Towards a Frequentist Interpretation of Sets of Measures

The American Statistician, 1986

Commentaries are informative essays dealing with viewpoints of statistical practice, statistical education, and other topics considered to be of general interest to the broad readership of The American Srarisrician. Commentaries are similar in spirit to Letters to the Editor, but they involve longer discussions of background, issues, and perspectives. All commentaries will be refereed for their merit and compatibility with these criteria.

Two philosophers who disagree about a point should, instead of arguing fruitlessly and endlessly, be able to take out their pencils, sit down amicably at their desks, and say "Let us calculate." Gottfried Wilhelm von Leibniz (1646-1716) Lest men suspect your tale untrue, Keep probability in view. John Gay (1685-1732) Fables. Part 1: The Painter who pleased Nobody and Everybody. But to us, probability is the very guide of life.-Bishop Joseph Butler, The Analogy of Religion, Introduction It is seen in this essay that the theory of probabilities is at bottom only common sense reduced to calculus; it makes us appreciate with exactitude that which exact minds feel by a sort of instinct without being able ofttimes to give a reason for it.

Studia Logica, 1978

The three volumes of this book contain the proceedings of a colloquium oil foundations of probability, statistics and statistical theolies of science, held in May 1973 at the University of Western Ontario. They give a very colourful picture of the recent state of the discussions in this field. A central theme is the controversy between the Bayesian and non-Bayesian approaches to the foundations of probability. Many of the pa.pers are followed by parts of the discnssion which facilitate ,~ much deeper undcrst,~nding of the relations between the sep~trate papers. Certainly, this book will have a strong influence on the d(,veh)pment of the foundations of statistics. Subsequently the papers will 1)~' r(,vicwcd separately. The extent to which they arc reviewed in detail depends primarily ()u the presumed fi~tcrests of the readcr.~ of "Studio Legion" and seco~(ialily, of course, oil lhe rcvi(;wer's iHfcrcst ~tnd knowh'dgc. Thus the papers are reviewed undo1 the aspect of their possible izlfluence on logics, while statistical or philosophical aspects are kept rather short.

Information and Computation, 1996

The Bayesian program in statistics starts from the assumption that an individual can always ascribe a definite probability to any event. It will be demonstrated that this assumption is incompatible with the natural requirement that the individual's subjective probability distribution should be computable. We shall construct a probabilistic algorithm producing with probability extremely close to 1 an infinite binary sequence which is not random with respect to any computable probability distribution (we use Dawid's notion of randomness, computable calibration, but the results hold for other widely known notions of randomness as well). Since the Bayesian knows the algorithm, he must believe that this sequence will be noncalibrable. On the other hand, it seems that the Bayesian must believe that the sequence is random with respect to his own probability distribution. We hope that the discussion of this apparent paradox will clarify the foundations of Bayesian statistics. We analyse also the time of computation and the place of``losing randomness.'' We show that we need only polynomial time and space to demonstrate non-calibration effects on finite sequences.

1984

Preface page xi Acknowledgments xxiv 10 From probability theory to statistical inference* 10.1 Introduction 10.2 Interpretations of probability Contents vii 10.3 Attempts to build a bridge between probability and observed data 10.4 Toward a tentative bridge 10.5 The probabilistic reduction approach to specification 10.6 Parametric versus non-parametric models 10.7 Summary and conclusions 10.8 Exercises An introduction to statistical inference References Index Contents ix

Theory of Computing Systems, 2013

Kolmogorov complexity furnishes many useful tools for studying different natural processes that can be expressed using sequences of symbols from a finite alphabet (texts), such as genetic texts, literary and music texts, animal communications, etc. Although Kolmogorov complexity is not algorithmically computable, in a certain sense it can be estimated by means of data compressors. Here we suggest a method of analysis of sequences based on ideas of Kolmogorov complexity and mathematical statistics, and apply this method to biological (ethological) "texts." A distinction of the suggested method from other approaches to the analysis of sequential data by means of Kolmogorov complexity is that it belongs to the framework of mathematical statistics, more specifically, that of hypothesis testing. This makes it a promising candidate for being included in the toolbox of standard biological methods of analysis of different natural texts, from DNA sequences to animal behavioural patterns (ethological "texts"). Two examples of analysis of ethological texts are considered in this paper. Theses examples show that the proposed method is a useful tool for Theory Comput Syst distinguishing between stereotyped and flexible behaviours, which is important for behavioural and evolutionary studies.

2003

arXiv (Cornell University), 2018

Mathematical Theory of Evidence (MTE) is known as a foundation for reasoning when knowledge is expressed at various levels of detail. Though much research effort has been committed to this theory since its foundation, many questions remain open. One of the most important open questions seems to be the relationship between frequencies and the Mathematical Theory of Evidence. The theory is blamed to leave frequencies outside (or aside of) its framework. The seriousness of this accusation is obvious: no experiment may be run to compare the performance of MTE-based models of real world processes against real world data. In this paper we develop a frequentist model of the MTE bringing to fall the above argument against MTE. We describe, how to interpret data in terms of MTE belief functions, how to reason from data about conditional belief functions, how to generate a random sample out of a MTE model, how to derive MTE model from data and how to compare results of reasoning in MTE model and reasoning from data. It is claimed in this paper that MTE is suitable to model some types of destructive processes

Andrés Rivadulla: Éxito, razón y cambio en física, Madrid: Ed. Trotta, 2004

In these pages I offer my solution to the problem of inductive probability of theories. Against the existing expectations in certain areas of the current philosophy of science, I argue that Bayes’s Theorem does not constitute an appropriate tool to assess the probability of theories and that we would do well to banish the question about how likely a certain scientific theory is to be true, or to what extent one theory is more likely true than another. Although I agree with Popper that inductive probability is impossible, I disagree with him in the way Sir Karl presents his argument, as I have showed elsewhere, so my proof is completely different. The argument I present in this paper is based on applying Bayes’s Theorem to specific situations that show its inefficiency both in the case of whether a hypothesis becomes all the more likely true the greater the empirical evidence that supports it, as whether the probability calculus allows to identify a given hypothesis from a set of hypotheses incompatible with each other as the most likely true.