Fermi's Golden Rule says that the transition rate W is given by a transition matrix element ( or " amplitude " ) M _ { i, f } weighted by the phase space and Planck's constant \ hbar such that
42.
For a continuous-time Markov chain with transition rate matrix " Q ", the uniformized discrete-time Markov chain has probability transition matrix P : = ( p _ { ij } ) _ { i, j }, which is defined by
43.
In fact, Game B is a Markov chain, and an analysis of its state transition matrix ( again with M = 3 ) shows that the steady state probability of using coin 2 is 0.3836, and that of using coin 3 is 0.6164.
44.
For a discrete time Markov chain with transition matrix " Q " and equilibrium distribution \ pi, the global balance equation has the same form as above, except that q _ { ij } should be interpreted as the transition probability and not the transition rate.
45.
While at the University of Toronto, Hastings also supervised his one Ph . D . student, Peter Peskun ( now at York University ), whose 1970 dissertation " The Choice Of Transition Matrix In Monte Carlo Sampling Methods Using Markov Chains " developed the Peskun ordering on Markov chain kernels.
46.
Even with time-inhomogeneous Markov chains, where multiple transition matrices are used, if each such transition matrix exhibits detailed balance with the desired "'? "'distribution, this necessarily implies that "'? "'is a steady-state distribution of the Markov chain.
47.
The matrix \ theta _ a is then a square matrix, whose entries are zero or one, indicating whether a transition q \ stackrel { a } { \ rightarrow } q ^ \ prime is allowed by the NFA . Such a transition matrix is always defined for a non-deterministic finite automaton.
48.
Hence, a two-level model such as just described allows independent control over ( 1 ) the overall density of the transition matrix, and ( 2 ) the density of states to which transitions are likely ( i . e . the density of the prior distribution of states in any particular hidden variable x _ i ).
49.
Consider a weighted network either directed or undirected with n nodes denoted by j = 1, &, n; and a random walk process on this network with a transition matrix M . The m _ { jk } element of M describes the probability of the random walker that has reached node i, proceeds directly to node j.
50.
With state vector " x " and transition matrix " A ", " x " converges asymptotically to the steady state vector " x " * if and only if all eigenvalues of the transition matrix " A " ( whether real or complex ) have an absolute value which is less than 1.
