Concept
Markov chain
Related concepts (47)
Memorylessness
In probability and statistics, memorylessness is a property of certain probability distributions. It usually refers to the cases when the distribution of a "waiting time" until a certain event does not depend on how much time has elapsed already. To model memoryless situations accurately, we must constantly 'forget' which state the system is in: the probabilities would not be influenced by the history of the process. Only two kinds of distributions are memoryless: geometric distributions of non-negative integers and the exponential distributions of non-negative real numbers.
Entropy rate
In the mathematical theory of probability, the entropy rate or source information rate of a stochastic process is, informally, the time density of the average information in a stochastic process. For stochastic processes with a countable index, the entropy rate is the limit of the joint entropy of members of the process divided by , as tends to infinity: when the limit exists. An alternative, related quantity is: For strongly stationary stochastic processes, .
Statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process. When referring specifically to probabilities, the corresponding term is probabilistic model. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables.
Markov blanket
In statistics and machine learning, when one wants to infer a random variable with a set of variables, usually a subset is enough, and other variables are useless. Such a subset that contains all the useful information is called a Markov blanket. If a Markov blanket is minimal, meaning that it cannot drop any variable without losing information, it is called a Markov boundary. Identifying a Markov blanket or a Markov boundary helps to extract useful features. The terms of Markov blanket and Markov boundary were coined by Judea Pearl in 1988.
Markov information source
In mathematics, a Markov information source, or simply, a Markov source, is an information source whose underlying dynamics are given by a stationary finite Markov chain. An information source is a sequence of random variables ranging over a finite alphabet , having a stationary distribution. A Markov information source is then a (stationary) Markov chain , together with a function that maps states in the Markov chain to letters in the alphabet .
Markov kernel
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space. Let and be measurable spaces. A Markov kernel with source and target is a map with the following properties: For every (fixed) , the map is -measurable For every (fixed) , the map is a probability measure on In other words it associates to each point a probability measure on such that, for every measurable set , the map is measurable with respect to the -algebra .
Stationary distribution
Stationary distribution may refer to: A special distribution for a Markov chain such that if the chain starts with its stationary distribution, the marginal distribution of all states at any time will always be the stationary distribution. Assuming irreducibility, the stationary distribution is always unique if it exists, and its existence can be implied by positive recurrence of all states. The stationary distribution has the interpretation of the limiting distribution when the chain is irreducible and aperiodic.
Stochastic cellular automaton
Stochastic cellular automata or probabilistic cellular automata (PCA) or random cellular automata or locally interacting Markov chains are an important extension of cellular automaton. Cellular automata are a discrete-time dynamical system of interacting entities, whose state is discrete. The state of the collection of entities is updated at each discrete time according to some simple homogeneous rule. All entities' states are updated in parallel or synchronously.
Deterministic system
In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state. Physical laws that are described by differential equations represent deterministic systems, even though the state of the system at a given point in time may be difficult to describe explicitly.
Diffusion equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation, when bulk velocity is zero.