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For example, a series of simple observations, such as a person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the Hierarchical hidden Markov model [2] and the Abstract Hidden Markov Model. [3]
A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies. [6] For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless ...
For a continuous time Markov chain (CTMC) with transition rate matrix, if can be found such that for every pair of states and π i q i j = π j q j i {\displaystyle \pi _{i}q_{ij}=\pi _{j}q_{ji}} holds, then by summing over j {\displaystyle j} , the global balance equations are satisfied and π {\displaystyle \pi } is the stationary ...
A finite-state machine can be used as a representation of a Markov chain. Assuming a sequence of independent and identically distributed input signals (for example, symbols from a binary alphabet chosen by coin tosses), if the machine is in state y at time n , then the probability that it moves to state x at time n + 1 depends only on the ...
[1] [2] Such models are often described as M/G/1 type Markov chains because they can describe transitions in an M/G/1 queue. [ 3 ] [ 4 ] The method is a more complicated version of the matrix geometric method and is the classical solution method for M/G/1 chains.
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 equilibrium distribution matches the target distribution. The more steps ...
Poisson process, an example of a jump process; Continuous-time Markov chain (CTMC), an example of a jump process and a generalization of the Poisson process; Counting process, an example of a jump process and a generalization of the Poisson process in a different direction than that of CTMCs; Interacting particle system, an example of a jump ...
The mixing time of a Markov chain is the number of steps needed for this convergence to happen, to a suitable degree of accuracy. A family of Markov chains is said to be rapidly mixing if the mixing time is a polynomial function of some size parameter of the Markov chain, and slowly mixing otherwise. This book is about finite Markov chains ...