From Wikipedia, the free encyclopedia
Bayesian probability is one
of the concept of . The Bayesian interpretation of probability can be seen as an extension of
that enables
with hypotheses, i.e., the
Bayesian probability belongs to the category of evid to evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then updated in the light of new, relevant
(evidence). The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation.
In contrast to interpreting
of some phenomenon, Bayesian probability is a quantity that we assign for the purpose of representing a state of knowledge, or a state of belief. In the Bayesian view, a probability is assigned to a hypothesis, whereas under the , a hypothesis is typically
without being assigned a probability.
The term "Bayesian" refers to the 18th century mathematician and theologian , who provided the first mathematical treatment of a non-trivial problem of . Mathematician
pioneered and popularised what is now called Bayesian probability.
Broadly speaking, there are two views on Bayesian probability that interpret the probability concept in different ways. According to the objectivist view, the rules of Bayesian statistics can be justified by
and interpreted as an extension of . According to the subjectivist view, probability quantifies a "personal belief".
Bayesian methods are characterized by the following concepts and procedures:
The use of random variables, or, more generally, unknown quantities, to model all sources of uncertainty in statistical models. This also includes uncertainty resulting from lack of information (see also the ).
The need to determine the prior probability distribution taking into account the available (prior) information.
The sequential use of the : when more data becomes available, calculate the posterior distribution using the Bayes' subsequently, the posterior distribution becomes the next prior.
For the frequentist a
(which must be ), so that the frequentist probability of a hypothesis is either one or zero. In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the truth value is uncertain.
Broadly speaking, there are two views on Bayesian probability that interpret the 'probability' concept in different ways. For , probability objectively measures the plausibility of propositions, i.e. the probability of a proposition corresponds to a reasonable belief everyone (even a "robot") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by . For , probability corresponds to a 'personal belief'. For subjectivists, rationality and coherence constrain the probabilities a subject may have, but allow for substantial variation within those constraints. The objective and subjective variants of Bayesian probability differ mainly in their interpretation and construction of the prior probability.
Main article:
The term Bayesian refers to
(), who proved a special case of what is now called
in a paper titled "". In that special case, the prior and posterior distributions were
and the data came from . It was
() who introduced a general version of the theorem and used it to approach problems in , medical statistics, , and . Early Bayesian inference, which used uniform priors following Laplace's , was called "" (because it
backwards from observations to parameters, or from effects to causes). After the 1920s, "inverse probability" was largely supplanted by a collection of methods that came to be called .
In the 20th century, the ideas of Laplace were further developed in two different directions, giving rise to objective and subjective currents in Bayesian practice. ' Theory of Probability (first published in 1939) played an important role in the revival of the Bayesian view of probability, followed by works by
(1950) and
(1954). The adjective Bayesian itself dates to the 1950s; the derived Bayesianism, neo-Bayesianism is of 1960s coinage. In the objectivist stream, the statistical analysis depends on only the model assumed and the data analysed. No subjective decisions need to be involved. In contrast, "subjectivist" statisticians deny the possibility of fully objective analysis for the general case.
In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of
methods, which removed many of the computational problems, and an increasing interest in nonstandard, complex applications. Despite the growth of Bayesian research, most undergraduate teaching is still based on frequentist statistics.[] Nonetheless, Bayesian methods are widely accepted and used, such as in the field of .
The use of Bayesian probabilities as the basis of
has been supported by several arguments, such as the , the , arguments based on
showed that Bayesian updating follows from several axioms, including two
and a controversial hypothesis of differentiability. It is known that Cox's 1961 development (mainly copied by ) is non-rigorous, and in fact a counterexample has been found by Halpern. The assumption of differentiability or even continuity is questionable since the Boolean algebra of statements may only be finite. Other axiomatizations have been suggested by various authors to make the theory more rigorous.
The Dutch book argument was proposed by de Finetti, and is based on betting. A
is made when a clever gambler places a set of bets that guarantee a profit, no matter what the outcome of the bets. If a
follows the rules of the Bayesian calculus in the construction of his odds, a Dutch book cannot be made.
noted that traditional Dutch book arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books. For example,
writes "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour."
In fact, there are non-Bayesian updating rules that also avoid Dutch books (as discussed in the literature on "probability kinematics" following the publication of ' rule, which is itself regarded as Bayesian ). The additional hypotheses sufficient to (uniquely) specify Bayesian updating are substantial, complicated, and unsatisfactory.
justification of the use of Bayesian inference (and hence of Bayesian probabilities) was given by , who proved that every
statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures. Conversely, every Bayesian procedure is .
Following the work on
and , decision-theorists have accounted for
using a probability distribution for the . Johann Pfanzagl completed the
by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and : their original theory supposed that all the agents had the same probability distribution, as a convenience. Pfanzagl's axiomatization was endorsed by Oskar Morgenstern: "Von Neumann and I have anticipated" the question whether probabilities "might, perhaps more typically, be subjective and have stated specifically that in the latter case axioms could be found from which could derive the desired numerical utility together with a number for the probabilities (cf. p. 19 of The ). We di it was demonstrated by Pfanzagl ... with all the necessary rigor".
Ramsey and
noted that the individual agent's probability distribution could be objectively studied in experiments. The role of judgment and disagreement in science has been recognized since
and even more clearly with . The objectivity of science lies not in the psychology of individual scientists, but in the process of science and especially in statistical methods, as noted by . Recall that the objective methods for falsifying propositions about personal probabilities have been used for a half century, as noted previously. Procedures for
about probabilities (using finite samples) are due to
(1931) and
(, ). Both
acknowledge[] their debts to , particularly (for Ramsey) to .
The "Ramsey test" for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century. This work demonstrates that Bayesian-probability propositions can be , and so meet an empirical criterion of , whose work inspired Ramsey. (This -criterion was popularized by .)
Modern work on the experimental evaluation of personal probabilities uses the randomization, , and Boolean-decision procedures of the Peirce-Jastrow experiment. Since individuals act according to different probability judgments, these agents' probabilities are "personal" (but amenable to objective study).
Personal probabilities are problematic for science and for some applications where decision-makers lack the knowledge or time to specify an informed probability-distribution (on which they are prepared to act). To meet the needs of science and of human limitations, Bayesian statisticians have developed "objective" methods for specifying prior probabilities.
Indeed, some Bayesians have argued the prior state of knowledge defines the (unique) prior probability-distribution for "regular"
cf. . Finding the right method for constructing such "objective" priors (for appropriate classes of regular problems) has been the quest of statistical theorists from Laplace to , , and : These theorists and their successors have suggested several methods for constructing "objective" priors:
Each of these methods contributes useful priors for "regular" one-parameter problems, and each prior can handle some challenging
(with "irregularity" or several parameters). Each of these methods has been useful in Bayesian practice. Indeed, methods for constructing "objective" (alternatively, "default" or "ignorance") priors have been developed by avowed subjective (or "personal") Bayesians like
(), simply because such priors are needed for Bayesian practice, particularly in science. The quest for "the universal method for constructing priors" continues to attract statistical theorists.
Thus, the Bayesian statistician needs either to use informed priors (using relevant expertise or previous data) or to choose among the competing methods for constructing "objective" priors.
A Bayesian average is a method of estimating the
of a population consistent with Bayesian interpretation, where instead of estimating the mean strictly from any or all available data set, other existing information related to that data set may also be incorporated into the calculation in order to minimize the impact of large deviations, or to assert a default value when the data set is small.
Calculating the Bayesian average uses the prior mean m and a constant C. C is assigned a value that is proportional to the typical data set size. The value is larger when the expected variation between data sets (within the larger population) is small. It is smaller, when the data sets are expected to vary substantially from one another.
 — a paradox in classical probability, solved by
in the context of Bayesian probability
 — a procedure for evaluating someone's subjective probability
 — a controversial application of Bayesian probabilities to
Paulos, John Allen.
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(1978). . Annals of Statistics 6 (March): 239–265 esp. p. 248. :.  .  .
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2nd Edition, Routledge
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(1994). Bayesian Theory. Wiley.  .
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(December 1967). "Slightly More Realistic Personal Probability". Philosophy of Science 34 (4): 311–325. :.  . Partly reprinted in:
and Sahlin, Nils-Eric. (1988) Decision, Probability, and Utility: Selected Readings. 1988. Cambridge University Press.
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: Hidden categories:音节划分：ap?proach
高频词，一定要记住哦！
[?'pr??t?]
[?'pro?t?]
接近，走近，靠近
着手处理；
试图贿赂（或影响，疏通）
过去分词：
现在分词：
第三人称单数：
大家都在背：
1. We have a very communicative approach to teaching languages.
我们在语言教学中非常强调交际教学法。
来自柯林斯例句
2. At their approach the little boy scurried away and hid.
他们走近时，小男孩急忙跑开藏了起来。
来自柯林斯例句
3. The traffic on the approach road slowed to a crawl.
引路上车辆行驶缓慢。
来自柯林斯例句
4. The city of Memphis is promoting a populist approach to culture.
孟菲斯市在推行文化的民粹主义道路。
来自柯林斯例句
5. The path serves as an approach to the boat house.
这条小路通往船屋。
来自柯林斯例句
可接近的,随和的 (approach靠近,接近+able可以……的→可接近的)
应责备的 (reproach[n.&v.责备,指责]+able……的→adj.应责备的)
接近;途径,入门;方式,方法 (ap一再+proach接近→v.靠近,接近n.接近;途径,入门;方式,方法)
责备,指责 (re反+proach接近→不接近→指责)
靠近,接近 (ap一再+proach接近→v.靠近,接近n.接近;途径,入门;方式,方法)
1. “通道,入口”释义下的同义词
1. “接近,靠近”释义下的同义词
其他释义下的同义词
1. “接近;靠近;开始对付”释义下的反义词
其他释义下的反义词
： 多指行动的特殊方式或独特的方法。
： 指有系统、有条理地办事或解决问题的方法。
： 普通用词，可指一般的方法，有时也指个人的方法或方式，也可指特殊的方式或方法。
： 书面用词，常指因个人爱好或传统习俗等因素而遵循的方法。
： 着重独特的程序或方式，尤指个人的偏爱或习惯。
： 指为达到某种目的或目标而采用的方法、手段或途径。
： 指从事某事的特别方法、途径。
靠近;接近;走近 When you approach something, you get closer to it.&
【语法信息】：V n
【语法信息】：V
【语法信息】：V-ing
【搭配模式】：usu N of n
He didn't approach the front door at once...
他没有马上走向前门。
When I approached, they grew silent...
当我走近时，他们就不说话了。
We turned to see the approaching car slow down.
我们转身看见驶近的车慢慢停下。
Approach is also a noun.
At their approach the little boy ran away and hid.
他们一走近，小男孩就逃走藏了起来。
...the approach of a low-flying helicopter.
一架低空飞行的直升机的逼近
路径;途径;道路 An approach to a place is a road, path, or other route that leads to it.&
【搭配模式】：usu N to n
The path serves as an approach to the boat house.
这条小路通往船屋。
与…接洽；找…商量 If you approach someone about something, you speak to them about it for the first time, often making an offer or request.&
【语法信息】：V n prep
【语法信息】：V n to-inf
【语法信息】：V n
【搭配模式】：no cont
When Chappel approached me about the job, my first reaction was of disbelief...
当查普尔为这份工作找我商量时，我的第一反应是不相信。
He approached me to create and design the restaurant...
他请我建造并设计该饭店。
Anna approached several builders and was fortunate to come across Eddie.
安娜与几个建筑商有过接洽，并且很幸运地碰到了埃迪。
Approach is also a noun.
There had already been approaches from buyers interested in the whole of the group.
已有对整组感兴趣的买家来进行洽谈。
探讨；处理；对待 When you approach a task, problem, or situation in a particular way, you deal with it or think about it in that way.&
【语法信息】：V n prep/adv
【语法信息】：V n
The Bank has approached the issue in a practical way...
银行已经实际解决了这个问题。
Employers are interested in how you approach problems.
雇主们对你如何处理问题感兴趣。
方法;态度;手段 Your approach to a task, problem, or situation is the way you deal with it or think about it.&
【搭配模式】：usu with supp
We will be exploring different approaches to gathering information.
我们将探索收集信息的不同方法。
...the adversarial approach of the British legal system.
英国法律制度采用的辩论式审判模式
临近；来临 As a future time or event approaches, it gradually gets nearer as time passes.&
【语法信息】：V
【语法信息】：V-ing
As autumn approached, the plants and colours in the garden changed.
秋天临近，花园中的草木和颜色也发生了变化。
...the approaching crisis.
步步逼近的危机
Approach is also a noun.
...the festive spirit that permeated the house with the approach of Christmas.
随着圣诞节临近房子里洋溢着的节日气氛
接近（未来时间或事件） As you approach a future time or event, time passes so that you get gradually nearer to it.&
【语法信息】：V n
We approach the end of the year with the economy slowing and little sign of cheer.
接近年末时，我们面临经济迟滞、处处萧条的局面。
近似；接近于 If something approaches a particular level or state, it almost reaches that level or state.&
【语法信息】：V n
Oil prices have approached their highest level for almost ten years...
石油价格已接近近10年中的最高水平。
Mansell will race at average speeds approaching 200mph.
曼塞尔将以接近每小时200英里的平均车速参赛。
1. 接近, 走近, 靠近
Christmas was approaching.
圣诞节快到了。
Walk softly as you approach the bed.
当你接近床时, 走路轻些。
1. 接洽, 交涉; 着手处理
He approached the question as a scientist.
他以一个科学家的眼光去处理这个问题。
2. 试图贿赂(或影响，疏通)
Those officials were approached with bribes.
那些官员被贿赂打通关节的。
3. (在性质、数量、质量、情形、时间等方面)近似，近于，接近；相似，类似
As a poet he hardly approaches John Milton.
他的诗才很难与弥尔顿媲美。
4. 使移近，使接近(某物)，把…挪近
to approach the magnet to this heap of filings
将那吸铁石移近这一个锉屑
1. 靠近，临近，逼近；接近，走近，行近，即将来临
Our approach drove away the wild animals.
我们一走近, 野兽全都跑开了。
2. 进路，通路；入门，入口；途径
Police are patrolling the major approach roads to the stadium.
警察正在通往运动场的主要道路上巡逻。
3. (处理问题、完成任务的)方法，方式；手段；步骤；态度
His approaches to the problem are wrong.
他处理这个问题的方法是错误的。
4. 接洽；建议；要求
We approach you today in the hope of establishing business relations with you.
我们同您接洽以希望能建立起合作关系。
5. 进场；进场着陆
6. 相似（或近似）的事物
7. 近似(值)，近于；相似，类似
8. [通常用于复数](对某人主动的)亲近(或接近)的表示；打交道；提议，建议；疏通行为
1. (在性质、时间、数量、质量等方面)接近，近似
to approach to the character of …
几乎具有…的性格
2. 【航空学】1). 进场着陆；进场；2).(战斗机)接敌；3). (轰炸机)进入目标
1. ideas or actions intended to deal with a
"his approach to every problem is to draw up a list of pros and cons"
"an attack on inflation"
"his plan of attack was misguided"
2. the act of drawing spatially
"the hunter's approach scattered the geese"
3. a way of
"he took a wrong turn on the access to the bridge"
4. the final path followed by an aircraft as it is landing
5. the event of one object coming closer to another
6. a tentative suggestion designed to elicit the
"she rejected his advances"
7. the temporal property of bec
"the approach of winter"
"the nearest approach to genius"
9. a relatively short golf shot intended to put the ball on
"he lost the hole when his approach rolled over the green"
"We were approaching our destination"
"They are drawing near"
"The enemy army came nearer and nearer"
，，，，，
2. come near or verge on, resemble, come nearer in quality,
"This borders on discrimination!"
"His playing approaches that of Horowitz"
"approach a task"
"go about a difficult problem"
"approach a new project"
"Winter is approaching"
"approaching old age"
5. make advances to someone, usually with a pr
"I was approached by the President to serve as his adviser in foreign matters"
1.进山路线
1.进路：在手术中，指显露器官或部分的特殊解剖步骤
1.湿球温差
2.为完成切入过程所必须附加的行程长度。
4.进给运动开始前，加工工具与工件相互接近的过程。
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