Two-Urn 모형의 비복원 추출 데이터에 근거한 사전 확률의 변화

Abstract

Urn 모형(urn model)은 확률과정(stochastic process)을 직관적으로 구현하는 개념적인 방법으로써, 다양한 현실적인 문제에 적용이 가능하여 지금까지 다양한 형태의 모형들이 연구되어왔다. Urn 모형은 기본적으로 Urn과 Urn에 포함된 공으로 구성되며, 모형을 구성하는 Urn의 수와 공의 종류, 공을 추출하는 방법에 따라 구분된다. 본 연구는 두 개의 Urn과 두 가지 종류의 공으로 구성된 Two-Urn 모형에서 추출 시행의 결과에 기반한 사전확률(prior probability)의 변화 과정을 제시한다. 사전확률은 첫 시행에 발생한 관심 사건(event)의 확률로 정의하며, 추가 데이터가 발생할 때 사전확률이 변화하는 과정을 설명한다. 확률의 변화 과정에서는 데이터에 관한 확률과 사전확률이 결합하여 사후확률(posterior probability)을 추정하는 베이지안(Bayesian) 접근법을 사용한다. 도출된 확률과정을 기반으로 Two-Urn 모형에서 사전확률이 특정 값으로 수렴되는 상황과 조건을 명제(proposition)로 제시하고 증명한다. 그리고 응용사례를 구축하여 본 연구에서 제시한 Two-Urn 모형과 사전확률 변화 과정의 응용 가능성을 제시한다.|This paper deals with the evolution of prior probabilities based on data from non-replacement extraction of the two-urn model. The urn model is a conceptual method for intuitively implementing the stochastic process and can be applied to various practical problems. Basically, the urn model consists of the urn and balls contained in the urn. The urn models are classified by the number of urn, type of ball, and method of extraction. A two-urn model consisting of two urns and two color balls is considered in this paper. The prior probability is defined as the probability that the selected white ball at the first trial came from urn 1. The Bayesian approach is used to estimate the posterior probability by combining the probability of data and the prior probability. Based on the stochastic process of the two-urn model, four propositions related to the prior probabilities, depending on the situation relevant to the number of each colored balls, are proven. In respective proposition, the prior probability is shown to converge on a specific constant. An application example is also given.; This paper deals with the evolution of prior probabilities based on data from non-replacement extraction of the two-urn model. The urn model is a conceptual method for intuitively implementing the stochastic process and can be applied to various practical problems. Basically, the urn model consists of the urn and balls contained in the urn. The urn models are classified by the number of urn, type of ball, and method of extraction. A two-urn model consisting of two urns and two color balls is considered in this paper. The prior probability is defined as the probability that the selected white ball at the first trial came from urn 1. The Bayesian approach is used to estimate the posterior probability by combining the probability of data and the prior probability. Based on the stochastic process of the two-urn model, four propositions related to the prior probabilities, depending on the situation relevant to the number of each colored balls, are proven. In respective proposition, the prior probability is shown to converge on a specific constant. An application example is also given.