Recently we proposed quantum language(or, measurement theory), which is characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. Also, we consider that this is a kind of system theory such that it is applicable to both classical and quantum systems. As far as classical systems, it should be noted that quantum language is similar to statistics. In this paper, we discuss the usual confidence interval methods in terms of quantum language. And we assert that three concepts (i.e.,"estimator" and "quantity" and "semi-distance)are indispensable for the theoretical understanding of the confidence interval methods. Since our argument is quite elementary, we hope that the readers acquire a new viewpoint of statistics, and agree that our proposal is, from the pure theoretical point of view, the true confidence interval methods. (Key words: Confidence interval, Chi-squared distribution, Student's t-distribution)
Recently we proposed quantum language(or, measurement theory), which is characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. Also, we consider that this is a kind of system theory such that it is applicable to both classical and quantum systems. As far as classical systems, it should be noted that quantum language is similar to statistics. In this paper, we discuss the usual confidence interval methods in terms of quantum language. And we assert that three concepts (i.e.,"estimator" and "quantity" and "semi-distance)are indispensable for the theoretical understanding of the confidence interval methods. Since our argument is quite elementary, we hope that the readers acquire a new viewpoint of statistics, and agree that our proposal is, from the pure theoretical point of view, the true confidence interval methods. (Key words: Confidence interval, Chi-squared distribution, Student's t-distribution)
Although there are many ideas for the formulations of statistical hypothesis testing, we consider that the likelihood ratio test is the most reasonable and orthodox. However, it is not handy, and thus, it is not usual in elementary books. That is, the statistical hypothesis testing written in elementary books is different from the likelihood ratio test. Thus, from the theoretical point of view, we have the following question: "What is the statistical hypothesis testing written in elementary books?" For example, we consider that even the difference between "one sided test" and "two sided test" is not clear yet. In this paper, we give an answer to this question. That is, we propose a new formulation of statistical hypothesis testing, which is contrary to the confidence interval methods. In other words, they are two sides of the same coin. This will be done in quantum language (or, measurement theory), which is characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics, and also, a kind of system theory such that it is applicable to both classical and quantum systems. Since quantum language is suited for theoretical arguments, we believe that our results are essentially final as a general theory.
Since the problem: "What is statistics?" is most fundamental in sceince, in order to solve this problem, there is every reason to believe that we have to start from the proposal of a worldview. Recently we proposed measurement theory (i.e., quantum language, or the linguistic interpretation of quantum mechanics), which is characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. This turn from physics to language does not only extend quantum theory to classical theory but also yield the quantum mechanical world view (i.e., the (quantum) linguistic world view, and thus, a form of quantum thinking, in other words, quantum philosophy). Thus, we believe that the quantum lingistic formulation of statistics gives an answer to the question: "What is statistics?". In this paper, this will be done through the studies of inference interval, statistical hypothesis testing, Fisher maximum likelihood method, Bayes method and regression analysis in meaurement theory.
Recently, we proposed measurement theory ( or. quantum language) as a linguistic turn of quantum mechanics (with the Copenhagen interpretation). This theory has a great power of scientific descriptions. In fact, we have continued asserting that even statistics can be described in terms of measurement theory. Thus, we believe that quantum language is future statistics (i.e., statistics will develop into quantum language). However, now we think that our arguments were too abstract and philosophical, that is, we should have presented concrete examples much more. Thus, in this paper, we show that the calculation of Kalman filter is more understandable in terms of quantum language than in terms of usual statistics. For this, we devote ourselves to statistical measurement theory, in which the Bertrand paradox is discussed.
Recently, we proposed quantum language (or, measurement theory), which is characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. We believe that this language has a great powet of description, and therefore, even statistics can be described by quantum language. In this paper, we show that ANOVA (analysis of variance (one-way and two-way)) can be formulated in quantum language. Since quantum language is suited for theoretical arguments, we believe that our results are visible and understandable. For example, we can answer the question "What kind of role does Kolmogorov's probability theory play in ANOVA?" That is, the readers find that Kolmogorov's probability theory is merely used in order to calculate multi-dimenstional Gauss integrals, and thus, they can avoid to confuse the relation between Kolmogorov's probability theory and statistics.
Although regression analysis has a great history, we consider that it has always continued being confused. For example, the fundamental terms in regression analysis (e.g., "regression", "least-squares method", "explanatory variable", "response variable", etc.) seem to be historically conventional, that is, these words do not express the essence of regression analysis. Recently, we proposed quantum language (or, classical and quantum measurement theory), which is characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. We believe that this language has a great power of description, and therefore, even statistics can be described by quantum language. Therefore, in this paper, we discuss the regression analysis and the generalized linear model (i.e., multiple regression analysis) in quantum language, and clarify that the terms "explanatory variable" and "response variable" is respectively characterized as a kind of causality and the measured value.
It is a matter of course that Kolmogorov’s probability theory is a very useful mathematical tool for the analysis of statistics. However, this fact never means that statistics is based on Kolmogorov’s probability theory, since it is not guaranteed that mathematics and our world are connected. In order that mathematics asserts some statements concerning our world, a certain theory (so called “world view”) mediates between mathematics and our world. Recently we propose measurement theory (i.e., the theory of the quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. In this paper, we assert that statistics is based on measurement theory. And, for example, we show, from the pure theoretical point of view (i.e., from the measurement theoretical point of view), that regression analysis can not be justified without Bayes’ theorem. This may imply that even the conventional classification of (Fisher’s) statistics and Bayesian statistics should be reconsidered.