Bernoulli scheme

 ( Markov Models ) 

In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes.P. Shields, The theory of Bernoulli shifts, Univ. Chicago Press (1973)Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as Chapter 2 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of . Many important dynamical systems (such as Axiom A systems) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift.Pierre Gaspard, Chaos, scattering and statistical mechanics (1998), Cambridge University press This is essentially the Markov partition. The term shift is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal.

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Definition A Bernoulli scheme is a discrete-time stochastic process where each independent random variable may take on one of N distinct possible values, with the outcome i occurring with probability $p\_i$, with i = 1, ...,  N, and

$\backslash sum\_\{i=1\}^N\; p\_i\; =\; 1.$

The sample space is usually denoted as

$X=\backslash \{1,\backslash ldots,N\; \backslash \}^\backslash mathbb\{Z\}$

as a shorthand for

$X=\backslash \{\; x=(\backslash ldots,x\_\{-1\},x\_0,x\_1,\backslash ldots)\; :$

x_k \in \{1,\ldots,N\} \; \forall k \in \mathbb{Z} \}.

The associated measure is called the Bernoulli measure

$\backslash mu\; =\; \backslash \{p\_1,\backslash ldots,p\_N\backslash \}^\backslash mathbb\{Z\}$

The sigma-algebra $\backslash mathcal\{A\}$ on X is the product sigma algebra; that is, it is the (countable) direct product of the σ-algebras of the finite set {1, ...,  N}. Thus, the triplet

$(X,\backslash mathcal\{A\},\backslash mu)$

is a measure space. A basis of $\backslash mathcal\{A\}$ is the . Given a cylinder set $x\_0,$, its measure is

$\backslash mu\backslash left(x\_0,\backslash right)=$

\prod_{i=0}^n p_{x_i} The equivalent expression, using the notation of probability theory, is

$\backslash mu\backslash left(x\_0,\backslash right)=$

\mathrm{Pr}(X_0=x_0, X_1=x_1, \ldots, X_n=x_n) for the random variables $\backslash \{X\_k\backslash \}$

The Bernoulli scheme, as any stochastic process, may be viewed as a dynamical system by endowing it with the shift operator T where

$T(x\_k)\; =\; x\_\{k+1\}.$

Since the outcomes are independent, the shift preserves the measure, and thus T is a measure-preserving transformation. The quadruplet

$(X,\backslash mathcal\{A\},\backslash mu,\; T)$

is a measure-preserving dynamical system, and is called a Bernoulli scheme or a Bernoulli shift. It is often denoted by

$BS(p)=BS(p\_1,\backslash ldots,p\_N).$

The N = 2 Bernoulli scheme is called a Bernoulli process. The Bernoulli shift can be understood as a special case of the Markov shift, where all entries in the adjacency matrix are one, the corresponding graph thus being a clique.

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Matches and metrics The Hamming distance provides a natural metric on a Bernoulli scheme. Another important metric is the so-called $\backslash overline\; f$ metric, defined via a supremum over string matches.

Let $A\; =\; a\_1a\_2\backslash cdots\; a\_m$ and $B\; =\; b\_1b\_2\backslash cdots\; b\_n$ be two strings of symbols. A match is a sequence M of pairs $(i\_k,\; j\_k)$ of indexes into the string, i.e. pairs such that $a\_\{i\_k\}=b\_\{j\_k\},$ understood to be totally ordered. That is, each individual subsequence $(i\_k)$ and $(j\_k)$ are ordered:

The $\backslash overline\; f$- distance between $A$ and $B$ is

$\backslash overline\; f(A,B)\; =\; 1-\backslash frac\{2\backslash sup\; |M|\}\{m+n\}$

where the supremum is being taken over all matches $M$ between $A$ and $B$. This satisfies the triangle inequality only when $m=n,$ and so is not quite a true metric; despite this, it is commonly called a "distance" in the literature.

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Generalizations Most of the properties of the Bernoulli scheme follow from the countable direct product, rather than from the finite base space. Thus, one may take the base space to be any standard probability space $(Y,\backslash mathcal\{B\},\backslash nu)$, and define the Bernoulli scheme as

$(X,\; \backslash mathcal\{A\},\; \backslash mu)=(Y,\backslash mathcal\{B\},\backslash nu)^\backslash mathbb\{Z\}$

This works because the countable direct product of a standard probability space is again a standard probability space.

As a further generalization, one may replace the integers $\backslash mathbb\{Z\}$ by a countable discrete group $G$, so that

$(X,\; \backslash mathcal\{A\},\; \backslash mu)=(Y,\backslash mathcal\{B\},\backslash nu)^G$

For this last case, the shift operator is replaced by the group action

$gx(f)=x(g^\{-1\}f)$

for group elements $f,g\backslash in\; G$ and $x\backslash in\; Y^G$ understood as a function $x:G\backslash to\; Y$ (any direct product $Y^G$ can be understood to be the set of functions $G\backslash to$, as this is the exponential object). The measure $\backslash mu$ is taken as the Haar measure, which is invariant under the group action:

$\backslash mu(gx)=\backslash mu(x).\; \backslash ,$

These generalizations are also commonly called Bernoulli schemes, as they still share most properties with the finite case.

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Properties Ya. Sinai demonstrated that the Kolmogorov entropy of a Bernoulli scheme is given byYa.G. Sinai, (1959) "On the Notion of Entropy of a Dynamical System", Doklady of Russian Academy of Sciences 124, pp. 768–771.Ya. G. Sinai, (2007) " Metric Entropy of Dynamical System"

$H\; =\; -\backslash sum\_\{i=1\}^N\; p\_i\; \backslash log\; p\_i\; .$

This may be seen as resulting from the general definition of the entropy of a Cartesian product of probability spaces, which follows from the asymptotic equipartition property. For the case of a general base space $(Y,\; \backslash mathcal\{B\},\; \backslash nu)$ ( i.e. a base space which is not countable), one typically considers the relative entropy. So, for example, if one has a countable partition $Y\text{'}\backslash subset\; Y$ of the base Y, such that $\backslash nu(Y\text{'})=1$, one may define the entropy as

$H\_\{Y\text{'}\}\; =\; -\backslash sum\_\{y\text{'}\backslash in\; Y\text{'}\}\; \backslash nu(y\text{'})\; \backslash log\; \backslash nu(y\text{'})\; .$

In general, this entropy will depend on the partition; however, for many , it is the case that the symbolic dynamics is independent of the partition (or rather, there are isomorphisms connecting the symbolic dynamics of different partitions, leaving the measure invariant), and so such systems can have a well-defined entropy independent of the partition.

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Ornstein isomorphism theorem The Ornstein isomorphism theorem states that two Bernoulli schemes with the same entropy are isomorphic. The result is sharp, in that very similar, non-scheme systems, such as Kolmogorov automorphisms, do not have this property.

The Ornstein isomorphism theorem is in fact considerably deeper: it provides a simple criterion by which many different measure-preserving dynamical systems can be judged to be isomorphic to Bernoulli schemes. The result was surprising, as many systems previously believed to be unrelated proved to be isomorphic. These include all finite stationary stochastic processes, subshifts of finite type, finite , , and Sinai's billiards: these are all isomorphic to Bernoulli schemes.

For the generalized case, the Ornstein isomorphism theorem still holds if the group G is a countably infinite amenable group.

(2025). 9780821869222 ISBN 9780821869222

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Bernoulli automorphism An invertible, measure-preserving transformation of a standard probability space (Lebesgue space) is called a Bernoulli automorphism if it is isomorphic to a Bernoulli shift.Peter Walters (1982) An Introduction to Ergodic Theory, Springer-Verlag,

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Loosely Bernoulli A system is termed "loosely Bernoulli" if it is Kakutani-equivalent to a Bernoulli shift; in the case of zero entropy, if it is Kakutani-equivalent to an irrational rotation of a circle.

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See also

• Shift of finite type
• Markov chain
• Hidden Bernoulli model

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