Statistical Inference: The Big Picture

📝 Abstract

Statistics has moved beyond the frequentist-Bayesian controversies of the past. Where does this leave our ability to interpret results? I suggest that a philosophy compatible with statistical practice, labeled here statistical pragmatism, serves as a foundation for inference. Statistical pragmatism is inclusive and emphasizes the assumptions that connect statistical models with observed data. I argue that introductory courses often mischaracterize the process of statistical inference and I propose an alternative “big picture” depiction.

💡 Analysis

Statistics has moved beyond the frequentist-Bayesian controversies of the past. Where does this leave our ability to interpret results? I suggest that a philosophy compatible with statistical practice, labeled here statistical pragmatism, serves as a foundation for inference. Statistical pragmatism is inclusive and emphasizes the assumptions that connect statistical models with observed data. I argue that introductory courses often mischaracterize the process of statistical inference and I propose an alternative “big picture” depiction.

📄 Content

arXiv:1106.2895v2 [stat.OT] 22 Jun 2011 Statistical Science 2011, Vol. 26, No. 1, 1–9 DOI: 10.1214/10-STS337 c ⃝Institute of Mathematical Statistics, 2011 Statistical Inference: The Big Picture1 Robert E. Kass Abstract. Statistics has moved beyond the frequentist-Bayesian con- troversies of the past. Where does this leave our ability to interpret re- sults? I suggest that a philosophy compatible with statistical practice, labeled here statistical pragmatism, serves as a foundation for infer- ence. Statistical pragmatism is inclusive and emphasizes the assump- tions that connect statistical models with observed data. I argue that introductory courses often mischaracterize the process of statistical in- ference and I propose an alternative “big picture” depiction. Key words and phrases: Bayesian, confidence, frequentist, statistical education, statistical pragmatism, statistical significance.

1. INTRODUCTION The protracted battle for the foundations of statis- tics, joined vociferously by Fisher, Jeffreys, Neyman, Savage and many disciples, has been deeply illumi- nating, but it has left statistics without a philoso- phy that matches contemporary attitudes. Because each camp took as its goal exclusive ownership of inference, each was doomed to failure. We have all, or nearly all, moved past these old debates, yet our textbook explanations have not caught up with the eclecticism of statistical practice. The difficulties go both ways. Bayesians have de- nied the utility of confidence and statistical signifi- cance, attempting to sweep aside the obvious success of these concepts in applied work. Meanwhile, for their part, frequentists have ignored the possibility of inference about unique events despite their ubiq- uitous occurrence throughout science. Furthermore, Robert E. Kass is Professor, Department of Statistics, Center for the Neural Basis of Cognition and Machine Learning Department, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA e-mail: kass@stat.cmu.edu. 1Discussed in 10.1214/11-STS337C, 10.1214/11-STS337A, 10.1214/11-STS337D and 10.1214/11-STS337B; rejoinder at 10.1214/11-STS337REJ. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in Statistical Science, 2011, Vol. 26, No. 1, 1–9. This reprint differs from the original in pagination and typographic detail. interpretations of posterior probability in terms of subjective belief, or confidence in terms of long-run frequency, give students a limited and sometimes confusing view of the nature of statistical inference. When used to introduce the expression of uncer- tainty based on a random sample, these caricatures forfeit an opportunity to articulate a fundamental attitude of statistical practice. Most modern practitioners have, I think, an open- minded view about alternative modes of inference, but are acutely aware of theoretical assumptions and the many ways they may be mistaken. I would sug- gest that it makes more sense to place in the center of our logical framework the match or mismatch of theoretical assumptions with the real world of data. This, it seems to me, is the common ground that Bayesian and frequentist statistics share; it is more fundamental than either paradigm taken separately; and as we strive to foster widespread understanding of statistical reasoning, it is more important for be- ginning students to appreciate the role of theoret- ical assumptions than for them to recite correctly the long-run interpretation of confidence intervals. With the hope of prodding our discipline to right a lingering imbalance, I attempt here to describe the dominant contemporary philosophy of statistics.
2. STATISTICAL PRAGMATISM I propose to call this modern philosophy statisti- cal pragmatism. I think it is based on the following attitudes: 1 2 R. E. KASS
3. Confidence, statistical significance, and posterior probability are all valuable inferential tools.
4. Simple chance situations, where counting argu- ments may be based on symmetries that generate equally likely outcomes (six faces on a fair die; 52 cards in a shuffled deck), supply basic intuitions about probability. Probability may be built up to important but less immediately intuitive situ- ations using abstract mathematics, much the way real numbers are defined abstractly based on in- tuitions coming from fractions. Probability is use- fully calibrated in terms of fair bets: another way to say the probability of rolling a 3 with a fair die is 1/6 is that 5 to 1 odds against rolling a 3 would be a fair bet.
5. Long-run frequencies are important mathemati- cally, interpretively, and pedagogically. However, it is possible to assign probabilities to unique events, including rolling a 3 with a fair die or having a confidence interval cover the true mean, without considering long-run frequency. Long-run frequencies may be regarded as consequences of the law of large numbers rather than as part of the definition of probability or confidence.

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