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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]
MCMC methods can be described in three steps: first using a stochastic mechanism a new state for the Markov chain is proposed. Secondly, the probability of this new state to be correct is calculated. Thirdly, a new random variable (0,1) is proposed.
Many theoretical studies ask how the nervous system could implement Bayesian algorithms. Examples are the work of Pouget, Zemel, Deneve, Latham, Hinton and Dayan. George and Hawkins published a paper that establishes a model of cortical information processing called hierarchical temporal memory that is based on Bayesian network of Markov chains ...
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...
In 1953 the term Markov chain was used for stochastic processes with discrete or continuous index set, living on a countable or finite state space, see Doob. [1] or Chung. [2] Since the late 20th century it became more popular to consider a Markov chain as a stochastic process with discrete index set, living on a measurable state space. [3] [4] [5]
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 ...
A semi-Markov process (defined in the above bullet point) in which all the holding times are exponentially distributed is called a continuous-time Markov chain. In other words, if the inter-arrival times are exponentially distributed and if the waiting time in a state and the next state reached are independent, we have a continuous-time Markov ...
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, their stationary distributions and mixing times, and methods for determining whether Markov chains are rapidly or slowly mixing. [1] [4]