Stochastic chains with memory of variable length are a family of stochastic chains of finite order in a finite alphabet, such as, for every time pass, only one finite suffix of the past, called context, is necessary to predict the next symbol. These models were introduced in the information theory literature by Jorma Rissanen in 1983, as a universal tool to data compression, but recently have been used to model data in different areas such as biology, linguistics and music.

A stochastic chain with memory of variable length is a stochastic chain ${\displaystyle (X_{n})_{n\in Z}}$, taking values in a finite alphabet ${\displaystyle A}$, and characterized by a probabilistic context tree ${\displaystyle (\tau ,p)}$, so that

The class of stochastic chains with memory of variable length was introduced by Jorma Rissanen in the article A universal data compression system. Such class of stochastic chains was popularized in the statistical and probabilistic community by P. Bühlmann and A. J. Wyner in 1999, in the article Variable Length Markov Chains. Named by Bühlmann and Wyner as “variable length Markov chains” (VLMC), these chains are also known as “variable-order Markov models" (VOM), “probabilistic suffix trees” and “context tree models”. The name “stochastic chains with memory of variable length” seems to have been introduced by Galves and Löcherbach, in 2008, in the article of the same name.

Consider a system by a lamp, an observer and a door between both of them. The lamp has two possible states: on, represented by 1, or off, represented by 0. When the lamp is on, the observer may see the light through the door, depending on which state the door is at the time: open, 1, or closed, 0. such states are independent of the original state of the lamp.

Let ${\displaystyle (X_{n})_{n\geq 0}}$ a Markov chain that represents the state of the lamp, with values in ${\displaystyle A={0,1}}$ and let ${\displaystyle p}$ be a probability transition matrix. Also, let ${\displaystyle (\xi _{n})_{n\geq 0}}$ be a sequence of independent random variables that represents the door's states, also taking values in ${\displaystyle A}$, independent of the chain ${\displaystyle (X_{n})_{n\geq 0}}$ and such that

where ${\displaystyle 0<\epsilon <1}$. Define a new sequence ${\displaystyle (Z_{n})_{n\geq 0}}$ such that

In order to determine the last instant that the observer could see the lamp on, i.e. to identify the least instant ${\displaystyle k}$, with ${\displaystyle k<n}$ in which ${\displaystyle Z_{k}=1}$.

Using a context tree it's possible to represent the past states of the sequence, showing which are relevant to identify the next state.