科学哲学视野下的贝叶斯方法
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摘要
贝叶斯主义是目前最具优势的研究纲领之一。它广泛运用于统计学、经济学、心理学和人工智能等领域,相应的推理模型称为贝叶斯推理。自凯恩斯开创现代归纳逻辑的现时代,建立第一个概率归纳逻辑系统以后,贝叶斯主义成为归纳逻辑的有力捍卫者,并且逐步发展为一套一般性的科学推理理论和方法。
在科学哲学框架内研究贝叶斯方法,包含两个方面：其一,从逻辑上探索贝叶斯方法的推理机制和程序；其二,在方法论意义上回答贝叶斯方法的合理性问题。由此需要讨论的议题有：贝叶斯推理机制的优越性体现；主观概率与客观概率之争；旧证据问题和简单性等难题；以及贝叶斯方法的可能发展进路。
首先,笔者从归纳合理性与贝叶斯方法的兴起出发,分别从纵向和横向对这种方法进行了梳理总结。现代逻辑学者对数学上的贝叶斯方法赋予逻辑意义,源于休谟问题引发的归纳逻辑合理性质疑。休谟问题在逻辑上否定了简单枚举归纳法这种定性的推理模型,而贝叶斯方法的定量特性,使得贝叶斯推理成为现代归纳逻辑的主要推理模型。对于贝叶斯方法的理论内核,其哲学意义体现在概率的解释上。根据概率的不同解释,可以分为逻辑贝叶斯主义和主观贝叶斯主义两种类型。由于主观主义避免了逻辑主义中存在的无差别悖论,所以主观主义表现出明显的优越性,进而成为贝叶斯方法的核心理论依据。从这个意义上理解,本文提及的贝叶斯方法实际上是主观贝叶斯主义的研究方法；而且如无特别说明,本文出现的贝叶斯主义专指主观贝叶斯主义。另外,笔者从科学方法论意义上论证了贝叶斯方法是一种逻辑—历史方法,进而彰显其合理性。
其次,笔者用两章的内容通过介绍经典统计推理的类型和局限性,以及贝叶斯方法在统计推理中的运用,来表现贝叶斯推理机制的优越性。贝叶斯主义的复兴出现在统计推理领域。贝叶斯统计推理避免了经典统计推理中的主观因素问题以及先验回避问题,因而是推理方法的一次革命。在假说检验上,贝叶斯统计推理解决了经典统计中存在的检验统计量的选择难题;避免了停止法则带来的困难并凸显了归纳特性。在面对估计问题时,贝叶斯估计用可信区间代替置信区间,为经典置信区间下的直觉提供了一个概念性的解释和合理说明。贝叶斯定理的应用,克服了经典估计无法引入先验信息的困难。在归纳意义上,经典统计方法受其方法论根源——证伪主义方法的影响,面临缺乏归纳性的质疑。而贝叶斯方法作为归纳推理机制的定量进路,具备显著的归纳意义。
再次,笔者分析了贝叶斯方法面临的困难和挑战,指出这种方法遭遇的合理性质疑问题。贝叶斯方法存在主观性、简单性与旧证据问题等难题。对贝叶斯主义的主观性诘难,主要在于贝叶斯推理机制的先验概率的约束问题。简单性问题就是对简单性原则的质疑,表现在简单性假设与概率公理之间的不一致上。福斯特和索伯指出简单性只是一种“特设方法”,而豪森主张回避简单性问题,他认为简单性不应该被视为理论选择时的一条重要指导准则。旧证据问题表明,在贝叶斯主义框架内,一个旧证据不能对理论或假说提供任何确证。这与我们的直觉相悖。豪森用证据的相关性表明,“证据支持”隐含一个关于数据、假说和背景知识k之间的三元关系。只有当e被判定为证据,且k包含e时,才会出现旧证据问题。这就表明贝叶斯推理需要进一步改进和发展。
最后,笔者根据逻辑、哲学与认知的视角,利用这三个维度多层次探析贝叶斯方法的可能发展进路。贝叶斯方法面临的合理性问题也为其进一步发展留下了宽广空间。从归纳逻辑的发展历程来看,非帕斯卡概率逻辑的兴起,为贝叶斯方法揭示了一条可能进路：修改或弱化贝叶斯主义坚持的帕斯卡概率规则,形成一种非帕斯卡贝叶斯主义。在哲学层面上讨论贝叶斯方法,主要集中于主观概率解释和客观概率解释的选择问题。一方面,笔者认为概率在应用上是多元的,主观解释和客观解释都有其适用范围。另一方面,在哲学上,主体交互解释更满足科学推理要求的客观性。贝叶斯方法在认知科学领域的研究和运用,为其本身发展提供了一些启示。特别是认知心理学对贝叶斯推理模型的成功运用,为贝叶斯推理研究的认知转向提供了契机；同时为这种方法的发展提供了可能进路：寻求频率主义与贝叶斯主义的统一；引入内涵因素,尝试外延性与非外延性的融合。具体而言,吉仁泽和霍夫拉格提出了频率格式的贝叶斯推理模型,即用频率格式代替概率格式来对问题进行信息表征,进而改进贝叶斯推理方法。图文斯基等人提出了主观概率判断的支持理论,建立了一个主观概率的非外延归纳推理理论。从发展趋势看,实现归纳逻辑的认知转向,可能是归纳逻辑未来的重要发展方向之一。
Bayesianism is one of the front edge research methods. It has been widely applied in statistics, economics, psychology and artificial intelligence, etc. Its reasoning model has been called Bayesian inference. Since Keynes started the modern age of induction and created the first probabilistic induction system, Bayesianism has become a strong defender of the induction and gradually developed to be a general scientific reasoning theory and method.
There are two aspects for studying Bayesian method under the spectrum of science philosophy:firstly, to explore the inference mechanism and procedure of Bayesian method from the aspect of reasoning; secondly, to answer reasonable questions of Bayesian method from the aspect of methodology. Therefore, questions to be discussed include the superiority of Bayesian inference mechanism, the argument of subjective probability and objective probability, the puzzle of the old evidence and simplicity, and the possible developing direction of Bayesian method.
First of all, to start from the perspectives of rationality and originality of Bayesian method, the author outlines the Bayesian method both longitudinally and horizontally. The modern logistic scholars give mathematical Bayesian method logistic meaning. Its origin was the reasonable questioning of induction in Hume's problem. Hume's problem vetoed the rigid reasoning pattern of the induction of simple enumeration method from logistics. The predetermined characteristics of Bayesian method make itself the main reasoning model in the modern induction. The philosophical core of Bayesian method represents in the interpretation of probability. According to the interpretation of probability, there are two categories of Bayesianism:the logistic Bayesianism and subjective Bayesianism. Because the subjectivism has obvious superiority since it avoids the paradoxes of indifference in logistism, it becomes the core theoretical basis of Bayesian method. For this reason, the Baeysian method discussed in this paper is the research method of subjective Bayesianism. Without special indication, Bayesianism mentioned in this paper refers to subjective Bayesianism. In addition, the author proves that Bayesian method is a logical historical method from the perspective of scientific methodology and therefore presents its rationality.
Secondly, the author discusses the types and limitations of classical statistic induction and the application of Bayesian method in statistic induction in two chapters to prove the superiority of Bayesian induction mechanism. The resurrection of Bayesianism happens in the statistical induction area. Bayesian statistical induction avoids the subjectivity and ignore-to-prior in classical statistical inference, and therefore is a revolution of inference method. In the review of hypothesis, the Bayesian statistic inference solves the problem of deciding test statistic in classical statistic inference, averts the problem of stopping rule and its specialty in induction. When solving the estimation problem, Bayesian estimation utilizes credible interval to substitute the confidence interval, and provides a definition and reasonable explanation for classical confidence interval intuition. The application of Bayesian theory conquers the difficulty of introducing prior information in classical estimation. From the perspective of induction, the classical statistic method has been limited by the origin of its methodology-falsification, and has been faced with the questioning of lack of induction. The Bayesian method, however, as a quantification of induction and reasoning mechanism has the advantage of induction.
Thirdly, the author analyzes the difficulties and challenges that the Bayesian method faces with, and points out the reasonable questioning. The Bayesian method has the difficulties of subjectivity, simplicity and the problem of old evidence. The difficulty of subjectivity lies in the problem of constraint of prior probability in Bayesian inference mechanism. The difficulty of simplicity is questioning the principle of simplicity, representing in the inconsistency of simplicity postulate and probability axioms. Foster and Sober stated that simplicity is an "ad hoc method," while Howson advocated to avoid the simplicity and thought simplicity should not be considered as an important guidance for choosing a theory. The old evidence indicates that an old evidence does not provide any confirmation for theory or assumption in the Bayesionism structure. It contradicts with our intuition. Howson uses the relatedness of evidence to prove that "evidence support" implies the relationship among data, assumption and background knowledge k. The problem of old evidence exists, only when e is made the evidence and k includes e. This is the exact reason why Bayesian inference needs further improvement and development.
Last but not the least, the author explores the possible development of Bayesian method from the perspectives of logic, philosophy, and cognition. The rationality of Bayesian method leaves much space for its further development. To view from the developing process of induction logic, the rise of the logic of non-Pascal probability reveals a possible way for Bayesian method, that is, to revise or weaken the Pascal probability that Bayesianism has been insist on, and to form the non-Pascal Bayesianism. The philosophical discussion of Bayesian focuses on the selection of subjective probability interpretation and objective probability interpretation. On one hand, the author believes probability is multi-facets in application and both subjective interpretation and objective interpretation have their own areas for application. On the other hand, the subject interactive interpretation satisfies the objectivity of science inference. Through the research and application in cognitive science, Bayesian method provides hints for its own development. Especially the successful application of Bayesian inference model in cognitive phycology provides opportunity of perception redirection for Bayesian inference study. It also provides possible access to the development of Bayesian method, that is, to pursue the integrity of the probability and Bayesianism, and to attempt the blending of extensionality and non-extensionality by introducing connotation. To speak more specifically, Gigerenzer and Hoffrage brought forward a Bayesian model with frequency representation of information, which means to improve the Bayesian inference method by substituting probability representation with frequency representation. Tversky and others put forward a supporting theory for subjective probability, and built a non-extention logic for the support theory for subjective probability. It is therefore very likely that the cognitive turn of induction logic will be one of the important direction for induction logic's development in the future.
在科学哲学框架内研究贝叶斯方法,包含两个方面：其一,从逻辑上探索贝叶斯方法的推理机制和程序；其二,在方法论意义上回答贝叶斯方法的合理性问题。由此需要讨论的议题有：贝叶斯推理机制的优越性体现；主观概率与客观概率之争；旧证据问题和简单性等难题；以及贝叶斯方法的可能发展进路。
首先,笔者从归纳合理性与贝叶斯方法的兴起出发,分别从纵向和横向对这种方法进行了梳理总结。现代逻辑学者对数学上的贝叶斯方法赋予逻辑意义,源于休谟问题引发的归纳逻辑合理性质疑。休谟问题在逻辑上否定了简单枚举归纳法这种定性的推理模型,而贝叶斯方法的定量特性,使得贝叶斯推理成为现代归纳逻辑的主要推理模型。对于贝叶斯方法的理论内核,其哲学意义体现在概率的解释上。根据概率的不同解释,可以分为逻辑贝叶斯主义和主观贝叶斯主义两种类型。由于主观主义避免了逻辑主义中存在的无差别悖论,所以主观主义表现出明显的优越性,进而成为贝叶斯方法的核心理论依据。从这个意义上理解,本文提及的贝叶斯方法实际上是主观贝叶斯主义的研究方法；而且如无特别说明,本文出现的贝叶斯主义专指主观贝叶斯主义。另外,笔者从科学方法论意义上论证了贝叶斯方法是一种逻辑—历史方法,进而彰显其合理性。
其次,笔者用两章的内容通过介绍经典统计推理的类型和局限性,以及贝叶斯方法在统计推理中的运用,来表现贝叶斯推理机制的优越性。贝叶斯主义的复兴出现在统计推理领域。贝叶斯统计推理避免了经典统计推理中的主观因素问题以及先验回避问题,因而是推理方法的一次革命。在假说检验上,贝叶斯统计推理解决了经典统计中存在的检验统计量的选择难题;避免了停止法则带来的困难并凸显了归纳特性。在面对估计问题时,贝叶斯估计用可信区间代替置信区间,为经典置信区间下的直觉提供了一个概念性的解释和合理说明。贝叶斯定理的应用,克服了经典估计无法引入先验信息的困难。在归纳意义上,经典统计方法受其方法论根源——证伪主义方法的影响,面临缺乏归纳性的质疑。而贝叶斯方法作为归纳推理机制的定量进路,具备显著的归纳意义。
再次,笔者分析了贝叶斯方法面临的困难和挑战,指出这种方法遭遇的合理性质疑问题。贝叶斯方法存在主观性、简单性与旧证据问题等难题。对贝叶斯主义的主观性诘难,主要在于贝叶斯推理机制的先验概率的约束问题。简单性问题就是对简单性原则的质疑,表现在简单性假设与概率公理之间的不一致上。福斯特和索伯指出简单性只是一种“特设方法”,而豪森主张回避简单性问题,他认为简单性不应该被视为理论选择时的一条重要指导准则。旧证据问题表明,在贝叶斯主义框架内,一个旧证据不能对理论或假说提供任何确证。这与我们的直觉相悖。豪森用证据的相关性表明,“证据支持”隐含一个关于数据、假说和背景知识k之间的三元关系。只有当e被判定为证据,且k包含e时,才会出现旧证据问题。这就表明贝叶斯推理需要进一步改进和发展。
最后,笔者根据逻辑、哲学与认知的视角,利用这三个维度多层次探析贝叶斯方法的可能发展进路。贝叶斯方法面临的合理性问题也为其进一步发展留下了宽广空间。从归纳逻辑的发展历程来看,非帕斯卡概率逻辑的兴起,为贝叶斯方法揭示了一条可能进路：修改或弱化贝叶斯主义坚持的帕斯卡概率规则,形成一种非帕斯卡贝叶斯主义。在哲学层面上讨论贝叶斯方法,主要集中于主观概率解释和客观概率解释的选择问题。一方面,笔者认为概率在应用上是多元的,主观解释和客观解释都有其适用范围。另一方面,在哲学上,主体交互解释更满足科学推理要求的客观性。贝叶斯方法在认知科学领域的研究和运用,为其本身发展提供了一些启示。特别是认知心理学对贝叶斯推理模型的成功运用,为贝叶斯推理研究的认知转向提供了契机；同时为这种方法的发展提供了可能进路：寻求频率主义与贝叶斯主义的统一；引入内涵因素,尝试外延性与非外延性的融合。具体而言,吉仁泽和霍夫拉格提出了频率格式的贝叶斯推理模型,即用频率格式代替概率格式来对问题进行信息表征,进而改进贝叶斯推理方法。图文斯基等人提出了主观概率判断的支持理论,建立了一个主观概率的非外延归纳推理理论。从发展趋势看,实现归纳逻辑的认知转向,可能是归纳逻辑未来的重要发展方向之一。
Bayesianism is one of the front edge research methods. It has been widely applied in statistics, economics, psychology and artificial intelligence, etc. Its reasoning model has been called Bayesian inference. Since Keynes started the modern age of induction and created the first probabilistic induction system, Bayesianism has become a strong defender of the induction and gradually developed to be a general scientific reasoning theory and method.
There are two aspects for studying Bayesian method under the spectrum of science philosophy:firstly, to explore the inference mechanism and procedure of Bayesian method from the aspect of reasoning; secondly, to answer reasonable questions of Bayesian method from the aspect of methodology. Therefore, questions to be discussed include the superiority of Bayesian inference mechanism, the argument of subjective probability and objective probability, the puzzle of the old evidence and simplicity, and the possible developing direction of Bayesian method.
First of all, to start from the perspectives of rationality and originality of Bayesian method, the author outlines the Bayesian method both longitudinally and horizontally. The modern logistic scholars give mathematical Bayesian method logistic meaning. Its origin was the reasonable questioning of induction in Hume's problem. Hume's problem vetoed the rigid reasoning pattern of the induction of simple enumeration method from logistics. The predetermined characteristics of Bayesian method make itself the main reasoning model in the modern induction. The philosophical core of Bayesian method represents in the interpretation of probability. According to the interpretation of probability, there are two categories of Bayesianism:the logistic Bayesianism and subjective Bayesianism. Because the subjectivism has obvious superiority since it avoids the paradoxes of indifference in logistism, it becomes the core theoretical basis of Bayesian method. For this reason, the Baeysian method discussed in this paper is the research method of subjective Bayesianism. Without special indication, Bayesianism mentioned in this paper refers to subjective Bayesianism. In addition, the author proves that Bayesian method is a logical historical method from the perspective of scientific methodology and therefore presents its rationality.
Secondly, the author discusses the types and limitations of classical statistic induction and the application of Bayesian method in statistic induction in two chapters to prove the superiority of Bayesian induction mechanism. The resurrection of Bayesianism happens in the statistical induction area. Bayesian statistical induction avoids the subjectivity and ignore-to-prior in classical statistical inference, and therefore is a revolution of inference method. In the review of hypothesis, the Bayesian statistic inference solves the problem of deciding test statistic in classical statistic inference, averts the problem of stopping rule and its specialty in induction. When solving the estimation problem, Bayesian estimation utilizes credible interval to substitute the confidence interval, and provides a definition and reasonable explanation for classical confidence interval intuition. The application of Bayesian theory conquers the difficulty of introducing prior information in classical estimation. From the perspective of induction, the classical statistic method has been limited by the origin of its methodology-falsification, and has been faced with the questioning of lack of induction. The Bayesian method, however, as a quantification of induction and reasoning mechanism has the advantage of induction.
Thirdly, the author analyzes the difficulties and challenges that the Bayesian method faces with, and points out the reasonable questioning. The Bayesian method has the difficulties of subjectivity, simplicity and the problem of old evidence. The difficulty of subjectivity lies in the problem of constraint of prior probability in Bayesian inference mechanism. The difficulty of simplicity is questioning the principle of simplicity, representing in the inconsistency of simplicity postulate and probability axioms. Foster and Sober stated that simplicity is an "ad hoc method," while Howson advocated to avoid the simplicity and thought simplicity should not be considered as an important guidance for choosing a theory. The old evidence indicates that an old evidence does not provide any confirmation for theory or assumption in the Bayesionism structure. It contradicts with our intuition. Howson uses the relatedness of evidence to prove that "evidence support" implies the relationship among data, assumption and background knowledge k. The problem of old evidence exists, only when e is made the evidence and k includes e. This is the exact reason why Bayesian inference needs further improvement and development.
Last but not the least, the author explores the possible development of Bayesian method from the perspectives of logic, philosophy, and cognition. The rationality of Bayesian method leaves much space for its further development. To view from the developing process of induction logic, the rise of the logic of non-Pascal probability reveals a possible way for Bayesian method, that is, to revise or weaken the Pascal probability that Bayesianism has been insist on, and to form the non-Pascal Bayesianism. The philosophical discussion of Bayesian focuses on the selection of subjective probability interpretation and objective probability interpretation. On one hand, the author believes probability is multi-facets in application and both subjective interpretation and objective interpretation have their own areas for application. On the other hand, the subject interactive interpretation satisfies the objectivity of science inference. Through the research and application in cognitive science, Bayesian method provides hints for its own development. Especially the successful application of Bayesian inference model in cognitive phycology provides opportunity of perception redirection for Bayesian inference study. It also provides possible access to the development of Bayesian method, that is, to pursue the integrity of the probability and Bayesianism, and to attempt the blending of extensionality and non-extensionality by introducing connotation. To speak more specifically, Gigerenzer and Hoffrage brought forward a Bayesian model with frequency representation of information, which means to improve the Bayesian inference method by substituting probability representation with frequency representation. Tversky and others put forward a supporting theory for subjective probability, and built a non-extention logic for the support theory for subjective probability. It is therefore very likely that the cognitive turn of induction logic will be one of the important direction for induction logic's development in the future.
引文
1 Howson,C. and Urbach.P. (1989). Scientific Reasoning:The Bayesian Approach. La Salle,:Open Court.10.
2 Stuart, A.1962. Basic Ideas of Scientific Sampling. London:Griffin.12.
3 Howson,C. and Urbach,P. (2006). Scientific Reasoning:The Bayesian Approach (3rd ed.).Chicago:Open Court,253.
4 江天骥,归纳逻辑导论,长沙：湖南人民出版社,1987,第251页。
5 Corfield David and Williamson Jon, Foundations of Bayesianism. Dordrecht:Kluwer Academic Puhlishers, 2001.
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8 Glymour,C. Theory and Evidence. Princeton:Princeton University Press,1980.
9 Earman, J. Bayes or bust? A Critical Examination of Bayesian Confirmation Theory. Cambridge, MA:The MIT Press,1992.135.
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11 Bertrand Russell, Human Knowledge:Its Scope and Limits (London,1948),p.429.
12 K.R.Popper,'Note,'Atture (1983):687.
13 Ken Gemes, Australasian Journal of Philosophy (1983):434-438.
14 任晓明主编：《新编归纳逻辑导论》,河南人民出版社2009年第一版,第59-60页。
15 桂起权、任晓明等：《机遇与冒险的逻辑》,石油大学出版社1996年第一版,第9页。
16 Howson, C.2000. Hume's Problem:Induction and the Justification of Belief. Claredon Press:Oxford,166.
17 John Maynard Keynes, A Treatise on Probability, London:Macmillan and co., limited,1921,3.
18 John Maynard Keynes, A Treatise on Probability, London:Macmillan and co., limited,1921,129-130.
19 John Maynard Keynes, A Treatise on Probability, London:Macmillan and co., limited,1921,10-11.
0 John Maynard Keynes, A Treatise on Probability, London:Macmillan and co., limited,1921,420.
21 Laplace, P.S. de.1820. Essai Philosophique sur les Probabilites. Page references are to Philosophical Essay on Probabilities,1951, New York:Dover Publications.
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