Stochastic chains with memory of variable length

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

• $\backslash tau$ is the group of all contexts. A context $X\_\{n-l\},\backslash ldots,X\_\{n-1\}$, being $l$ the size of the context, is a finite portion of the past $X\_\{-\backslash infty\},\backslash ldots,X\_\{n-1\}$, which is relevant to predict the next symbol $X\_\{n\}$;
• $p$ is a family of transition probabilities associated with each context.

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”.{{cite journal |vauthors= Galves A, Garivier A, Gassiat E|date=2012|title= Joint estimation of intersecting context tree models|journal = Scandinavian Journal of Statistics|volume = 40|issue = 2|pages = 344–362|doi = 10.1111/j.1467-9469.2012.00814.x|arxiv= 1102.0673}} 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.{{cite journal |vauthors= Galves A, Löcherbach E|date=2008|title= Stochastic chains with memory of variable length|journal = TICSP Series|volume = 38|pages = 117–133|arxiv=0804.2050 |url = https://hal.archives-ouvertes.fr/hal-00798528/document}}

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

: $\backslash mathbb\{P\}(\backslash xi\_n\; =\; 1)\; =\; 1\; -\; \backslash varepsilon$

where . Define a new sequence $(Z\_n)\_\{n\; \backslash ge\; 0\}$ such that

: $Z\_n\; =\; X\_n\; \backslash xi\_n$ for every $(Z\_n)\_\{n\; \backslash ge\; 0\}.$

In order to determine the last instant that the observer could see the lamp on, i.e. to identify the least instant $k$, with

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

The stochastic chain $(Z\_n)\_\{n\backslash in\backslash mathbb\{Z\}\}$ is, then, a chain with memory of variable length, taking values in $A$ and compatible with the probabilistic context tree $(\backslash tau,p)$, where

: $\backslash tau\; =\; \backslash \{1,10,100,\backslash cdots\backslash \}\; \backslash cup\; \backslash \{0^\backslash infty\backslash \}.$

Given a sample $X\_\{l\},\backslash ldots,X\_\{n\}$, one can find the appropriated context tree using the following algorithms.

In the article A Universal Data Compression System, Rissanen introduced a consistent algorithm to estimate the probabilistic context tree that generates the data. This algorithm's function can be summarized in two steps:

1. Given the sample produced by a chain with memory of variable length, we start with the maximum tree whose branches are all the candidates to contexts to the sample;
2. The branches in this tree are then cut until you obtain the smallest tree that's well adapted to the data. Deciding whether or not shortening the context is done through a given gain function, such as the ratio of the log-likelihood.

Be $X\_\{0\},\backslash ldots,X\_\{n-1\}$ a sample of a finite probabilistic tree $(\backslash tau,p)$. For any sequence $x\_\{-j\}^\{-1\}$ with $j\; \backslash leq\; n$, it is possible to denote by $N\_n(x\_\{-j\}^\{-1\})$ the number of occurrences of the sequence in the sample, i.e.,

: $N\_n(x\_\{-j\}^\{-1\})\; =\; \backslash sum\_\{t=0\}^\{n-j\}\; \backslash mathbf\{1\}\backslash left\backslash \{X\_t^\{t+j-1\}\; =\; x\_\{-j\}^\{-1\}\backslash right\backslash \}$

Rissanen first built a context maximum candidate, given by $X\_\{n-K(n)\}^\{n-1\}$, where $K(n)=C\backslash log\{n\}$ and $C$ is an arbitrary positive constant. The intuitive reason for the choice of $C\backslash log\{n\}$ comes from the impossibility of estimating the probabilities of sequence with lengths greater than $\backslash log\{n\}$ based in a sample of size $n$.

From there, Rissanen shortens the maximum candidate through successive cutting the branches according to a sequence of tests based in statistical likelihood ratio. In a more formal definition, if bANnxk1b0 define the probability estimator of the transition probability $p$ by

: $\backslash hat\{p\}\_n(a\backslash mid\; x\_\{-k\}^\{-1\})\; =\; \backslash frac\{N\_n(x\_\{-k\}^\{-1\}a)\}\{\backslash sum\_\{b\; \backslash in\; A\}\; N\_n\; (x\_\{-k\}^\{-1\}\; b)\}$

where $x\_\{-j\}^\{-1\}a=(x\_\{-j\},\; \backslash ldots,\; x\_\{-1\},a)$. If $\backslash sum\_\{b\; \backslash in\; A\}N\_n(x\_\{-k\}^\{-1\}b)\; \backslash ,=\backslash ,0$, define $\backslash hat\{p\}\_n(a\backslash mid\; x\_\{-k\}^\{-1\})\; \backslash ,=\backslash ,\; 1/|A|$.

To $i\; \backslash geq\; 1$, define

: $\backslash Lambda\_n\; (x\_\{-\; i\; \}^\{-1\}\; )\; \backslash ,=\backslash ,\; 2\; \backslash ,\; \backslash sum\_\{y\; \backslash in\; A\}\; \backslash sum\_\{a\; \backslash in\; A\}\; N\_n(y\; x\_\{-i\}^\{-1\}a)\; \backslash log\backslash left[\backslash frac\{\backslash hat\{p\}\_n(a\backslash mid\; x\_\{-i\}^\{-1\}\; y)\}\; \{\backslash hat\{p\}\_n(a\backslash mid\; x\_\{-i\}^\{-1\})\}\; \backslash right]\backslash ,$

where $y\; x\_\{-i\}^\{-1\}=(y,x\_\{-i\},\; \backslash ldots\; ,\; x\_\{-1\})$ and

: $\backslash hat\{p\}\_n(a\backslash mid\; x\_\{-i\}^\{-1\}y)=\; \backslash frac\{N\_n(y\; x\_\{-i\}^\{-1\}a)\}\{\backslash sum\_\{b\; \backslash in\; A\}\; N\_n\; (y\; x\_\{-i\}^\{-1\}b)\}.$

Note that $\backslash Lambda\_n\; (x\_\{-\; i\; \}^\{-1\})$ is the ratio of the log-likelihood to test the consistency of the sample with the probabilistic context tree $(\backslash tau,p)$ against the alternative that is consistent with $(\backslash tau\text{'},p\text{'})$, where $\backslash tau$ and $\backslash tau\text{'}$ differ only by a set of sibling knots.

The length of the current estimated context is defined by

: $\backslash hat\{\backslash ell\}\_n(X\_0^\{n-1\})=\; \backslash max\; \backslash left\backslash \{i=1,\backslash ldots,\; K(n):\; \backslash Lambda\_n$

(X_{n-i}^{n-1}) \,>\, C \log n \right\}\,

where $C$ is any positive constant. At last, by Rissanen, there's the following result. Given $X\_0,\backslash ldots,\; X\_\{n-1\}$ of a finite probabilistic context tree $(\backslash tau,p)$, then

: $P\backslash left(\; \backslash hat\{\backslash ell\}\_n(X\_0^\{n-1\})\; \backslash neq\; \backslash ell(X\_0^\{n-1\})\; \backslash right)\; \backslash longrightarrow\; 0,$

when $n\; \backslash rightarrow\; \backslash infty$.

The estimator of the context tree by BIC with a penalty constant $c>0$ is defined as

: $\backslash hat\{\backslash tau\}\_\backslash mathrm\{BIC\}=\backslash underset\{\backslash tau\; \backslash in\; \backslash mathcal\{T\}\_n\}\{\backslash arg\; \backslash max\}\backslash \{\backslash log\; L\_\backslash tau\; (X\_1^n)-c\backslash ,\backslash textrm\{d\}f\; (\backslash tau)\backslash log\; n\; \backslash \}$

The smallest maximizer criterion is calculated by selecting the smallest tree τ of a set of champion trees C such that

: $\backslash lim\_\{n\backslash to\; \backslash infty\}\; \backslash frac\{\backslash log\; L\_\backslash tau\; (X^n\_1)\; -\; \backslash log\; L\_\{\backslash hat\{\backslash tau\}\}(X^n\_1)\}\{n\}\; =\; 0$

• Variable-order Markov model
• Markov chain
• Stochastic process

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Category:Stochastic models