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Markov Chain Mixing Time
In probability theory, the mixing time of a Markov chain is the time until the Markov chain is "close" to its steady state distribution. More precisely, a fundamental result about Markov chains is that a finite state irreducible aperiodic chain has a unique stationary distribution and, regardless of the initial state, the time-''t'' distribution of the chain converges to as ''t'' tends to infinity. Mixing time refers to any of several variant formalizations of the idea: how large must ''t'' be until the time-''t'' distribution is approximately ? One variant, ''total variation distance mixing time'', is defined as the smallest ''t'' such that the total variation distance of probability measures is small: :t_(\varepsilon) = \min \left\. Choosing a different \varepsilon, as long as \varepsilon < 1/2, can only change the mixing time up to a constant factor (depending on ) and so one often fixes and simply writes [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Markov Chain
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Steady State
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p'' of the system, the partial derivative with respect to time is zero and remains so: : \frac = 0 \quad \text t. In discrete time, it means that the first difference of each property is zero and remains so: : p_t-p_=0 \quad \text t. The concept of a steady state has relevance in many fields, in particular thermodynamics, economics, and engineering. If a system is in a steady state, then the recently observed behavior of the system will continue into the future. In stochastic systems, the probabilities that various states will be repeated will remain constant. For example, see ' for the derivation of the steady state. In many systems, a steady state is not achieved until some time after the system is started or initiated. This initial sit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Probability Distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical description of a Randomness, random phenomenon in terms of its sample space and the Probability, probabilities of Event (probability theory), events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that fair coin, the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names. Introduction A prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Markov Chains
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, finance, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Total Variation Distance Of Probability Measures
In probability theory, the total variation distance is a statistical distance between probability distributions, and is sometimes called the statistical distance, statistical difference or variational distance. Definition Consider a measurable space (\Omega, \mathcal) and probability measures P and Q defined on (\Omega, \mathcal). The total variation distance between P and Q is defined as :\delta(P,Q)=\sup_\left, P(A)-Q(A)\. This is the largest absolute difference between the probabilities that the two probability distributions assign to the same event. Properties The total variation distance is an ''f''-divergence and an integral probability metric. Relation to other distances The total variation distance is related to the Kullback–Leibler divergence by Pinsker’s inequality: :\delta(P,Q) \le \sqrt. One also has the following inequality, due to Bretagnolle and Huber (see also ), which has the advantage of providing a non-vacuous bound even when \textstyle D_(P\pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Shuffle
Shuffling is a technique used to randomize a deck of playing cards, introducing an element of chance into card games. Various shuffling methods exist, each with its own characteristics and potential for manipulation. One of the simplest shuffling techniques is the overhand shuffle, where small packets of cards are transferred from one hand to the other. This method is easy to perform but can be manipulated to control the order of cards. Another common technique is the riffle shuffle, where the deck is split into two halves and interleaved. This method is more complex but minimizes the risk of exposing cards. The Gilbert–Shannon–Reeds model suggests that seven riffle shuffles are sufficient to thoroughly randomize a deck, although some studies indicate that six shuffles may be enough. Other shuffling methods include the Hindu shuffle, commonly used in Asia, and the pile shuffle, where cards are dealt into piles and then stacked. The Mongean shuffle involves a specific sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Randomized Algorithms
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses Uniform distribution (discrete), uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. There is a distinction between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result (Monte Carlo algorithms, for example the Monte Carlo algorithm for the Minimum feedback arc set, MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Graph Coloring
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the Vertex (graph theory), vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an ''edge coloring'' assigns a color to each Edge (graph theory), edges so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each Face (graph theory), face (or region) so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Markov Chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that is, the Markov chain's Discrete-time Markov chain#Stationary distributions, equilibrium distribution matches the target distribution. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Markov chain Monte Carlo methods are used to study probability distributions that are too complex or too highly N-dimensional space, dimensional to study with analytic techniques alone. Various algorithms exist for constructing such Markov chains, including the Metropolis–Hastings algorithm. General explanation Markov chain Monte Carlo methods create samples from a continuous random variable, with probability density proportional to a known function. These samples can be used to e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Conductance (probability)
In theoretical computer science, graph theory, and mathematics, the conductance is a parameter of a Markov chain that is closely tied to its mixing time, that is, how rapidly the chain converges to its stationary distribution, should it exist. Equivalently, the conductance can be viewed as a parameter of a directed graph, in which case it can be used to analyze how quickly random walks in the graph converge. The conductance of a graph is closely related to the Cheeger constant of the graph, which is also known as the edge expansion or the isoperimetic number. However, due to subtly different definitions, the conductance and the edge expansion do not generally coincide if the graphs are not regular. On the other hand, the notion of electrical conductance that appears in electrical networks is unrelated to the conductance of a graph. History The conductance was first defined by Mark Jerrum and Alistair Sinclair in 1988 to prove that the permanent of a matrix with entries fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
