贝叶斯推理的逻辑哲学研究
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摘要
贝叶斯推理理论是归纳逻辑理论和应用研究的热点,本文在借鉴国内外上归纳逻辑最新研究成果的基础上,系统阐述了归纳疑难和归纳悖论、贝叶斯主义和贝叶斯概率逻辑的恰当性、贝叶斯统计推理的恰当性以及贝叶斯网络的哲学意义等问题,重点论证了归纳逻辑哲学的中心问题是逻辑形式系统与其现实原型的恰当相符性问题,分析了认知科学中著名的蒙提霍尔问题、从因明论式的角度对确证悖论做了全新的解读。从逻辑、哲学与认知三个方面总结了归纳逻辑发展的趋势和展望了现代归纳逻辑的未来。
论文首先在比较解决休谟问题的不同方案的前提下,指出贝叶斯方案在解决休谟问题上的优越性,论证贝叶斯方法是如何实现客观性和主观性的统一,私人性和公共性的统一的。其次,阐明了贝叶斯推理的形式化发展对现代归纳逻辑发展的贡献。认为贝叶斯推理的形式化使得现代归纳推理获得并具备了与演绎推理大致相等的地位,按照贝叶斯主义的路径和方法所进行的推理研究,将是科学的、合理的推理方法。第三,指出贝叶斯推理可以更好地处理各种疑难问题。主张在经验与先验、主观与客观之间保持必要的张力,以避免和克服贝叶斯推理所面临的困难,预测了贝叶斯推理未来发展的方向和前景。第四,对三种主要的概率解释进行解读,认为概率解释是多元的,是一个从客观到主观的连续谱系。第五,从贝叶斯推理的视角分析了非经典统计推理的显著性检验和估计理论,认为经典“估计”理论和显著性检验理论都不具有归纳意义,不能用于裁定竞争理论。论文最后指出,不断追求恰当相符是贝叶斯主义归纳逻辑系统产生发展的原动力。
在对贝叶斯推理和非贝叶斯推理比较研究的基础上,论文探讨了贝叶斯统计推理中的应用问题。首先,探讨了贝叶斯统计推理的可应用性问题,指出贝叶斯统计推理具有主观与客观相结合的特性。其次,探讨了贝叶斯推理的另一个重要的应用——贝叶斯网络,认为这种不确定推理可以广泛应用于机器学习、人工智能以及无人机等领域,具有较大的可应用性。最后,论文认为贝叶斯推理在认知科学领域及在因明理论中的分析表明,贝叶斯推理不仅要考虑外延因素,而且要探讨涉及背景信息之类的因素。
Bayesianism has already gained a prominent. role as a theory for probability interpretation. Keynes'formal inductive system helped it to evolve into a new era: Bayesian reasoning is now remodeling statistics, economics, psychology and artificial intelligence, and its influence is still expanding.
Such a trend can be traced back into the renewal of Bayesian method inside statistics. The revolutionary view Bayesianism took successfully eliminated the problem of ignore-to-prior, as well any doubt against subjective factor in induction. By solving the problem of deciding test statistic it averted the problem stopping rule has generated. When it comes to estimation, Bayesianism has replaced confidence interval with credible interval in order to obtain the prediction of the true value of a parameter. It also employs the a priori information through Bayes theorem on conditional probability.
Despite its popularity, several challenges were raised against Bayesianism, specifically on its subjectivity, simplicity and on the problem of old-evidence. And as such, more space has been left for the further development of Bayesianism. The debate of subjectivity is focused on a priori-constraint, while the simplicity question is echoed by the inconsistency between probability axiom and the simplicity postulate. Forster and Sober remarked that the postulate itself is an ad hoc method; Howson argued that it should not be taken as an essential guideline. Additionally, an old-evidence in the Bayesian framework could not serve as affirming the current hypothesis, which contradicts with our instinct. Howson insightfully revealed that "evidence support" machanism is actually a ternary relation between data e, hypothesis h and background knowledge k, and the problem of old-evidence only become evident when e was decided as evidence and e was included in k. It shows that the further improvement and development were needed in Bayesian reasoning.
Recent developments on the application of Bayesianism in cognitive psychology have invoked a possible cognitive turn in Bayesianism, and it reveals that the various possible approaches of Bayesian reasoning:the integration between Frequentism and Bayesianism; to conjoin the intentional factor with its logical indicative nature, it makes an attempt on the combination of extensionality and non-extensionality. Gigerenzer and Hoffrage has already offered a Bayesian model with frequency representation of information; Tversky, etc. proposed their non-extention logic for the support theory for subjective probability. This seems to be a prospective path for inductive reasoning and Bayesianism that worth further exploring.
论文首先在比较解决休谟问题的不同方案的前提下,指出贝叶斯方案在解决休谟问题上的优越性,论证贝叶斯方法是如何实现客观性和主观性的统一,私人性和公共性的统一的。其次,阐明了贝叶斯推理的形式化发展对现代归纳逻辑发展的贡献。认为贝叶斯推理的形式化使得现代归纳推理获得并具备了与演绎推理大致相等的地位,按照贝叶斯主义的路径和方法所进行的推理研究,将是科学的、合理的推理方法。第三,指出贝叶斯推理可以更好地处理各种疑难问题。主张在经验与先验、主观与客观之间保持必要的张力,以避免和克服贝叶斯推理所面临的困难,预测了贝叶斯推理未来发展的方向和前景。第四,对三种主要的概率解释进行解读,认为概率解释是多元的,是一个从客观到主观的连续谱系。第五,从贝叶斯推理的视角分析了非经典统计推理的显著性检验和估计理论,认为经典“估计”理论和显著性检验理论都不具有归纳意义,不能用于裁定竞争理论。论文最后指出,不断追求恰当相符是贝叶斯主义归纳逻辑系统产生发展的原动力。
在对贝叶斯推理和非贝叶斯推理比较研究的基础上,论文探讨了贝叶斯统计推理中的应用问题。首先,探讨了贝叶斯统计推理的可应用性问题,指出贝叶斯统计推理具有主观与客观相结合的特性。其次,探讨了贝叶斯推理的另一个重要的应用——贝叶斯网络,认为这种不确定推理可以广泛应用于机器学习、人工智能以及无人机等领域,具有较大的可应用性。最后,论文认为贝叶斯推理在认知科学领域及在因明理论中的分析表明,贝叶斯推理不仅要考虑外延因素,而且要探讨涉及背景信息之类的因素。
Bayesianism has already gained a prominent. role as a theory for probability interpretation. Keynes'formal inductive system helped it to evolve into a new era: Bayesian reasoning is now remodeling statistics, economics, psychology and artificial intelligence, and its influence is still expanding.
Such a trend can be traced back into the renewal of Bayesian method inside statistics. The revolutionary view Bayesianism took successfully eliminated the problem of ignore-to-prior, as well any doubt against subjective factor in induction. By solving the problem of deciding test statistic it averted the problem stopping rule has generated. When it comes to estimation, Bayesianism has replaced confidence interval with credible interval in order to obtain the prediction of the true value of a parameter. It also employs the a priori information through Bayes theorem on conditional probability.
Despite its popularity, several challenges were raised against Bayesianism, specifically on its subjectivity, simplicity and on the problem of old-evidence. And as such, more space has been left for the further development of Bayesianism. The debate of subjectivity is focused on a priori-constraint, while the simplicity question is echoed by the inconsistency between probability axiom and the simplicity postulate. Forster and Sober remarked that the postulate itself is an ad hoc method; Howson argued that it should not be taken as an essential guideline. Additionally, an old-evidence in the Bayesian framework could not serve as affirming the current hypothesis, which contradicts with our instinct. Howson insightfully revealed that "evidence support" machanism is actually a ternary relation between data e, hypothesis h and background knowledge k, and the problem of old-evidence only become evident when e was decided as evidence and e was included in k. It shows that the further improvement and development were needed in Bayesian reasoning.
Recent developments on the application of Bayesianism in cognitive psychology have invoked a possible cognitive turn in Bayesianism, and it reveals that the various possible approaches of Bayesian reasoning:the integration between Frequentism and Bayesianism; to conjoin the intentional factor with its logical indicative nature, it makes an attempt on the combination of extensionality and non-extensionality. Gigerenzer and Hoffrage has already offered a Bayesian model with frequency representation of information; Tversky, etc. proposed their non-extention logic for the support theory for subjective probability. This seems to be a prospective path for inductive reasoning and Bayesianism that worth further exploring.
引文
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① Wesley C. Salmon. The Foundations of Scientific Inference. Pittsburgh:University of Pittsburgh Press.1967; Peter Walley Statistical Reasoning with Imprecise Probabilities. Chapman & Hall.1991
① Keynes,1.M. ATreatise on Probability. London:Macmillan.1921; Jeffreys, H. Scientific inference,2ed.Cambridge Press.1957; Carnap, R. Logical Foundations of Probability,2nd ed. Chicago:University of Chicago Press.1962
② R.Carnap, Intelectual Autobiography, in P.A.Schilpp(ed), The Philosophy of Rudalf Carnap, Cambridge University Press,1963:72
① R.Carnap, Logical Foundations of Probability, the University of Chicago press,1950 and 1962:211
② K.R.Popper, Conjectures and Refutations. London:Routledge.1963:283
① W.施特格米勒：《归纳问题：休谟提出的挑战和当前的回答》载洪谦主编：《逻辑经验主义》上卷,商务印书馆,1982：260
② 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：41
① 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：48
② 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：52
③ 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：84
④ 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：53
⑤ 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：84
① 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：142
② 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：142
③ 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：107
① [美]罗·格勃尔.《哲学逻辑》,张清宇等译.北京：中国人民大学出版社,2008：420
① 参见[美]罗格勃尔：《哲学逻辑》中国人民大学出版社2008年版,第423-437页
① [美]格勃尔.《哲学逻辑》,张清宇等译.北京：中国人民大学出版社,2008：420
① 参见[美]格勃尔.《哲学逻辑》,张清宇等译.北京：中国人民大学出版社,2008：425
① 蒙提霍尔问题,亦称为蒙特霍问题或三门问题(英文：Monty Hall problem),是一个源自
① 参见：NRC-Handelsblad,1995.5.18.
② 参见：NRC,1995,6.1
③ Wouters P. Filosofische puzzle[J]. Parados 1991,(29):28-43.
① Vos Savant Of Course I'm for Monogamy:I'm Also for Everlasting Peace and an End to Taxes. St Martins Pr,1997:15.
① 参见董英东.动态认知的概率逻辑研究[D].博士论文.南开大学.2011.06.
① Kooi B P. Probabilistic dynamic epistemic logic[J]. Logic Lang. Inform. Special issue on connecting the different faces of information.2003,12(4):381-408
① 庄朝晖：：《西方知识论与佛教知识论的若干比较》,天涯社区(www.tianya.cn)。
Nelson Goodman,Fact,Fiction and Forecast. Cambridge, Mass:Harvard University Press,1955.
① 参见庄朝晖：《因明论视野下的渡鸦悖论,葛梯尔问题和绿蓝悖论》,百度禅吧[2006-10-27,19：37：00]。
② 参见庄朝晖：《因明论视野下的渡鸦悖论,葛梯尔问题和绿蓝悖论》,百度禅吧[2006-10-2719：37：00]
① 纳尔逊·古德曼：《事实,虚构和预测》,北京：商务印书馆,第3章。
② 参见Brian Skyrms, Choice and chance (4thed), Wadsworth Publishing Company,2000
① 邓生庆,归纳逻辑：从古典向现代类型的演进,四川人民出版社,1991：14
② 邓生庆,归纳逻辑：从古典向现代类型的演进,四川人民出版社,1991：14-15
① 任晓明等.新编归纳逻辑导论.郑州：河南人民出版社,2009：66
② 邓生庆、任晓明.归纳逻辑百年历程.北京：中央编译出版社,2006年版第六章
① 参见：桂起权.当代数学哲学与逻辑哲学入门.华东师范大学出版社,1991：97-116
② 桂起权、任晓明、朱志方.机遇与冒险的逻辑,石油大学出版社,1955年
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[1].萨维奇.重新考虑统计学基础[J].选自江天骥主编.《科学哲学著作选读》,.湖北人民出版社,1988：148.
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① John Earman. Bayes or Bust?A critical examination of Bayesian Confirmation Theory.Lodon MIT Press.1992:32.
② 程献礼,牛翠波：《贝叶斯方法和频率方法比较研究》,山西高等社会科学学报,2012年第6期。
① 陈晓平.贝叶斯方法与科学合理性——对休谟问题的思考.北京：人民出版社.2010年
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② 江天骥主编.科学哲学著作选读.武汉：湖北人民出版社,1988：1
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① 原文为the future will look like the past,见休谟.人类理解研究.北京：商务印书馆,1997.33
② B.玛吉.鲍波尔的哲学观与政治观——与卡尔·鲍波尔的对话,杨祯钦译.《哲学译丛》,1980,(6):54-55
③ B.玛吉.鲍波尔的哲学观与政治观——与卡尔·鲍波尔的对话,杨祯钦译.《哲学译丛》,1980,(6)：54-55
① Popper, K.R. The Propensity Interpretation of Probability. British Journal of the Philosophy of Science.1959,(10):25-42
① Poincare, H. Science and Hypothesis. New York:Dover.1952(1905):186
② Poincare, H. Science and Hypothesis. New York:Dover.1952(1905):183-84
③ Poincare, H. Science and Hypothesis. New York:Dover.1952(1905):186
④ Jevons, W.S. The Principles of Science. London:Macmillan.1874,(vol.1):263
① Lakatos, L Philosophical Papers. Two volumes, Edited by Worrell and G. Currie. Cambridge: Cambridge University Press.1978,(1):1
① 洪谦主编.《逻辑经验主义》(上).北京：商务印书馆,1982：257
① 程献礼.弗朗西斯培根的逻辑思想探析[D].河南大学,2009：18-19
① Jaynes, E.T. The Well-Posed Problem. Foundations of Physics, 1973,(Vol.3):413-500
① 这个案例选自：Ian Hacking, Probability and Inductive Logic Cambridge University Press,2001.
① Daid M. Eddy. (1982):"Probabilistic Reasoning in Clinical Medicine:Problems and Opportunities." In D. Kahneman, P. Slovic, and A. Tversky, eds, Judgment under Uncertainty: Heuristics and Biases. Cambridge, England:Cambridge University Press,1982:249-267
① Gerd Gigerenzer & Ulrich Hoffrage. (1995):"How to Improve Bayesian Reasoing without Instruction:Frequency Formats." Psychological Review.102:684-704
① 参见李章吕.支持理论对标准概率理论的超越和发展——逻辑与认知研究的新视角[J].学术论坛,2010,(8)
① 参见程献礼.概率解释探析[J].南开大学学报(增刊),2012：144-146
② Keynes(1921)和卡尔纳普(1950)是20世纪该解释的领军人物；前者把这一思想的起源归功于莱布尼茨。
① 也有非贝叶斯表示定理,这种偏好以某种方式违反约束集C,并产生该类偏好的非贝叶斯表示。请参阅(卡尼曼和特沃斯基,1979)和(Wakker和特沃斯基,1993)。
② Hubbard, Raymond, and M.J. Bayarri.2003. C onfusion over measures of evidence (p's) versus errors (a's) in classical statistical testing (with comments). The American Statistician 57 (August):171-82.
① Hubbard, Raymond, and M.J. Bayarri. Confusion over measures of evidence (p's) versus errors (a's) in classical statistical testing (with comments). The American Statistician 2003.(57-8): 171-82.
② Keuth,H.The Philosophy of Karl Popper[M]. Cambridge, UK:Cambridge University Press. 2005.324.
① Popper,K.R.The Logic of Scientific Discovery[M].London:Hutchinson.1959a:191
② Cournot,A.A.Exposition de la Theories des Chances et des Probabilites[M].Paris.1843:155
① WAINERH, ROBINSONDH. Shaping up the practice of null hypothesis significance testing[J]. Educational Researcher,2003,32 (7):22
② Cowgrecd Statistical significance tests:Scientific ritualism or scientific method? [J]. Social Service Review, 1984,58(3):369
③ 鲍贵席雁.统计显著性检验：问题与思考[J],南京工程学院学报,2010,(4):27
① Kendall, M. G.,and A. Stuart. The Advanced Theory of Statistics[M]. Fourth edition. London:Griffin.1979,(2): 457.
② Hays, W.L. and R.L. Winkler. Statistics:Probability, inference, and Decision, Vol. I. New York:Holt, Rinehart. 1970:195.
③ Cochran, w.G. The x2 Test of Goodness of Fit. Annals of Mathematical Statistics,1952,(23):315-345.
① Good, I.J. Some Logic and History of Hypothes is Testing. In Philosophical r(J/lnda tions o(Economics, edited by.I.e. Pitt (Dordrecht:Reidel) 1981:161.
② Fisher, R.A. Statistical Methods for Research Workers. Fourteenth edition. Edinburgh:Oliver and Boyd.1970 [1925]:Chapter 20.
③ Stuart, A. Too Good to Be True. Applied Statistics.1954,(3):29-32.
① Mood A.M.. and E.A. Gravbill. Introduction to the Theory of Statistics[M]. New York:McGraw-Hill,1963:168.
② Fisher, R.A. On the Mathematical Foundations of Theoretical Statistics. Philosophical Transactions of the Royal Society of London.1922,(A222):316)
① Seidenfeld, T. Philosophical Problems of Statistical Inference[M]. Dordrecht:Reidel, 1979:1973.
Kyburg, H.E., Jr.,and E. Smokier, eds.1980. Studies ill Subjective Probability. Huntington:Krieger.1974:26-35)
① Fisher, R. A The Design of Experiments[M]. Fourth edition. Edinburgh:Oliver and Boyd..1947[1926]:16.
② Bland, M. An Introduction to Medical Statistics. Oxford:Oxford University Press.1987:158.
③ Fisher, R.A. Statistical Tests. Nature.1935,(136):474.
④ Fisher, R.A. Statistical Tests. Nature.1935,(136):474
① Hacking, L Logic of Statistical Inference[M]. Cambridge:Cambridge University Press. 1965.81.
② Lindgren, B.W. Statistical Theory. Third edition. New York:Macmillan 1976:306.
③ 这是我们需强调的Neyman认为没有归纳推理得到抽样信息的许可,这一观点将在下文讨论。
④ Neyman, J., and E.S. Pearson. On the Problem of the Most Efficient Tests of Statistical Hypotheses. Philosophical Transactions of the Royal Society.1933,(231A):289-337.
⑤ Neyman, J. Fiducial Argument and the Theory of Confidence Intervals. Biometrika, Volume 32, 128--150. (Page references are to the reprint in Neyman 1967).1941:380.
① Cramer, H. Mathematical Methods of Statistics[M]. Princeton:Princeton University Press. 1946:421-23.
② Wood, M. Making Sense of Statistics. New York:Palgrave Macmillan.2003:134.
③ Cramer, H. Mathematical Methods of Statistics[M]. Princeton:Princeton University Press.1946:421-23.
Weinberg, S. and K. Goldberg. Statistics for the Behavioral Sciences[M]. Cambridge:Cambridge University Press.1990:291.
① Lindley,D.V A Statistical Paradox[J]. Biometrika,1957,(44):187-192.
② Freeman, P.R. The Role of P-values in Analysing Trial Results[J]. Statistics in Medicine, 1993,(12):1433 1459.
① Freeman, P.R. The Role of P-values in Analysing Trial Results[J]. Statistics in Medicine, 1993,(12):1444
① Lindley, D.V, and L.D. Phillips. Inference for a Bernoulli Process (a Bayesian View). American Statistician,1976,(30):112-19.
② Whitehead. The Case for Frequentism in Clinical Trials. Statistics in Medicine,1993, (12): 1405-413.
① Gillies,D.A.Bayesianism versus Falsificationism[J].Ratio,1990,(3):94
② Gillies,D.A. Bayesianism versus Falsificationism[J].Ratio,1990,(3):94
① 样本范围是最大测量值与最小测量值的差别：如果用递增的顺序排列样本测量值,且如果n是奇数,它们的中位数就是这个数列的中间元素：如果n是偶数,中位数就是两位中间元素中较大的那位。
② Carnap, R. On the Applications of Inductive Logic[J]. Philosophy and Phenomenological Research,1947,(8):133 148
① 参见Hays的事例,1963年,第196页
② Barnett, V Comparative Statistical Inference. New York:Wiley.1973:120
① Kendall, M.G., and A. Stuart. The Advanced Theory of Statistics. Volume 2. Fourth edition. London:Griffin.1979:4
② Rao, C.D. Linear Statistical Inference and its Applications. New York:Wiley.1965:283
③ Fisher, R.A. Statistical Methods and Statisticallnlerende. Edinburgh:Oliver and Boyd. 1956:141
④ Fisher, R.A. Statistical Methods for Research Workers. Fourteenth edition. Edinburgh:Oliver and Boyd. 1970 [1925]:11
⑤ Neyman, J. Lectures and Conferences on Mathematical Statistics and Probability. Second edition. Washington, D.C.:U.S. Department of Agriculture.1952:188
⑥ 可推测此处应该理解为：“估计量”。
① Fisher, R. A. Statistical Methods for Research Workers.14th edition. Edinburgh:Oliver and Boyd. 1970 [1925]:11
① Fisher, R.A. Statistical Methods for Research Workers.14th edition. Edinburgh:Oliver and Boyd.1970 [1925]:12
② Kendall, M.G., and A. Stuart. The Advanced Theory of Statistics. Volume 2.4th edition. London: Griffin.1979:7
③ Neyman, J. Lectures and Conferences on Mathematical Statistics and Probability.2nd edition. Washington, D.C.:U.S. Department of Agriculture.1952:159
① Neyman, J. Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability[J]. Philosophical Transactions of the Royal Society.1937,(236A):263
① Neyman, J. Fiducial Argument and the Theory of Confidence Intervals[J]. Biometrika, Volume32,128-150.请参照重印版Neyman 1967.1941:379
② Hays, W. L. Statistics[M]. London:Holt, Rinehart and Winston.1969 [1963]:290
③ Mood, A. M. Introduction to the Theory of Statistics[M]. New York:McGraw-Hill.1950:222
① Kendall, M.G., and A. Stuart. The Advanced Theory of Statistics[M]. Volume 2.4th edition. London:Griffin.1979:126
② Neyman, J. Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability[J]. Philosophical Transactions of the Royal Society. Volume 236A,1937:333-380
③ Mood, A. M. Introduction to the Theory of Statistics[M]. New York:McGraw-Hill.1950:222
① Chiang, C.L. Statistical Methods of Analysis[M]. Singapore:World Scientific Publishing.2003: 138-39
① Howson和Oddie(1979)年指出了直接概率原则在另一背景中的误用。
② 例如,Lee, P.M. Bayesian statistics:An introduction. New York:Oxford University Press.1989: 213
① Yates, F. Sampling Methodsfor Censuses and Surveys.4th edition. London:Gliffin.1981:11-16
① Downham,.L ed. Issues in Political Opinion Polling. In Occasional Papers on Market Research. London:The Market Research Society.1988:13
② Peto, R., et 01. Randomised Trial of Prophylactic Daily Aspirin in British Male Doctors[J]. British Medical Journal,1988,(296):313-331
③ 关于这部分中抽样的讨论,同样参见Urbach(1989)。
① 程献礼,牛翠波.论统计推理中贝叶斯归纳的哲学研究[J].毕节学院学报,2012,(05).
① 陈晓平.贝叶方法与科学合理性：对休谟问题的思考[M].北京：人民出版社,2010：310-
① Lindley D.V.Introduction to Probability and Statistics, from a Bayesian Viewpoint[M]. Cambridge:Cambridge University Press,1965:52
①. Lindley, D.V. Introduction to Probability and Statistics, from a Bayesian Viewpoint. Volume Two. Cambridge:Cambridge University Press.1965:24-25
② Edwards W. H., Lindman L. J. Savage. Bayesian Statistical Inference for Psychological Research[J]. Psychological Review.1963,(70):193-242
③ Blackwell D., Dubins. L.Merging of Opinions with Increasing Information[J]. Annals of
① Mayo D.G.. Error and the Growth of Experimental Knowledge[M]. Chicago:University of Chicago Press,1996:98
② Savage L. J.. Subjective Probability and Statistical Practice. The Foundations of Statistical Inference, edited by G.A. Barnard and D.R. Cox [M]. New York:Wiley,1962:9-35
① Kadane J.B. Seidenfeld T.. Randomization in a Bayesian Perspective[J]. Journal of Statistical Planning and Inference,1990,(25):329-34
② Stuart, A..Too Good to Be True [J]. Applied Statistics,1954,(3):12
① Korb K.B.. Infinitely Many Resolutions of Hempel's Paradox. In Theoretical Aspects of Reasoning about Knowledge [M]. R. Fagin. Asilomar:Morgan Kaufmann,1994:138-49
② Fisher RA..The Design of Experiments [M]. Edinburgh:Oliver and Boyd.1947:18
③ 参阅:Urbach P..Randomization and the Design of Experiments[J]. Philosophy of Science.1985, (52):256-273
① Lindley, D.VThe Role of Randomization in Inference[J]. Philosophy of Science Association, 1982,(2):431-446
① Mclntyre I.M.C..Tribulations for Clinical Trials [J]. British Medical Journal,1991(302):1099-1100
② Byar D.P..et al. Design Considerations for AIDS Trials. [J]. New England Journal of Medicine, 1990,(323):1343-1348
③ Byar D.P..et al. (22 co-authors). Design Considerations for AIDS Trials. [J]. New England Journal of Medicine,1990, (323):1343-1348
② Blasco A.. An Bayesian Controversy in Animal Breeding[J]. Journal Of Animal Science,2001, (79):2023-2046
③ Lindley D. V. The Future of Statistics: A Bayesian 21st Century. Advances in Applied Probability. Supplement:Proceedings of the Conference on Directions for Mathematical Statistics.1975, (7):106
① 胡玉胜等.基于贝叶斯网络的不确定性知识的推理方法[J].计算机集成制造系统,2001(5).
① Heckerman D. Learning Bayesian Network:the Combination of Knowledge and statistical data [J].Machine Learning,1995,20(3):197-243
① 肖秦琨,高嵩.贝叶斯网络在智能信息处理中的应用.北京：国防工业出版社,2012：11
② [美]Joseph C.Giarrantano Gary D.Riley著；专家系统原理与编程(原书第4版),陈忆群,刘星成译.北京：机械工业出版社,2006：312
① Wesley C. Salmon. The Foundations of Scientific Inference. Pittsburgh:University of Pittsburgh Press.1967; Peter Walley Statistical Reasoning with Imprecise Probabilities. Chapman & Hall.1991
① Keynes,1.M. ATreatise on Probability. London:Macmillan.1921; Jeffreys, H. Scientific inference,2ed.Cambridge Press.1957; Carnap, R. Logical Foundations of Probability,2nd ed. Chicago:University of Chicago Press.1962
② R.Carnap, Intelectual Autobiography, in P.A.Schilpp(ed), The Philosophy of Rudalf Carnap, Cambridge University Press,1963:72
① R.Carnap, Logical Foundations of Probability, the University of Chicago press,1950 and 1962:211
② K.R.Popper, Conjectures and Refutations. London:Routledge.1963:283
① W.施特格米勒：《归纳问题：休谟提出的挑战和当前的回答》载洪谦主编：《逻辑经验主义》上卷,商务印书馆,1982：260
② 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：41
① 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：48
② 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：52
③ 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：84
④ 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：53
⑤ 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：84
① 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：142
② 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：142
③ 江天骥主编,《科学哲学名著选读》,武汉：湖北人民出版社,1988：107
① [美]罗·格勃尔.《哲学逻辑》,张清宇等译.北京：中国人民大学出版社,2008：420
① 参见[美]罗格勃尔：《哲学逻辑》中国人民大学出版社2008年版,第423-437页
① [美]格勃尔.《哲学逻辑》,张清宇等译.北京：中国人民大学出版社,2008：420
① 参见[美]格勃尔.《哲学逻辑》,张清宇等译.北京：中国人民大学出版社,2008：425
① 蒙提霍尔问题,亦称为蒙特霍问题或三门问题(英文：Monty Hall problem),是一个源自
① 参见：NRC-Handelsblad,1995.5.18.
② 参见：NRC,1995,6.1
③ Wouters P. Filosofische puzzle[J]. Parados 1991,(29):28-43.
① Vos Savant Of Course I'm for Monogamy:I'm Also for Everlasting Peace and an End to Taxes. St Martins Pr,1997:15.
① 参见董英东.动态认知的概率逻辑研究[D].博士论文.南开大学.2011.06.
① Kooi B P. Probabilistic dynamic epistemic logic[J]. Logic Lang. Inform. Special issue on connecting the different faces of information.2003,12(4):381-408
① 庄朝晖：：《西方知识论与佛教知识论的若干比较》,天涯社区(www.tianya.cn)。
Nelson Goodman,Fact,Fiction and Forecast. Cambridge, Mass:Harvard University Press,1955.
① 参见庄朝晖：《因明论视野下的渡鸦悖论,葛梯尔问题和绿蓝悖论》,百度禅吧[2006-10-27,19：37：00]。
② 参见庄朝晖：《因明论视野下的渡鸦悖论,葛梯尔问题和绿蓝悖论》,百度禅吧[2006-10-2719：37：00]
① 纳尔逊·古德曼：《事实,虚构和预测》,北京：商务印书馆,第3章。
② 参见Brian Skyrms, Choice and chance (4thed), Wadsworth Publishing Company,2000
① 邓生庆,归纳逻辑：从古典向现代类型的演进,四川人民出版社,1991：14
② 邓生庆,归纳逻辑：从古典向现代类型的演进,四川人民出版社,1991：14-15
① 任晓明等.新编归纳逻辑导论.郑州：河南人民出版社,2009：66
② 邓生庆、任晓明.归纳逻辑百年历程.北京：中央编译出版社,2006年版第六章
① 参见：桂起权.当代数学哲学与逻辑哲学入门.华东师范大学出版社,1991：97-116
② 桂起权、任晓明、朱志方.机遇与冒险的逻辑,石油大学出版社,1955年
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