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In statistics, decision theory is also known as Bayesian Decision theory and Decagons theory which enables one to choose the optimum decisions and remove the preceding shortcoming. Decision is concerned with the financial management and business management. It refers to a set of methods that is used for obtaining the optimal solution. It is helpful in the various areas like mathematics, economics, psychology, statistics, etc.
Decision theory plays an important role in the Statistics field. It is extensively used in the managerial statistics and applied statistics. Statistical decision theory mainly deals with the various topics like Sensitivity analysis, Expected value of Perfect Information, Prior Probabilities, Shrinkage Estimation in Parametric Models, and many more.
Decision theory mainly used at that situation where there are a lot of options are available and user is not able to forecast their consequences with certainty. Strategy for performing the decision theory in various fields involves some main steps which are as follows:
- Objective: the objective or goal should be clearly decided.
- Path of activities: this step mainly concerned with the various path of the activities that they followed to achieve the goal.
- Several choices: it becomes easier to take the decisions if the value of the several solutions not be predictable.
Heuristic approach makes the decision making easier than step-by-step processing. It enables user to take the decisions quickly without wasting too much time on analyzing and researching the information. It mitigates the cognitive burdens that are associated with the decision making. Heuristic approach enables user to inspect the choices in decision making. Heuristics approach falls under the various categories which are as follows:
- Base-rate heuristics
Decision quality is mainly concerned with the quality of the decision at the time when the decision is made. It doesn’t concerns with the consequences of the decision. For analyzing the decision problems, it provides the assurance of the effectiveness and efficiency. By this, we can say that decision quality is the extension to decision making. The process which is responsible for the high-quality decision can be defined by the Decision quality. Decision quality process permits apprehend the utmost value in composite scenarios. Some of the major elements of Decision Quality are listed below:
- Sound reasoning
- Trade-offs and values
Mappings among the white-noise experiments and nonparametric regression are responsible for the establishment of asymptotic equivalence. Between the nonparametric regression and white-noise experiments, asymptotic equivalence provides the artificial examination to the one of the experiments which is identical to the true observations and provides asymptotic solutions to the remaining experiment. This result provides a consequence that the spontaneously asymptotically analogous outcomes in all nonparametric problems are provided by doing the inspection in one nonparametric problem.
Moreover, for determining that the hypothesis test we are operating has the best statistical power or not, we use a way which is named as Neyman-Paerson Lemma. Hypothesis test is of two types, including alternate hypothesis and null hypothesis. The power of the hypothesis test is that if the alternate hypothesis is true, it clearly rejects the null hypothesis. Main objective of hypothesis test is to maximize this power, so it can reject the null hypothesis whenever the second one type of hypothesis test is true.
Neyman-Paerson Lemma explained us that best hypothesis tests are likelihood ratio tests and it is based on the manageable hypothesis test. A hypothesis test where the unspecified parameters are defined as the single value is termed as manageable hypothesis test. A manageable (simple) hypothesis is optimized hypothesis which contains finite practical value and acts as an imperative hypothetical tool.
Bayesian hypothesis testing is that in which more than two hypotheses can take place and there is no compulsory situation that both of the hypotheses will stands in an asymmetric relationship. Bayesian hypothesis testing works like the Bayesian inference.
EVSI is one of the major concept that involved in the Statistical decision Theory which stand for expected Value of Sample Information. By gaining the access from a sample of subsidiary observations before making a decision, a decision-maker can acquire the expected enhancements in utility, this is termed as EVSI. The subsidiary observations are acquired from the sample data which leads to make the decision making better. EVSI also determines the estimations that what kinds of enhancement occurred in the sample data from before.
Nowadays, Bayesian Statistics is becoming the rapidly growing area in the statistics. It is mainly used when there are few of the components have sparsely collected data. Various approaches for Bayesian decision analysis are as follows:
- Organizing a decision problem
- Evoking probability distributions among variables
- Acquiring preferences and objectives of the clients
The result that we get by transferring the arbitrarily crude estimator into optimal estimator is termed as Rao–Blackwell–Kolmogorov theorem. Sometimes, it is known as Rao-Blackwell theorem. This theorem provides a specific process by which enhancement in the efficiency of an estimator can be acquired by taking its conditional expectation. The process of transferring an estimator with the help of Rao-Blackwell theorem is known as Rao-Blackwellization.
Furthermore, some of the researched and advanced topics that take place in the field of Statistical decision theory are:
- Conjugate Prior Distributions
- Utility theory
- Posterior robustness
- Hierarchical Bayes analysis
- Bayesian robustness
- Minimax theorem
- Bayes rule
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- A decision criterion
- The degree of certainty of possible future events
- A list of alternatives
- A list of possible future events
- Payoffs associated with each combination of alternatives and events like a payoff matrix.
- Behavioural Decision Making, Clinical judgment, Intuitive correlations as sources of error, Linear models as ways to overcome bias, Intuitive Statistics, Availability: A heuristic for estimating frequency, Intuitive Inferential Statistics.
- Bayesian Inference, Confidence, Calibration: The appropriateness of confidence, Learning from experience, Representativeness, Improving Judgment, Better estimation, Better decisions, Critiques, Normative Choice Models, Economic formulations, Psychological reflections, Formulating Decisions, Eliciting Beliefs & Values, Implicit contexts, Explicit contexts, Social Context of Decision Making.
- Decision Aiding, Process, Product, Normative and descriptive decision theory, Rational Choice, Mathematical Background., Games:, Prisoners Dilemma, Battle of the Sexes, Leader, Game of Chicken, The Tragedy of the Commons, Rendezvous problem.
- Judgement under Uncertainty, Heuristics and Biases, Prospect Theory, Creativity and Psychology, Decision Support Systems (DSS), The role of the human decision maker in DSS, Rule-based reasoning, Mathspeak, Basic tools.
- Applications to representation theorems, Revealed preference theory., Sequential rationalizability, Choice over time, Modelling anticipatory feelings, Modelling Imperfect Recall, Modelling temptation, Modelling the effect of reference points.
- Describing data,Discrete,continous & conditional probability,Normal curve,Central Limit Theorem,Business Decision-Making
- Hypothesis testing, condence intervals Notes,Combining condence intervals Notes, Combining intervals without data Notes, Multiple and dependent sources Notes, Bias in information sources Notes, Picking the winner when comparing many groups, Separating coincidence from systematic pattern Dawkins, Regression: model specication and confounding, Data mining, Regression: extrapolation and myths of interpolation.
- Time series methods for forecasting, Exponential smoothing and forecasting, Probability forecasts and calibration, risk and certainty equivalent value Notes, principal agent and incentivizing Notes, CAPM and beta, Hedging risk, Implied volatility and options Notes, Statistical Decision Theoryive condence intervals Dec Traps, Working with Statistical Decision Theoryive intervals, Foundations of game theory Axelrod, Cooperation and tit-for-tat.
- Auctions, statistical analysis of data, Concepts as event, probability, independence and expected value are defined, different discrete and continuous probability models are studied, binomial, poisson and normal models, The structure and basic concepts of decision analysis are studied,, Bayes’ theorem, decision-trees and sensitivity analysis, The foundations of descriptive statistics:, Principles of tabulation and graphs.
- measures of location, variation and association,, standardization, and index-theory, The meaning of concepts as statistical precision and statistical significance are discussed, The course will also give an introduction to Monte Carlo simulation, Decision Theory, Loss function and risk, Foundations of optimal decision making, Bayes rule, minimax rule,, admissibility Sufficiency and Rao-Blackwellization.
- Utility theory, Utility and loss, Personal utility function, Prior and posterior, Conjugate families, Noninformative priors, Generalized Bayes rules, Bayesian estimation Credible sets, Bayesian hypothesis testing, Bayesian prediction, Empirical Bayes rules, Parametric and nonparametric empirical Bayesian analysis, Hierarchical Bayes analysis, Bayesian robustness Posterior robustness, Gamma-minimaxity, Admissibility of Bayes rules Bayesian calculation, Game theory and the minimax theorem
- Averages and Variation:,Measures of Central Tendency (Mode, Median, Mean),Measures of Variation,Percentiles and Box-and-Whisker Plots;,Correlation and Regression:,Scatter Diagrams and Linear Correlation,Linear Regression and Coefficient of Determination;,Elementary Probability Theory:,Probability Rules—Compound Events,Tree Diagrams and Counting Techniques;,Random Variables and Probability Distributions,Binomial Probabilities,Additional Properties of the Binomial Distributions,Normal Curves and Sampling Distributions:,Continuous Random Variables, Graphs of Normal Probability Distributions,Standard Units and Areas Under the Standard Normal Distribution,Areas Under Any Normal Curve,Sampling Distribution,The Central Limit Theorem,Normal Approximation to the Binomial Distribution and to
- Empirical processes,
- Asymptotic efficiency
- Uniform convergence of measures
- Resampling methods
- Edgeworth expansions