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Paradoxes in Scientific Inference



Learning and writing can be a quite enjoyable experience! This book is about scientific inference and surrounding controversies. It presents a new way of learning about scientific inference through paradoxes. But what is a paradox? "Everyone is unique, just like everyone else." This statement is a simple example of paradox. A paradox is a statement or phenomenon that seems contradictory but in reality expresses a possible truth. It can be simply something that is counterintuitive. Paradoxes exist everywhere, in science, mathematics, philosophy, and in every corner of our lives. Paradoxes are poems of science and philosophy that collectively allow us to address broad multidisciplinary issues intriguingly and profoundly within a tiny volume, such as this book. A true paradox is a concise expression that delivers a profound idea. A paradox provokes indisputably a wild and endless imagination. Paradoxes are absolutely a source of creativity. The study of paradoxes leads to ultimate clarity and at the same time evokes one's mind, indisputably. The book analyzes paradoxes from many different perspectives: statistics, mathematics, philosophy, science, artificial intelligence, etc., elaborates findings, and reaches new and exciting conclusions. Probability and Statistics provide an essential and yet powerful tool for scientific research because weighing scientific evidence involves uncertainties. Studying statistical paradoxes helps us understand how scientific evidence is measured together with surrounding uncertainties and controversies. On the other hand, studying scientific paradoxes helps us truly understand the different paradigms and controversies in statistics, and advances the theory of statistics in the right direction. That is the main reason I gather together many of the paradoxes in statistics and science in this single volume. In additional to existing paradoxes from various sources, there are many paradoxes newly discovered during my study of this topic. Some are simple; others are complicated. Although I have tried to keep mathematical formulations minimal, I have not totally eliminated them, so as to avoid mathematical anxiety that might result from either approach (I have marked those difficult sections with an asterisk). Such a limitation in mathematics will not impair me from having an in-depth discussion of the topics; instead, it allows a reader to focus on the conceptual challenges. Truly, in many places, it should challenge your knowledge, your intuition and conventional wisdom, and require you to make a necessary adjustment. Since many of the paradoxes are poems of science you will have different feelings, awakenings, sentiments, and understandings every time you read them. It is not a book addressing the symmetric approaches in scientific inference (I don't even know if such a book exists), but a book outlining general principles and issues that everyone needs to know to be a rigorous and creative scientist. We will not focus on those most people know but on those that are often neglected. Although the book is written primarily for scientists and statisticians, many of the contents are also helpful for general audiences or those who have a college degree. With exercises at the end of each chapter, it can also be used as a textbook. A good scientist is always critical and open-minded. He is always ready to face challenges and adjust his belief as new evidence arises. He always carefully examines new findings and their implications carefully, and separates out whenever it is possible a fact from belief, a plausible solution from the truth. I always encourage everyone, especially those who do scientific research, to get training in statistics or applied statistics, not because of their mathematical aspect, but because of the ways they offer to approach problems. As stated earlier, the approach taken in this book is unique: learning scientific inference through paradoxes. I will feel so rewarded if this book can help a younger reader in some way to become a thinker, a creative scientist, a rigorous statistician, or a sophisticated person living a simplified life. Road Map The book consists of five chapters, organized as follows: Chapter 1, Joy of Paradoxes: A Random Walk We will walk you "randomly" into the world of paradoxes. We will present you with many interesting examples, including simple paradoxical quotes for life and provocative paradoxes from social networks, the judicial system, and games. The paradox of identity switching, paradox of leadership, paradox of the murderer, paradox of the court, and the omnipotence paradox, while amusing, are stimulating and challenging. We will further provide you with many motivating applications of paradoxes in accounting fraud detection, stock diversification, social economics, prediction of disease outbreak, the application of pesticides, machine maintenance, electric bell making, and internet package encryption. We will share with you dozens of fascinating mathematical paradoxes and explore intriguing paradoxes involving probability and statistics. Most of them are count-intuitive examples that are commonly encountered in daily life. Chapter 2, Mathematical and Plausible Reasoning We will begin with the review of the two different notions of probabilities: Frequentist and Bayesian. We then unify them by examining the hidden assumption in conception of Bayesian probability---causal space. We conceptualize causal space using examples in daily life and unify the two paradigms. We will introduce the very basics of formal logic before studying probability and probabilistic inference. To facilitate our discussion, we will utilize some well-known paradoxes, including the Boy or Girl Paradox, the Coupon Collector's Problem, Bertrand's Paradox, the Monty Hall Dilemma, the Tennis Game Paradox, the Paradox of Nontransitive Dice, the Paradox of Ruining Time, and the Paradox of Independence. We will not only provide fresh views on classical paradoxes but also present new paradoxes. Chapter 3, Statistical Measures of Scientific Evidence We advance probabilistic inference further to statistical inference. However, we will not focus on hands on statistical methodologies and expect you to perform statistical analyses. Instead, we will provide in-depth discussions and critiques on fundamental statistical principles and different statistical paradigms that are derived from these principles. Each of principles appears intuitive but the surrounding controversies are extremely complex. We will often use paradoxes in numerical form to effectively explain abstract concepts and complex issues. We will describe the three statistical paradigms frequentism, Bayesianism, and likelihoodism, and the Decision Approach as well. After reviewing the concepts of statistical model, point estimate, confidence interval, p-value, type-I and type-II errors, and level of significance, we will provide a detailed discussion of five essential statistical principles and surrounding controversies: the conditionality principle, the sufficiency principle, the likelihood principle, the law of likelihood, and Bayes' law. The discussion of controversies will be broadened to statistical analyses, including model fitting, data pooling, subgroup analyses, regression to the mean, and the issue of confounding. We put a large effort on the multiplicity issues due. We will make an attempt to unify the statistical paradigms under the proposed principle of evidential totality and the new concept of causal space. This chapter covers comprehensively and in depth most controversial issues in statistical inference. To make it effective, I will again borrow from the power of paradoxes. Chapter 4, Scientific Principles and Inferences We are going to study various principles, ideas, rules, methods, inferences, procedures, thought processes, intuitions, and controversies in scientific research. Some of the topics belong to scientific philosophy. We will use paradoxes to facilitate our discussion and make the abstract concepts tangible and easy to understand with minimum mathematical involvement. We will discuss several fundamental and provoking questions in scientific philosophy: the definition of science and the meaning of understanding, theories of truth, and discovery versus invention. We give you a fresh view on the topic of Determinism versus Free Will. You will be surprised by how these simple terms with which we are so familiar can still cause so much confusion and controversy. We will address the common issues in scientific research, share paradoxical stories about scientific research, and outline what the gold standard experiment should be. We will introduce a special kind of experiments that do not require a physical experiment and whereby no observations are needed. The thought experiments and examples including Galileo's leaning tower of Pisa discussion for free falling bodies, Maxwell's demon for a perpetual machine, and the Twin and Grandfather paradoxes in regard to time travel. Logical or axiom systems are usually considered the most rigorous schemes in science. Surprisingly, Gödel's Incompleteness theorem proved that an axiom system is, necessarily, either incomplete or inconsistent. Fitch proved that if all truths are knowable in principle then all truths are in fact known. Probably no scientific law is simpler than the pigeonhole principle. However, very strange things happen when a human brain is considered as a set of pigeonholes. We will discuss many more paradoxes like these. Game theory has great applications in economics and social science. We will stimulate our discussion with four different paradoxes in game theory. Chapter 5, Artificial Intelligence This chapter has two distinct parts, the paradox in artificial intelligent (AI) and the architecture for new AI agent. The later part is to provide a way to prove the "human-race" AI agent. Those who are not interested in AI building or do not have a computer science or AI background may choose to skip the second part. The idea of AI was proposed by Alan Turing in the Turing Machine. There are great debates on whether any computer can be built to perform like a human. I asset this can simply be a definitional problem, as illustrated by the "Ship of Theseus". The possibility of building "machine-race" humans is discussed through paradoxes of emotion, understanding, and discovery. The possibility is further illustrated by the paradox of dreams, the paradox of meta-cognition, and recent developments in swarm intelligence. There are several fundamental differences between the proposed AI approach and the existing AI approaches: (1) The new AI has very limited built-in initial concepts (only 14) ---virtually an initial empty brain, whereas traditional AI has a built-in knowledge database and/or complex algorithms. (2) The new AI is language-independent, whereas traditional AI is language-dependent (English and Chinese AIs have different architecture). (3) Unlike traditional AI, the new AI is based on the recursion of simple rules over multiple levels in learning. (4) The new AI's learning largely depends on progressive teaching: It can learn very broadly about many things but it is slow, whereas traditional methods have a fast learning speed but a narrow learning scope. Fascinating, Exotic, Provocative, Enlightening! Indeed, I was excited during the whole process of, as well as after, writing this book. So I hope you will be too, when you are reading it. My last note to readers: If you are an advanced reader you should read everything; if you are an intermediate reader, you can skip Chapter 3 and the second part of Chapter 5; if you are a naïve/quick reader, you might just read Chapters 1 and 4, and the first part of Chapter 5. Mark Chang
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... The two-children problem, that is sometimes referred to as the boy-or-girl problem, gained much popularity twenty-five years ago when it was discussed by the well-known columnist Marilyn vos Savant in Parade magazine (vos Savant, 1997). It has since been discussed in several monographs (Mlodinow, 2008;Chang, 2012) and scientific papers (among which D'Agostini, 2010;Lynch, 2011;Pollak, 2013, and the aforementioned ones). ...
Initially proposed by Martin Gardner in the 1950s, the famous two-children problem is often presented as a paradox in probability theory. A relatively recent variant of this paradox states that, while in a two-children family for which at least one child is a girl, the probability that the other child is a boy is $2/3$, this probability becomes $1/2$ if the first name of the girl is disclosed (provided that two sisters may not be given the same first name). We revisit this variant of the problem and show that, if one adopts a natural model for the way first names are given to girls, then the probability that the other child is a boy may take any value in $]0,2/3[$. By exploiting the concept of Schur-concavity, we study how this probability depends on model parameters.
... Patient-Statistician Paradox: Suppose new compounds A and B for cancer treatment were tested in the same clinical trial with a common control [57,58,139]. For drug A, only one analysis was performed at the final analysis with 500 patients treated in drug A and 500 patients in control; the null hypothesis was rejected and drug is claimed to be efficacious. ...
... Among various renowned paradoxes related to the probability theory and statistics (for example, Gardner, 1982;Székely, 1986;Mosteller, 1987;Smullyan, 1997;Chang, 2012;Eckhardt, 2013), the Two-Envelopes paradox is maybe one of the most famous problems. ...
Conference Paper
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Two Envelopes paradox presents a fascinating problem in probability and decision making. The player is presented with two envelopes and informed that one of them contains twice as much money as the other one. The player takes one of them without looking inside, and is then given the opportunity to change their mind and take the second envelope instead of the first one. Assuming that the 1 st envelope contains a value A, then the 2 nd one can have 2A or A/2 with equal probability, and its expected value is the mean 1.25A, or 25% profit from switching envelopes. It is a great result, but we could denote the amount in the 2 nd envelope as A and repeat the derivation outcome 1.25A already in 1 st envelope, so each of them is worth more than the other one. So, to switch or not to switch?-That is the question. As shown in this article, the ideas borrowed from the Analytic Hierarchy Process can help in resolving this paradox by transforming the ratio scale into the additive or logarithmic scales which correspond to application of the multiplicative utility function.
... The notion behind prediction is causality, whereas the notion of causality is the similarity principle (Chang, 2012(Chang, , 2014: Similar conditions will likely result in the same or similar outcomes. The principle can be stated differently for better understanding depending on the problem. ...
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We briefly discuss the philosophical basis of science, causality, and scientific evidence, by introducing the hidden but most fundamental principle of science: the similarity principle. The principle’s use in scientific discovery is illustrated with Simpson’s paradox and other examples. In discussing the value of null hypothesis statistical testing, the controversies in multiple regression, and multiplicity issues in statistics, we describe how these difficult issues should be handled based on our interpretation of the similarity principle.
This chapter introduces the procedure for statistical inference most used in practice: hypothesis testing. In this method, an assumption is made about the value of the parameter, then the data is used to determine if the assumption is reasonable. The chapter focuses on hypothesis tests about the mean of a population. It discusses hypothesis tests for population proportion and population variance. Hypothesis testing starts by making two statements about the world: null hypothesis and alternative hypothesis. The chapter defines some intuitive steps necessary to perform hypothesis testing. It presents some background on the use of the p‐value, discusses some issues with the measure, and provides alternatives when conducting statistical inference. The chapter also features practice problems designed to ensure that readers understand the concepts and can apply them using real data.
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