Combining statistical inference and decisions in ecology.

Statistical decision theory (SDT) is a sub-field of decision theory that formally incorporates statistical investigation into a decision-theoretic framework to account for uncertainties in a decision problem. SDT provides a unifying analysis of three types of information: statistical results from a data set, knowledge of the consequences of potential choices (i.e., loss), and prior beliefs about a system. SDT links the theoretical development of a large body of statistical methods, including point estimation, hypothesis testing, and confidence interval estimation. The theory and application of SDT have mainly been developed and published in the fields of mathematics, statistics, operations research, and other decision sciences, but have had limited exposure in ecology. Thus, we provide an introduction to SDT for ecologists and describe its utility for linking the conventionally separate tasks of statistical investigation and decision making in a single framework. We describe the basic framework of both Bayesian and frequentist SDT, its traditional use in statistics, and discuss its application to decision problems that occur in ecology. We demonstrate SDT with two types of decisions: Bayesian point estimation and an applied management problem of selecting a prescribed fire rotation for managing a grassland bird species. Central to SDT, and decision theory in general, are loss functions. Thus, we also provide basic guidance and references for constructing loss functions for an SDT problem.

[1]  Mevin B. Hooten,et al.  A guide to Bayesian model selection for ecologists , 2015 .

[2]  R. A. Fisher,et al.  Design of Experiments , 1936 .

[3]  SIMPLIFIED DECISION FUNCTIONS , 1954 .

[4]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[5]  T. Gneiting.  Making and Evaluating Point Forecasts , 2009, 0912.0902.

[6]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[7]  David Lindley,et al.  Bayesian Statistics, a Review , 1987 .

[8]  D. Cox.  Some problems connected with statistical inference , 1958 .

[9]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[10]  Xin Guo,et al.  On the optimality of conditional expectation as a Bregman predictor , 2005, IEEE Trans. Inf. Theory.

[11]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[12]  Jay M. Ver Hoef,et al.  Space—time zero‐inflated count models of Harbor seals , 2007 .

[13]  M. Degroot.  Optimal Statistical Decisions , 1970 .

[14]  Gerald S. Rogers,et al.  Mathematical Statistics: A Decision Theoretic Approach , 1967 .

[15]  E. Lehmann.  Testing Statistical Hypotheses , 1960 .

[16]  E. S. Pearson,et al.  On the Problem of the Most Efficient Tests of Statistical Hypotheses , 1933 .

[17]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[18]  E. S. Pearson,et al.  On the Problem of the Most Efficient Tests of Statistical Hypotheses , 1933 .

[19]  Perry J. Williams.  Methods for incorporating population dynamics and decision theory in cackling goose management , 2016 .

[20]  David R. Anderson,et al.  Model Selection and Multimodel Inference , 2003 .

[21]  Allan Birnbaum,et al.  The Neyman-Pearson theory as decision theory, and as inference theory; with a criticism of the Lindley-savage argument for Bayesian theory , 1977, Synthese.

[22]  George E. P. Box,et al.  Some Comments on “Bayes” Estimators , 1973 .

[23]  A. Spanos.  Why the Decision‐Theoretic Perspective Misrepresents Frequentist Inference: Revisiting Stein’s Paradox and Admissibility , 2017 .

[24]  E. S. Pearson,et al.  ON THE USE AND INTERPRETATION OF CERTAIN TEST CRITERIA FOR PURPOSES OF STATISTICAL INFERENCE PART I , 1928 .

[25]  Jean,et al.  Henri Poincare，为科学服务的一生 , 2006 .

[26]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .

[27]  Charles A Rohde.  Introductory Statistical Inference with the Likelihood Function , 2014 .

[28]  J. Andrew Royle,et al.  Space: Time Dynamic Design of Environmental Monitoring Networks , 1999 .

[29]  Robert L. Wolpert,et al.  Statistical Inference , 2019, Encyclopedia of Social Network Analysis and Mining.

[30]  C. Hennig,et al.  Some thoughts about the design of loss functions , 2007 .

[31]  D. Cox.  Some problems connected with statistical inference , 1958 .

[32]  H. Raiffa,et al.  Introduction to Statistical Decision Theory , 1996 .

[33]  L. J. Savage,et al.  The Foundations of Statistics , 1955 .

[34]  Byron K. Williams,et al.  Markov decision processes in natural resources management: Observability and uncertainty , 2009 .

[35]  James D. Nichols,et al.  Adaptive harvest management of North American waterfowl populations: a brief history and future prospects , 2007, Journal of Ornithology.

[36]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[37]  J. Andrew Royle,et al.  Dynamic design of ecological monitoring networks for non‐Gaussian spatio‐temporal data , 2005 .

[38]  Abraham Wald,et al.  Statistical Decision Functions , 1951 .

[39]  J. Tukey.  Conclusions vs Decisions , 1960 .

[40]  F. Ramsey.  The Foundations of Mathematics and Other Logical Essays , 2001 .

[41]  Alan E. Gelfand,et al.  Model choice: A minimum posterior predictive loss approach , 1998, AISTATS.

[42]  Mevin B Hooten,et al.  Animal movement constraints improve resource selection inference in the presence of telemetry error. , 2015, Ecology.

[43]  John K Kruschke,et al.  Bayesian data analysis. , 2010, Wiley interdisciplinary reviews. Cognitive science.

[44]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[45]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[46]  Aris Spanos,et al.  Why the Decision Theoretic Perspective Misrepresents Frequentist Inference: 'Nuts and Bolts' vs. Learning from Data , 2012, 1211.0638.

[47]  Mevin B. Hooten,et al.  Bayesian Models: A Statistical Primer for Ecologists , 2015 .

[48]  Fred A. Johnson,et al.  BAYESIAN INFERENCE AND DECISION THEORY—A FRAMEWORK FOR DECISION MAKING IN NATURAL RESOURCE MANAGEMENT , 2003 .

[49]  F. Ramsey.  Truth and Probability , 2016 .

[50]  B. Williams.  Resolving structural uncertainty in natural resources management using POMDP approaches , 2011 .

[51]  C. T. Moore,et al.  Uncertainty and the management of mallard harvests , 1997 .

[52]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[53]  J. Dickey,et al.  Bayesian Decision Theory and the Simplification of Models , 1980 .

[54]  J. Andrew Royle,et al.  Hierarchical Modeling and Inference in Ecology: The Analysis of Data from Populations, Metapopulations and Communities , 2008 .

[55]  Mevin B. Hooten,et al.  Optimal spatio-temporal hybrid sampling designs for ecological monitoring , 2009 .

[56]  R. Fisher,et al.  STATISTICAL METHODS AND SCIENTIFIC INDUCTION , 1955 .

[57]  Mevin B. Hooten,et al.  Design and Analysis of Long-term Ecological Monitoring Studies: Optimal spatio-temporal monitoring designs for characterizing population trends , 2012 .

[58]  L. Goddard.  Information Theory , 1962, Nature.