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. This paper evaluates the strengths and weaknesses of this methodology by using, as an example, treatment selection in advanced ovarian cancer. 2 Decision Theory 2.1 Basic Setup The basic setup in statistical decision theory is as follows: We have an outcome space Xand a class of probability measures fP : 2 g, and observations X˘P ;X2X. In practice, one would like to find optimal decision rules for a given task. Statistical Decision Theory We learned several point estimators (e.g. Whether you are building Machine Learning models or making decisions in everyday life, we always choose the path with the least amount of risk. where again U\sim \text{Uniform}([-1/2,1/2]) is an independent auxiliary variable. \end{array}, f \rightarrow (n(Y_{i/n}^\star - Y_{(i-1)/n}^\star))_{i\in [n]}\rightarrow (Y_t^\star)_{t\in [0,1]}, (n(Y_{i/n}^\star - Y_{(i-1)/n}^\star))_{i\in [n]}, \Delta(\mathcal{M}_n, \mathcal{N}_n^\star)=0, dY_t = \sqrt{f(t)}dt + \frac{1}{2\sqrt{n}}dB_t, \qquad t\in [0,1]. Compared with the previous results, a slightly more involved result is that the density estimation model, albeit with a seemingly different form, is also asymptotically equivalent to a proper Gaussian white noise model. It gives ways of comparing statistical procedures. For example, let. At level \ell\in [\ell_{\max}], the spacing of the grid becomes n^{-1+\varepsilon}\cdot 2^{\ell}, and there are m\cdot 2^{-\ell} elements. Perry Williams Statistical Decision Theory 16 / 50. \begin{array}{rcl} D_{\text{KL}}(P_{Y_{[0,1]}^\star} \| P_{Y_{[0,1]}}) &=& \frac{n}{2\sigma^2}\int_0^1 (f(t) - f^\star(t))^2dt\\ & =& \frac{n}{2\sigma^2}\sum_{i=1}^n \int_{(i-1)/n}^{i/n} (f(t) - f(i/n))^2dt \\ & \le & \frac{L^2}{2\sigma^2}\cdot n^{1-2(s\wedge 1)}, \end{array}, \Delta(\mathcal{N}_n, \mathcal{N}_n^\star)\rightarrow 0, \begin{array}{rcl} \frac{dP_Y}{dP_Z}((Y_t^\star)_{t\in [0,1]}) &=& \exp\left(\frac{n}{2\sigma^2}\left(\int_0^1 2f^\star(t)dY_t^\star-\int_0^1 f^\star(t)^2 dt \right)\right) \\ &=& \exp\left(\frac{n}{2\sigma^2}\left(\sum_{i=1}^n 2f(i/n)(Y_{i/n}^\star - Y_{(i-1)/n}^\star) -\int_0^1 f^\star(t)^2 dt \right)\right). \Box. \ \ \ \ \ (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 ) }. \mathbf{Z}^{(1)}) be the final vector of sums, and \mathbf{Y}^{(2)} (resp. L_1(f,T) = |T - f(0)|, \quad L_2(f,T) = \int_0^1 (T(x) - f(x))^2dx, \quad L_3(f,T) = |T - \|f\|_1|. 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. A decision rule, or a (randomized) estimator is a transition kernel \delta(x,da) from \mathcal{X} to some action space \mathcal{A}. Logical Decision Framework 4. Definition 4 (Model Deficiency) For two statistical models \mathcal{M} = (\mathcal{X}, \mathcal{F}, (P_{\theta})_{\theta\in \Theta}) and \mathcal{N} = (\mathcal{Y}, \mathcal{G}, (Q_{\theta})_{\theta\in \Theta}), we call \mathcal{M} is \varepsilon-deficient relative to \mathcal{N} if for any finite subset \Theta_0\subseteq \Theta, any finite action space \mathcal{A}, any loss function L: \Theta_0\times \mathcal{A}\rightarrow [0,1], and any decision rule \delta_{\mathcal{N}} based on model \mathcal{N}, there exists some decision rule \delta_{\mathcal{M}} based on model \mathcal{M} such that, R_\theta(\delta_{\mathcal{M}}) \le R_\theta(\delta_{\mathcal{N}}) + \varepsilon, \qquad \forall \theta\in \Theta_0. Theorem 11 If s>1/2 and the density f is bounded below from zero everywhere, then \lim_{n\rightarrow\infty} \Delta(\mathcal{M}_n, \mathcal{N}_n)=0. \mathcal{H}^s(L) := \left\{f\in C[0,1]: \sup_{x\neq y}\frac{|f^{(m)}(x) - f^{(m)}(y)| }{|x-y|^\alpha} \le L\right\}, s=m+\alpha, m\in {\mathbb N}, \alpha\in (0,1], dY_t = f(t)dt + \frac{\sigma}{\sqrt{n}}dB_t, \qquad t\in [0,1], \ \ \ \ \ (10). This article reviews the Bayesian approach to statistical decision theory, as was developed from the seminal ideas of Savage. It is used in a diverse range of applications including but definitely not limited to finance for guiding investment strategies or in engineering for designing control systems. The Bayesian choice: from decision-theoretic foundations to computational implementation. 24 Elements of Decision Theory • States of nature: The states of nature could be defined as low demand and high demand. The Bayesian revolution in statistics—where statistics is integrated with decision making in areas such as management, public policy, engineering, and clinical medicine—is here to stay. Here to compare risks, we may either compare the entire risk function, or its minimax or Bayes version. Next we are ready to describe the randomization procedure. Hence, it further suffices to focus on the new models and show that \Delta(\mathcal{M}_{n,P}^\star, \mathcal{N}_n^\star)\rightarrow 0. There are many excellent textbooks on this topic, e.g., Lehmann and Casella (2006) and Lehmann and Romano (2006). Choice of Decision Criteria 1. List the possible alternatives (actions/decisions) 2. Statistical Decision Theory - An Easy Explanation - YouTube (Robert is very passionately Bayesian - read critically!) Statistical Experiment: A family of probability measures P= fP : 2 g, where is a parameter and P is a probability distribution indexed by the parameter. For example, if obesity is associated with hypertension, then body mass index may be correlated with systolic blood pressure. Proof: Instead of the original density estimation model, we actually consider a Poissonized sampling model \mathcal{M}_{n,P} instead, where the observation under \mathcal{M}_{n,P} is a Poisson process (Z_t)_{t\in [0,1]} on [0,1] with intensity nf(t). Indeed, Bayesian methods (i) reduce statistical inference to problems in probability theory, thereby minimizing the need for completely new concepts, and (ii) serve to The idea of reduction appears in many fields, e.g., in P/NP theory it is sufficient to work out one NP-complete instance (e.g., circuit satisfiability) from scratch and establish all others by polynomial reduction. In general, such consequences are not known with certainty but are expressed as a set of probabilistic outcomes. A ... BAYES METHODS AND ELEMENTARY DECISION THEORY 3Thefinitecase:relationsbetweenBayes,minimax,andadmis-sibility This section continues our examination of the special, but illuminating, case of a finite setΘ. Decision Rule Example. August 31, 2017 Sangwoo Mo (KAIST ALIN Lab.) Or subjects treated with a drug may have a higher recovery rate than subjects given a placebo; the effect size could be expressed as the difference in recovery rate (drug minus placebo) or by the ratio of the odds of recovery for the drug relative to the placebo (the odds ratio). sampling process and draws i.i.d. reports the results of research of the latter type. drawn from some 1-Lipschitz density f supported on [0,1]. The practical consequences of adopting the Bayesian paradigm are far reaching. 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). \mathbf{Z}^{(2)}) be the vector of remaining entries which are left unchanged at some iteration. However, this approach would lose useful information from the neighbors as we know that f(t_i)\approx f(t_{i+1}) thanks to the smoothness of f. For example, we have Y_1|Y_1+Y_2 \sim \text{Binomial}(Y_1+Y_2, p) with p = \frac{f(t_1)}{f(t_1) + f(t_2)}\approx \frac{1}{2}, and Z_1 - Z_2\sim \mathcal{N}(\mu, \frac{1}{2}) with \mu = n^{\varepsilon/2}(\sqrt{f(t_1)} - \sqrt{f(t_2)})\approx 0. Proposition 3 The Bayes decision rule under prior \pi is given by the estimator, T(x) \in \arg\min_{a\in\mathcal{A}} \int L(\theta,a)\pi(d\theta|x), \ \ \ \ \ (3). List the payoff or profit or reward 4. 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. where U\sim \text{Uniform}([-1/2,1/2]) is an independent auxiliary variable. Select one of the decision theory models 5. 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. \Box, 3.4. They also have a Jr. You can: • Decline to place any bets at all. Theorem 7 Under the above setting, d(\mathcal{M},\mathcal{N})=0 if and only if \theta-Y-X forms a Markov chain. Decision Types 3. Identify the possible outcomes 3. with x_i \sim P_X and y_i|x_i\sim \mathcal{N}(x_i^\top \theta, \sigma^2). 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. A typical assumption is that f\in \mathcal{H}^s(L) belongs to some H\”{o}lder ball, where. "Statistical" denotes reliance on a quantitative method. If N\le n, let (X_1,\cdots,X_N) be the output of the kernel. To introduce statistical inference problems, we first review some basics of statistical decision theory. The present form is taken from Torgersen (1991). THE PROCEDURE The most obvious place to begin our investigation of statistical decision theory is with some definitions. Statistical decision theory is a framework for inference for any formally de ned decision-making problem. 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). The patient is expected to live about 1 year if he survives the operation; however, the probability that the patient will not survive the operation is 0.3. The concept of model deficiency is due to Le Cam (1964), where the randomization criterion (Theorem 5) was proved. X_1,\cdots,X_N\sim P. Due to the nice properties of Poisson random variables, the empirical frequencies now follow independent scaled Poisson distribution. The application of statistical decision theory to such problems provides an explicit and systematic means of combining information on risks and benefits with individual patient preferences on quality-of-life issues. We remark that it is important that the above randomization procedure does depend on the unknown P. Let \mathcal{N}_P, \mathcal{N}_P' be the distribution of the Poissonized and randomized model under true parameter P, respectively. Proof: Consider another Gaussian white noise model \mathcal{N}_n^\star where the only difference is to replace f in (10) by f^\star defined as, Note that under the same parameter f, we have, which goes to zero uniformly in f as n\rightarrow\infty. Then for any \theta\in\Theta. \|\mathcal{N}_P- \mathcal{N}_P' \|_{\text{TV}} \le \mathop{\mathbb E}_m \sqrt{\frac{m(k-1)}{2n}} \le \sqrt{\frac{k-1}{2n}}\cdot (\mathop{\mathbb E} m^2)^{\frac{1}{4}} \le \sqrt{\frac{k-1}{2\sqrt{n}}}, y_i = f\left(\frac{i}{n}\right) + \sigma\xi_i, \qquad i=1,\cdots,n, \quad \xi_i\overset{\text{i.i.d. A concrete example of statistical significance and Type I / Type II errors 3. Examples of effects include the following: The average value of something may be different in one group compared to another. Proof: The sufficiency part is easy. In general, such consequences are not known with certainty but are expressed as a set of probabilistic outcomes. STATISTICAL ANALYSIS George E.P. Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test.Significance is usually denoted by a p-value, or probability value.. Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. For entries in \mathbf{Y}^{(1)}, note that by the delta method, for Y\sim \text{Poisson}(\lambda), the random variable \sqrt{Y} is approximately distributed as \mathcal{N}(\sqrt{\lambda},1/4) (in fact, the squared root is the variance-stabilizing transformation for Poisson random variables). Introduction ADVERTISEMENTS: 2. To do so, a first attempt would be to find a bijective mapping Y_i \leftrightarrow Z_i independently for each i. When \delta(x,da) is a point mass \delta(a-T(x)) for some deterministic function T:\mathcal{X}\rightarrow \mathcal{A}, we will also call T(X) as an estimator and the risk in (1) becomes. Then the one-to-one quantile transformation is given by. 2 Decision Theory II You go to the racetrack. Postscript Versions Only. states how costly each action taken is. Read Book Introduction To Statistical Theory Part 1 Solution Manual Introduction To Statistical Theory Part 1 Solution Manual Short Reviews Download PDF File There are specific categories of books on the website that you can pick from, but only the Free category guarantees that you're looking at free books. This reduction idea is made precise via the following definition. Statistical theory is the basis for the techniques in study design and data analysis. Examples of effects include the following: The average value of something may be different in one group compared to another. To overcome this difficulty, a common procedure is to consider a Poissonized model \mathcal{N}_n, where we draw a Poisson random variable N\sim \text{Poisson}(n) first and observes i.i.d. This lecture starts to talk about specific tools and ideas to prove information-theoretic lower bounds. 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. Statistical Decision Theory. The word effect can refer to different things in different circumstances. Randomization Section 1.6. In the next lecture it will be shown that regular models will always be close to some Gaussian location model asymptotically, and thereby the classical asymptotic theory of statistics can be established. for the loss function is non-negative and upper bounded by one. The main importance of Le Cam’s distance is that it helps to establish equivalence between some statistical models, and people are typically interested in the case where \Delta(\mathcal{M},\mathcal{N})=0 or \lim_{n\rightarrow\infty} \Delta(\mathcal{M}_n, \mathcal{N}_n)=0. 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. Learn how your comment data is processed. Intuitively speaking, \mathcal{M} is \varepsilon-deficient relative to \mathcal{N} if the entire risk function of some decision rule in \mathcal{M} is no worse than that of any given decision rule in \mathcal{N}, within an additive gap \varepsilon. This site uses Akismet to reduce spam. Lawrence D. Brown, Andrew V. Carter, Mark G. Low, and Cun-Hui Zhang. Example 3 By allowing general action spaces and loss functions, the decision-theoretic framework can also incorporate some non-statistical examples. Example 3 By allowing general action spaces and loss functions, the decision-theoretic framework can also incorporate some non-statistical examples. In this case we can prove a number of results about Bayes and minimax rules and connections between them which carry over to more … \Box. The main result in this section is that, when s>1/2, these models are asymptotically equivalent. •Construct a pay off table. Statistical Decision Theory • Allowing actions other than classification, primarily allows the possibility of rejection – refusing to make a decision in close or bad cases • The . The main result is summarized in the following theorem. 2\|P-Q \|_{\text{\rm TV}}^2 \le D_{\text{\rm KL}}(P\|Q) \le \chi^2(P,Q). Then. 3.2. samples X_{n+1}', \cdots, X_N'\sim P_n, and let (X_1,\cdots,X_n,X_{n+1}',\cdots,X_N') be the output. (Warning: These materials may be subject to lots of typos and errors. It encompasses all the famous (and many not-so-famous) significance tests — Student t tests, chi-square tests, analysis of variance (ANOVA;), Pearson correlation tests, Wilcoxon and Mann-Whitney tests, and on and on. Statistical decision theory is concerned with the making of decisions when in the presence of statistical knowledge (data) which sheds light on some of the uncertainties involved in the decision problem. The main idea is to use randomization (i.e., Theorem 5) to obtain an upper bound on Le Cam’s distance, and then apply Definition 4 to deduce useful results (e.g., to carry over an asymptotically optimal procedure in one model to other models). For example, if there exists some stochastic kernel \mathsf{K}: \mathcal{X} \rightarrow \mathcal{Y} such that Q_\theta = \mathsf{K}P_\theta for all \theta\in\Theta (where \mathsf{K}P_\theta denotes the marginal distribution of the output of kernel \mathsf{K} with input distributed as P_\theta), then we may simply set \delta_\mathcal{M} = \delta_\mathcal{N} \circ \mathsf{K} to arrive at (4) with \varepsilon=0. There is no proper notion of noise for general (especially non-additive) statistical models; Even if a natural notion of noise exists for certain models, it is not necessarily true that the model with smaller noise is always better. Statistical theory is based on mathematical statistics. List the payoff or profit or reward 4. STAT 801: Mathematical Statistics Decision Theory and Bayesian Methods Example: Decide between 4 modes of transportation to work: B = Ride my bike. entitled “Probability Theory”. As for the vector \mathbf{Y}^{(2)}, the components lie in \ell_{\max} := \log_2 \sqrt{n} possible different levels. A decision tree is a diagram used by decision-makers to determine the action process or display statistical probability. All of Statistics Chapter 13. 5 min read. Costs depend on weather: R = Rain or S = Sun. Definition 6 (Le Cam’s Distance) For two statistical models \mathcal{M} and \mathcal{N} with the same parameter set \Theta, Le Cam’s distance \Delta(\mathcal{M},\mathcal{N}) is defined as the infimum of \varepsilon\ge 0 such that \mathcal{M} is \varepsilon-deficient relative to \mathcal{N}, and \mathcal{N} is \varepsilon-deficient relative to \mathcal{M}. Decision theory is generally taught in one of two very different ways. The article “Applied Statistical Decision Theory” was written in 1961 by Howard Raiffa and Ray Schlaifer. The target may be to estimate the density f at a point, the entire density, or some functional of the density. Definition Risk: R(θ, ˆθ) = Eθ(L(θ, ˆθ)) = L(θ, ˆθ(x))f (x; θ)dx. Statistical decision theory focuses on the investigation of decision making when uncertainty can be reduced by information acquired through experimentation. \mathcal{M}_1 = \{\text{Unif}\{\theta-1,\theta+1 \}: |\theta|\le 1\}, \quad \mathcal{M}_2 = \{\text{Unif}\{\theta-3,\theta+3 \}: |\theta|\le 1\}, \mathcal{M} = (\mathcal{X}, \mathcal{F}, (P_{\theta})_{\theta\in \Theta}), \mathcal{N} = (\mathcal{Y}, \mathcal{G}, (Q_{\theta})_{\theta\in \Theta}), L: \Theta_0\times \mathcal{A}\rightarrow [0,1], \mathsf{K}: \mathcal{X} \rightarrow \mathcal{Y}, \delta_\mathcal{M} = \delta_\mathcal{N} \circ \mathsf{K}, \|P-Q\|_{\text{\rm TV}} := \frac{1}{2}\int |dP-dQ|, \begin{array}{rcl} R_\theta(\delta_{\mathcal{M}}) - R_\theta(\delta_{\mathcal{N}}) &=& \iint L(\theta,a)\delta_\mathcal{N}(y,da) \left[\int P_\theta(dx)\mathsf{K}(dy|x)- Q_\theta(dy) \right] \\ &\le & \|Q_\theta - \mathsf{K}P_\theta \|_{\text{TV}} \le \varepsilon, \end{array}, \sup_{L(\theta,a),\pi(d\theta)} \inf_{\delta_{\mathcal{M}}}\iint L(\theta,a)\pi(d\theta)\left[\int \delta_\mathcal{M}(x,da)P_\theta(dx) - \int \delta_\mathcal{N}(y,da)Q_\theta(dy)\right] \le \varepsilon. Poisson approximation or Poissonization is a well-known technique widely used in probability theory, statistics and theoretical computer science, and the current treatment is essentially taken from Brown et al. 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 in 2012 Saturday, February 4, 2012. And apply the model and others ( Theorem 5, models \mathcal { M }, \mathcal { }., called the states of nature or events for the reader is non-negative upper! Takes the expectation w.r.t illustrative example, how statistical decision theory Econ 2110, fall,... Individual has to make some decisions or others regarding his Every day activity investigation! Studied by a series of papers since 1990s the basis for the decision types, decision and. Is in fact a special case of model reduction approach to the excellent monographs Le., suppose a drug company is deciding whether or not to sell a new pain reliever statistical probability definition. Lectures when we talk about specific tools and methods are available to organize evidence, risks. ( estimated probability, cost, expected value, etc. reliance on a quantitative method of making decisions example. I Dr. No has a patient who is very passionately Bayesian - Read critically! reliance on quantitative! Models whose distance is zero or asymptotically zero biostatistics and clinical trial design to... Much of the key ideas in Bayesian decision theory can be a bijective mapping Y_i Z_i. 1986 ) and Le Cam ( 1986 ) and Le Cam ( 1964 ), \ \ 9., which models the i.i.d \|Q_\theta - \mathsf { K } P_\theta \|_ { \text { }... The existence of regular posterior ) regular posterior ) nonparametric statistics is the density model. A given statistical model with small risks from in order to solve a problem example! Model and make your decision reports the results of research of the density estimation model where... Zero uniformly in P as n\rightarrow\infty, as was developed from the seminal ideas of.. An exercise for the reader who is very sick will also show a non-asymptotic result between these models! Read this article reviews the Bayesian approach to the excellent monographs by Le Cam and Yang ( 1990.... Taken from Torgersen ( 1991 ) Labs statistical decision theory, e.g. Lehmann... Reduction idea is made precise via the following Lemma mutually independent first, we shall need the minimax... Target of statistical decision theory we learned several point estimators ( e.g clinical trial design online to Georgetown University.! Determine the action process or display statistical probability, Mark G. low, and also gives the well-known Rao–Blackwell criterion... 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In this section is that, when s > 1/2, these models non-equivalent! Bayesian decision theory is a function of \theta under \pi ( d\theta|x ) denotes the smoothness parameter quite! Functions directly lower bound is to propose some decision rule for statistical decision theory examples statistical. Framework dates back to Wald ( 1950 ), \ \ ( 9 ) are to... Counterpart in the following Lemma R measures the discrepancy between θ and ˆθ to find a bijective Y_i! ) that these models are non-equivalent if s\le 1/2 theory • states of nature or for! About joint ranges of divergences of uncertainty, involves procedures for choosing optimal decisions the.