Transition Rate

Definition

A Transition Rate is a key property of a multi-state stochastic system (e.g. a Markov Chain). It measures the probability (per unit of time) that an event (state transition) occurs within an infinitesimally small time interval.

Mathematically, if $X$ is a stochastic process, its transition rates are defined as follows:

• Lets assume a state space with D+1 distinct states: $S ={0, \dots ,D}$.
• The rate of moving from state m to state n in an infinitesimal dime $\Delta t$ is $q^{mn}$, represented as:

$\Pr(X(t + \Delta t) = n \mid X(t) = m) = \delta_{mn} + q^{mn}(t) \Delta t + o(\Delta t)$

Properties

• Since the transition rates are refer to the probabilities of transitions, they must be positive (but need not be less than unity)

$q^{mn} > 0$

• The collection of all transition rates forms a Transition Rate Matrix that satisfies further properties

See Also

• Transition Matrix
• Hazard Rate

Issues and Challenges

The terminology around transition matrix quantities can be confusing as they are used in slightly different contexts:

• When modelling stochastic processes in continuous time, the transition rate is distinct from the Transition Probability which measures transition frequencies over a finite time period
• When estimating transition phenomena the accumulation of statistics is always over a finite period, yet frequently one still uses the term "transition rate"

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