{"title":"量子力学中的贝叶斯、条件概率和拉普拉斯演替定律","authors":"Tsubasa Ichikawa","doi":"10.1007/s10701-025-00842-5","DOIUrl":null,"url":null,"abstract":"<div><p>We present a comparative study between classical probability and quantum probability from the Bayesian viewpoint, where probability is construed as our rational degree of belief on whether a given statement is true. From this viewpoint, including conditional probability, three issues are discussed: (i) given a measure of the rational degree of belief, does it satisfy the axioms of the probability? (ii) Given the probability satisfying these axioms, is it seen as the measure of the rational degree of belief? (iii) Can the measure of the rational degree of belief be evaluated in terms of the relative frequency of events occurring? Here we show that as with the classical probability, all these issues can be resolved affirmatively in the quantum probability, provided that the relation to the relative frequency is slightly modified from the Laplace law of succession in case of a small number of observations. This implies that the relation between the Bayesian probability and the relative frequency in quantum mechanics is the same as that in the classical probability theory, including conditional probability.</p></div>","PeriodicalId":569,"journal":{"name":"Foundations of Physics","volume":"55 3","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Bayesianism, Conditional Probability and Laplace Law of Succession in Quantum Mechanics\",\"authors\":\"Tsubasa Ichikawa\",\"doi\":\"10.1007/s10701-025-00842-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We present a comparative study between classical probability and quantum probability from the Bayesian viewpoint, where probability is construed as our rational degree of belief on whether a given statement is true. From this viewpoint, including conditional probability, three issues are discussed: (i) given a measure of the rational degree of belief, does it satisfy the axioms of the probability? (ii) Given the probability satisfying these axioms, is it seen as the measure of the rational degree of belief? (iii) Can the measure of the rational degree of belief be evaluated in terms of the relative frequency of events occurring? Here we show that as with the classical probability, all these issues can be resolved affirmatively in the quantum probability, provided that the relation to the relative frequency is slightly modified from the Laplace law of succession in case of a small number of observations. This implies that the relation between the Bayesian probability and the relative frequency in quantum mechanics is the same as that in the classical probability theory, including conditional probability.</p></div>\",\"PeriodicalId\":569,\"journal\":{\"name\":\"Foundations of Physics\",\"volume\":\"55 3\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2025-05-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Foundations of Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10701-025-00842-5\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s10701-025-00842-5","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Bayesianism, Conditional Probability and Laplace Law of Succession in Quantum Mechanics
We present a comparative study between classical probability and quantum probability from the Bayesian viewpoint, where probability is construed as our rational degree of belief on whether a given statement is true. From this viewpoint, including conditional probability, three issues are discussed: (i) given a measure of the rational degree of belief, does it satisfy the axioms of the probability? (ii) Given the probability satisfying these axioms, is it seen as the measure of the rational degree of belief? (iii) Can the measure of the rational degree of belief be evaluated in terms of the relative frequency of events occurring? Here we show that as with the classical probability, all these issues can be resolved affirmatively in the quantum probability, provided that the relation to the relative frequency is slightly modified from the Laplace law of succession in case of a small number of observations. This implies that the relation between the Bayesian probability and the relative frequency in quantum mechanics is the same as that in the classical probability theory, including conditional probability.
期刊介绍：
The conceptual foundations of physics have been under constant revision from the outset, and remain so today. Discussion of foundational issues has always been a major source of progress in science, on a par with empirical knowledge and mathematics. Examples include the debates on the nature of space and time involving Newton and later Einstein; on the nature of heat and of energy; on irreversibility and probability due to Boltzmann; on the nature of matter and observation measurement during the early days of quantum theory; on the meaning of renormalisation, and many others.
Today, insightful reflection on the conceptual structure utilised in our efforts to understand the physical world is of particular value, given the serious unsolved problems that are likely to demand, once again, modifications of the grammar of our scientific description of the physical world. The quantum properties of gravity, the nature of measurement in quantum mechanics, the primary source of irreversibility, the role of information in physics – all these are examples of questions about which science is still confused and whose solution may well demand more than skilled mathematics and new experiments.
Foundations of Physics is a privileged forum for discussing such foundational issues, open to physicists, cosmologists, philosophers and mathematicians. It is devoted to the conceptual bases of the fundamental theories of physics and cosmology, to their logical, methodological, and philosophical premises.
The journal welcomes papers on issues such as the foundations of special and general relativity, quantum theory, classical and quantum field theory, quantum gravity, unified theories, thermodynamics, statistical mechanics, cosmology, and similar.
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