# 마르코프 체인

## 예

- 다음의 문장이 주어져 있다
- 모든 국민은 학문과 예술의 자유를 가진다.
- 모든 국민은 사생활의 비밀과 자유를 침해받지 아니한다.

- 두 문장에 나오는 단어들은 이 확률과정의 상태공간 S를 이룬다
- S={"모든", "국민은", "비밀과", "예술의", "자유를", "학문과", "가진다.", "사생활의", "침해받지", "아니한다."}

- "모든" 이라는 단어(상태)에서 출발하여, 연결된 선을 따라 다음 단어로 이동하며 (전이), 마침표가 있는 단어에 이르면 이 과정을 종료한다
- 한 단어에서 다음 단어로 넘어갈 확률은 두 단어가 연결된 빈도로부터 얻어진다

- 이러한 확률과정을 통하여, 새로운 문장을 생성할 수 있게 된다
- 예: "모든 국민은 학문과 예술의 자유를 침해받지 아니한다"

- 이는 현재의 한 단어만이 다음 단어에 영향을 주는 '바이그램' 모형이다

## 관련된 항목들

## 매스매티카 파일

## 노트

### 위키데이터

- ID : Q176645

### 말뭉치

- Markov chains, named after Andrey Markov, are mathematical systems that hop from one "state" (a situation or set of values) to another.
^{[1]} - Of course, real modelers don't always draw out Markov chain diagrams.
^{[1]} - This means the number of cells grows quadratically as we add states to our Markov chain.
^{[1]} - One use of Markov chains is to include real-world phenomena in computer simulations.
^{[1]} - Markov chain analysis is concerned in general with how long individual entities spend, on average, in different states before being absorbed and on first passage times to absorbing states.
^{[2]} - Thus Markov chain analysis is ideal for providing insights on life history or anything related to timing.
^{[2]} - Life expectancy – The fundamental matrix in Markov chain analysis provides a measure of expected time in each state before being absorbed.
^{[2]} - Markov chains are a fairly common, and relatively simple, way to statistically model random processes.
^{[3]} - A popular example is r/SubredditSimulator, which uses Markov chains to automate the creation of content for an entire subreddit.
^{[3]} - Overall, Markov Chains are conceptually quite intuitive, and are very accessible in that they can be implemented without the use of any advanced statistical or mathematical concepts.
^{[3]} - This example illustrates many of the key concepts of a Markov chain.
^{[3]} - A (time-homogeneous) Markov chain built on states A and B is depicted in the diagram below.
^{[4]} - If the Markov chain in Figure 21.3 is used to model time-varying propagation loss, then each state in the chain corresponds to a different loss.
^{[5]} - Each number represents the probability of the Markov process changing from one state to another state, with the direction indicated by the arrow.
^{[6]} - A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC).
^{[6]} - A continuous-time process is called a continuous-time Markov chain (CTMC).
^{[6]} - However, many applications of Markov chains employ finite or countably infinite state spaces, which have a more straightforward statistical analysis.
^{[6]} - His analysis did not alter the understanding or appreciation of Pushkin’s poem, but the technique he developed—now known as a Markov chain—extended the theory of probability in a new direction.
^{[7]} - In physics the Markov chain simulates the collective behavior of systems made up of many interacting particles, such as the electrons in a solid.
^{[7]} - And Markov chains themselves have become a lively area of inquiry in recent decades, with efforts to understand why some of them work so efficiently—and some don’t.
^{[7]} - As Markov chains have become commonplace tools, the story of their origin has largely faded from memory.
^{[7]} - A Markov chain describes the transitions between a given set of states using transition probabilities.
^{[8]} - A Markov chain evolves in discrete time and moves step by step from state to state; the step size can be chosen arbitrarily, and depending on the application, it could be 1 day, or 1 month, or 1 year.
^{[8]} - Transition probabilities only depend on the current state the Markov chain is in at time t, and not on any previous states at t−1,t−2, ….
^{[8]} - Markov chains are usually analyzed in matrix notation.
^{[8]} - But the concept of modeling sequences of random events using states and transitions between states became known as a Markov chain.
^{[9]} - One of the first and most famous applications of Markov chains was published by Claude Shannon.
^{[9]} - For a Markov chain to be ergodic, two technical conditions are required of its states and the non-zero transition probabilities; these conditions are known as irreducibility and aperiodicity.
^{[10]} - It follows from Theorem 21.2.1 that the random walk with teleporting results in a unique distribution of steady-state probabilities over the states of the induced Markov chain.
^{[10]} - In the improved multivariate Markov chain model, Ching et al. incorporated positive and negative association parts.
^{[11]} - With the developments of Markov chain models and their applications, the number of the sequences may be larger.
^{[11]} - It is inevitable that a large categorical data sequence group will cause high computational cost in multivariate Markov chain model.
^{[11]} - In Section 2, we review two lemmas and several Markov chain models.
^{[11]} - The Season 1 episode "Man Hunt" (2005) of the television crime drama NUMB3RS features Markov chains.
^{[12]} - Along the way, you are welcome to talk with other students (don't forget we have a Blackboard Q&A discussion forum!) and explore how others have used Markov chains.
^{[13]} - A Markov chain is a mathematical model that describes movement (transitions) from one state to another.
^{[13]} - In a first-order Markov chain, this probability only depends on the current state, and not any further transition history.
^{[13]} - We can build more "memory" into our model by using a higher-order Markov chain.
^{[13]} - The paper also investigates the definition of stochastic monotonicity on a more general state space, and the properties of integer-valued stochastically monotone Markov Chains.
^{[14]} - Markov chain Monte Carlo using the Metropolis-Hastings algorithm is a general method for the simulation of stochastic processes having probability densities known up to a constant of proportionality.
^{[15]} - Consider the problem of modelling memory effects in discrete-state random walks using higher-order Markov chains.
^{[16]} - As an illustration of model selection for multistep Markov chains, this manuscript re-examines the hot hand phenomenon from a different analytical philosophy.
^{[16]} - The memoryless property of Markov chains refers to h = 1.
^{[16]} - This manuscript addressed general methods of degree selection for multistep Markov chain models.
^{[16]} - A Markov chain is a mathematical system usually defined as a collection of random variables, that transition from one state to another according to certain probabilistic rules.
^{[17]} - The algorithm known as PageRank, which was originally proposed for the internet search engine Google, is based on a Markov process.
^{[17]} - A discrete-time Markov chain involves a system which is in a certain state at each step, with the state changing randomly between steps.
^{[17]} - A Markov chain is represented using a probabilistic automaton (It only sounds complicated!).
^{[17]} - This unique guide to Markov chains approaches the subject along the four convergent lines of mathematics, implementation, simulation, and experimentation.
^{[18]} - An introduction to simple stochastic matrices and transition probabilities is followed by a simulation of a two-state Markov chain.
^{[18]} - The notion of steady state is explored in connection with the long-run distribution behavior of the Markov chain.
^{[18]} - Predictions based on Markov chains with more than two states are examined, followed by a discussion of the notion of absorbing Markov chains.
^{[18]} - We next give examples that illustrate various properties of quantum Markov chains.
^{[19]} - First, the probability sequence is treated as consisting of successive realizations of a stochastic process, and the stochastic process is modeled using Markov chain theory.
^{[20]} - The size and stationarity of the reforecast ensemble dataset permits straightforward estimation of the parameters of certain Markov chain models.
^{[20]} - Finally, opportunities to apply the Markov chain model to decision support are highlighted.
^{[20]} - In this same framework, the importance of recognizing inhomogeneity in the Markov chain parameters when specifying decision intervals is illustrated.
^{[20]} - The extra questions are interesting and off the well-beaten path of questions that are typical for an introductory Markov Chains course.
^{[21]} - CUP 1997 (Chapter 1, Discrete Markov Chains is freely available to download.
^{[21]} - (Each of these books contains a readable chapter on Markov chains and many nice examples.
^{[21]} - See also, Sheldon Ross and Erol Pekoz, A Second Course in Probability, 2007 (Chapter 5 gives a readable treatment of Markov chains and covers many of the topics in our course.
^{[21]} - Markov chains are used to model probabilities using information that can be encoded in the current state.
^{[22]} - More technically, information is put into a matrix and a vector - also called a column matrix - and with many iterations, a collection of probability vectors makes up Markov chains.
^{[22]} - Representing a Markov chain as a matrix allows for calculations to be performed in a convenient manner.
^{[22]} - A Markov chain is one example of a Markov model, but other examples exist.
^{[22]} - So far, we have discussed discrete-time Markov chains in which the chain jumps from the current state to the next state after one unit time.
^{[23]} - Markov chain Monte Carlo is one of our best tools in the desperate struggle against high-dimensional probabilistic computation, but its fragility makes it dangerous to wield without adequate training.
^{[24]} - Unfortunately the Markov chain Monte Carlo literature provides limited guidance for practical risk management.
^{[24]} - Before introducing Markov chain Monte Carlo we will begin with a short review of the Monte Carlo method.
^{[24]} - Finally we will discuss how these theoretical concepts manifest in practice and carefully study the behavior of an explicit implementation of Markov chain Monte Carlo.
^{[24]}

### 소스

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Markov Chains explained visually - ↑
^{2.0}^{2.1}^{2.2}Markov Chain - an overview - ↑
^{3.0}^{3.1}^{3.2}^{3.3}Introduction to Markov Chains - ↑ Brilliant Math & Science Wiki
- ↑ Markov Chain - an overview
- ↑
^{6.0}^{6.1}^{6.2}^{6.3}Markov chain - ↑
^{7.0}^{7.1}^{7.2}^{7.3}First Links in the Markov Chain - ↑
^{8.0}^{8.1}^{8.2}^{8.3}Estimating the number and length of episodes in disability using a Markov chain approach - ↑
^{9.0}^{9.1}Origin of Markov chains (video) - ↑
^{10.0}^{10.1}Definition: - ↑
^{11.0}^{11.1}^{11.2}^{11.3}A New Multivariate Markov Chain Model for Adding a New Categorical Data Sequence - ↑ Markov Chain -- from Wolfram MathWorld
- ↑
^{13.0}^{13.1}^{13.2}^{13.3}Mission 2: Markov Chains - ↑ Stochastically monotone Markov Chains
- ↑ Geyer : Practical Markov Chain Monte Carlo
- ↑
^{16.0}^{16.1}^{16.2}^{16.3}Predictive Bayesian selection of multistep Markov chains, applied to the detection of the hot hand and other statistical dependencies in free throws - ↑
^{17.0}^{17.1}^{17.2}^{17.3}(Tutorial) Markov Chains in Python - ↑
^{18.0}^{18.1}^{18.2}^{18.3}Markov Chains: From Theory to Implementation and Experimentation - ↑ Quantum Markov chains
- ↑
^{20.0}^{20.1}^{20.2}^{20.3}Markov Chain Modeling of Sequences of Lagged NWP Ensemble Probability Forecasts: An Exploration of Model Properties and Decision Support Applications - ↑
^{21.0}^{21.1}^{21.2}^{21.3}Markov Chains - ↑
^{22.0}^{22.1}^{22.2}^{22.3}Markov Chain - ↑ Introduction
- ↑
^{24.0}^{24.1}^{24.2}^{24.3}Markov Chain Monte Carlo in Practice