Lumpable Markov Chain
In probability theory, lumpability is a method for reducing the size of the state space of some continuous-time Markov chains, first published by Kemeny and Snell. Definition Suppose that the complete state-space of a Markov chain is divided into disjoint subsets of states, where these subsets are denoted by ''ti''. This forms a partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ... \scriptstyle of the states. Both the state-space and the collection of subsets may be either finite or countably infinite. A continuous-time Markov chain \ is lumpable with respect to the partition ''T'' if and only if, for any subsets ''ti'' and ''tj'' in the partition, and for any states ''n,n’'' in subset ''ti'', : \sum_ q(n,m) = \sum_ q(n',m) , where ''q''(''i,j'') is the transition rate f ...
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Probability Theory
Probability theory 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 not possible to perfectly p ...
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