베이지안 확률

빈도주의 : 

- 만약 우리가 f값을 안다면(실제로는 모르지만), 설문 결과로 얻게 될 F는 얼마가 될 것인가?

- 200명을 대상으로 '반복적으로' 설문조사를 시행한다면, 그 결과값 F는 95%의 확률로 33%와 48% 사이에 위치하게 될 것이다. 

베이지안 : 

- 만약 F라는 데이터가 주어진다면, 참값 f는 얼마가 될 것인가?

- 200명을 대상으로 '단 한 번의' 설문조사를 해서 F=40%라는 결과를 얻었다면, 참값 f는 95%의 확률로 33%와 48%사이에 위치할 것이다. 

Last time we derived the Frequentist result that when f=40%, if you repeatedly conduct surveys of N=200 people, then 95% of those surveys will produce values for F that are between 33% and 48%, which was our definition of F being "close" to f.

The Frequentist answers a subtly different question: if we knew f(which we don't), what would the survey result F be likely to be? So I drew it with a backward pointing arrow instead.

In contrast, the Bayesian result says: given that you actually conducted one survey of N=200 people in which you measured F=40%, then the true value f is 95% likely to be between 33% and 48%.

The Bayesian approach directly answers our question: given the data F, where is the true f likely to be? So I drew it with a forward arrow in the diagram. 

(출처 : http://meandering-through-mathematics.blogspot.kr/2011/05/bayesian-probability.html)

베이지안 확률론(Bayesian probability)은 확률을 '지식의 상태를 측정'하는 것이라고 해석하는 확률론이다. ^[1] 확률을 발생 빈도(frequency)나 어떤 시스템의 물리적 속성이라고 여기는 것과는 다른 해석이다. 이 분야의 선구자였던 18세기 통계학자 토마스 베이즈의 이름을 따서 명명되었다. 이어만(Earman 1992)에 따르면 베이지안 확률에는 두 가지 시점이 있는데 그 하나는 객관적 관점에서 베이지안 통계의 법칙은 이성적 보편적으로 증명될 수 있으며 논리의 확장으로 설명될 수 있다. 한편 주관주의 확률 이론의 관점으로 보면 지식의 상태는 개인적인 믿음의 정도(degree of belief)로 측정할 수 있다. 많은 현대적 기계 학습 방법은 객관적 베이지안 원리에 따라 만들어졌다. 베이지안 확률은 확률에 대한 여러 개념 중 가장 인기있는 것 중의 하나로 심리학, 사회학, 경제학이론에 많이 응용된다. 어떤 가설의 확률을 평가하기 위해서 사전 확률을 먼저 밝히고 새로운 관련 데이터에 의한 새로운 확률값을 변경한다. 베이즈 정리는 이러한 확률의 계산에 있어 필요한 절차와 공식의 표준을 제시하고 있다.

(출처 : http://ko.wikipedia.org/wiki/베이지안_확률론)

Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of the branch of mathematical logic known as propositional logic that enables reasoning with propositions whose truth or falsity is uncertain. To evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then updated in the light of new, relevant data.^[1]

The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation. Bayesian probability interprets the concept of probability as "an abstract concept, a quantity that we assign theoretically, for the purpose of representing a state of knowledge, or that we calculate from previously assigned probabilities,"^[2] in contrast to interpreting it as a frequency or "propensity" of some phenomenon.

The term "Bayesian" refers to the 18th century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of Bayesian inference.^[3] Nevertheless, it was the French mathematician Pierre-Simon Laplace who pioneered and popularised what is now called Bayesian probability.^[4]

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 requirements of rationality and consistency and interpreted as an extension of logic.^[2][5] According to the subjectivist view, probability quantifies a "personal belief".^[6] Many modern machine learning methods are based on objectivist Bayesian principles.^[7] In the Bayesian view, a probability is assigned to a hypothesis, whereas under the frequentist view, a hypothesis is typically tested without being assigned a probability.

Bayesian methodology [edit]

특징 

In general, Bayesian methods are characterized by the following concepts and procedures:

• The use of random variables to model all sources of uncertainty in statistical models. This includes not just sources of true randomness, but also uncertainty resulting from lack of information.
• The sequential use of the Bayes' formula: when more data become available after calculating a posterior distribution, the posterior becomes the next prior.
• For the frequentist a hypothesis is a proposition (which must be either true or false), 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 true value is uncertain.

(출처 : http://en.wikipedia.org/wiki/Bayesian_probability)