(1-p_i)}{np_i} = \frac{k-1}{n}. \ \ \ \ \ (7), \lim_{n\rightarrow\infty} \Delta(\mathcal{M}_n, \mathcal{N}_n)=0, \lim_{n\rightarrow\infty} \varepsilon_n=0, \|\mathcal{N}_P- \mathcal{N}_P' \|_{\text{TV}} = \mathop{\mathbb E}_m \mathop{\mathbb E}_{X^n} \|P_n^{\otimes m} - P^{\otimes m} \|_{\text{TV}}, \ \ \ \ \ (8), D_{\text{\rm KL}}(P\|Q) = \int dP\log \frac{dP}{dQ}, \begin{array}{rcl} \mathop{\mathbb E}_{X^n} \|P_n^{\otimes m} - P^{\otimes m} \|_{\text{TV}} & \le & \mathop{\mathbb E}_{X^n}\sqrt{\frac{1}{2} D_{\text{KL}}(P_n^{\otimes m},P^{\otimes m} ) }\\ &=& \mathop{\mathbb E}_{X^n}\sqrt{\frac{m}{2} D_{\text{KL}}(P_n,P ) } \\ &\le& \mathop{\mathbb E}_{X^n}\sqrt{\frac{m}{2} \chi^2(P_n,P ) }\\ &\le& \sqrt{\frac{m}{2} \mathop{\mathbb E}_{X^n}\chi^2(P_n,P ) }. \end{array}. Hence, people typically map the risk functions into scalars and arrive at the following minimax and Bayesian paradigm. The following theorem shows that model deficiency is in fact equivalent to approximate randomization. INTRODUCTION Automated agents often have several alternatives to choose from in order to solve a problem. Theory Keywords Decision theory 1. \Box, 3.3. The proof is completed. where U\sim \text{Uniform}([-1/2,1/2]) is an independent auxiliary variable. loss function . where \|P-Q\|_{\text{\rm TV}} := \frac{1}{2}\int |dP-dQ| is the total variation distance between probability measures P and Q. For example, if obesity is associated with hypertension, then body mass index may be correlated with systolic blood pressure. The equivalence of the density estimation model and others (Theorem 11) was established in Brown et al. 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