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optimal decision wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 formal mathematical description 2 under uncertainty in outcome 3 see also 4 references toggle the table of contents optimal decision 5 languages فارسی italiano polski русский українська edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia decision that leads to the best outcome in decision theory this article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations please help improve this article by introducing more precise citations september 2018 learn how and when to remove this message an optimal decision is a decision that leads to at least as good a known or expected outcome as all other available decision options it is an important concept in decision theory in order to compare the different decision outcomes one commonly assigns a utility value to each of them if there is uncertainty as to what the outcome will be but one has knowledge about the distribution of the uncertainty then under the von neumann morgenstern axioms the optimal decision maximizes the expected utility a probability weighted average of utility over all possible outcomes of a decision sometimes the equivalent problem of minimizing the expected value of loss is considered where loss is 1 times utility another equivalent problem is minimizing expected regret utility is only an arbitrary term for quantifying the desirability of a particular decision outcome and not necessarily related to usefulness for example it may well be the optimal decision for someone to buy a sports car rather than a station wagon if the outcome in terms of another criterion e g effect on personal image is more desirable even given the higher cost and lack of versatility of the sports car the problem of finding the optimal decision is a mathematical optimization problem in practice few people verify that their decisions are optimal but instead use heuristics and rules of thumb to make decisions that are good enough that is they engage in satisficing a more formal approach may be used when the decision is important enough to motivate the time it takes to analyze it or when it is too complex to solve with more simple intuitive approaches such as many available decision options and a complex decision outcome relationship formal mathematical description edit each decision d displaystyle d in a set d displaystyle d of available decision options will lead to an outcome o f d displaystyle o f d all possible outcomes form the set o displaystyle o assigning a utility u o o displaystyle u_ o o to every outcome we can define the utility of a particular decision d displaystyle d as u d d u o f d displaystyle u_ d d u_ o f d we can then define an optimal decision d o p t displaystyle d_ mathrm opt as one that maximizes u d d displaystyle u_ d d d o p t arg max d d u d d displaystyle d_ mathrm opt arg max limits _ d in d u_ d d solving the problem can thus be divided into three steps predicting the outcome o displaystyle o for every decision d displaystyle d assigning a utility u o o displaystyle u_ o o to every outcome o displaystyle o finding the decision d displaystyle d that maximizes u d d displaystyle u_ d d under uncertainty in outcome edit in case it is not possible to predict with certainty what will be the outcome of a particular decision a probabilistic approach is necessary in its most general form it can be expressed as follows given a decision d displaystyle d we know the probability distribution for the possible outcomes described by the conditional probability density p o d displaystyle p o d considering u d d displaystyle u_ d d as a random variable conditional on d displaystyle d we can calculate the expected utility of decision d displaystyle d as e u d d p o d u o d o displaystyle text e u_ d d int p o d u o do where the integral is taken over the whole set o displaystyle o degroot pp 121 an optimal decision d o p t displaystyle d_ mathrm opt is then one that maximizes e u d d displaystyle text e u_ d d just as above d o p t arg max d d e u d d displaystyle d_ mathrm opt arg max limits _ d in d text e u_ d d an example is the monty hall problem see also edit decision making decision making software two alternative forced choice references edit morris degroot optimal statistical decisions mcgraw hill new york 1970 isbn 0 07 016242 5 james o berger statistical decision theory and bayesian analysis second edition 1980 springer series in statistics isbn 0 387 96098 8 v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness sensitivity analysis frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio g test score lagrange multiplier wald z test normal specific tests parametric student s t test f test goodness of fit chi squared kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova analysis of variance ancova manova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal time series general decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickey fuller johansen q statistic ljung box durbin watson breusch godfrey time domain autocorrelation acf partial pacf cross correlation xcf arma model arima model box jenkins autoregressive conditional heteroskedasticity arch vector autoregression var autoregressive model ar frequency domain spectral density estimation fourier analysis least squares spectral analysis wavelet whittle likelihood survival survival function kaplan meier estimator product limit proportional hazards models accelerated failure time aft model first hitting time hazard function nelson aalen estimator test log rank test applications biostatistics bioinformatics clinical trials studies epidemiology medical statistics engineering statistics 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