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| In [[mathematics]], the '''Bernoulli scheme''' or '''Bernoulli shift''' is a generalization of the [[Bernoulli process]] to more than two possible outcomes.<ref>P. Shields, ''The theory of Bernoulli shifts'', Univ. Chicago Press (1973)</ref><ref>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). ISBN 0-19-853390-X</ref> Bernoulli schemes are important in the study of [[dynamical system]]s, as most such systems (such as [[Axiom A system]]s) 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.<ref>Pierre Gaspard, ''Chaos, scattering and statistical mechanics''(1998), Cambridge University press</ref> 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]]<ref>{{springer|author=D.S. Ornstein|title=Ornstein isomorphism theorem|id=Ornstein_isomorphism_theorem&oldid=17997}}</ref> shows that Bernoulli shifts are isomorphic when their [[Kolmogorov entropy|entropy]] is equal. Finite [[stationary stochastic process]]es are isomorphic to the Bernoulli shift; in this sense, Bernoulli shifts are [[universal property|universal]].
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| ==Definition==
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| A Bernoulli scheme is a [[discrete-time]] [[stochastic process]] where each [[statistical independence|independent]] [[random variable]] may take on one of ''N'' distinct possible values, with the outcome ''i'' occurring with probability <math>p_i</math>, with ''i'' = 1, ..., ''N'', and
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| :<math>\sum_{i=1}^N p_i = 1. </math>
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| The [[sample space]] is usually denoted as
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| :<math>X=\{1,\ldots,N \}^\mathbb{Z}</math>
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| as a shorthand for
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| :<math>X=\{ x=(\ldots,x_{-1},x_0,x_1,\ldots) :
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| x_k \in \{1,\ldots,N\} \; \forall k \in \mathbb{Z} \}.</math>
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| The associated [[measure (mathematics)|measure]] is called the '''Bernoulli measure'''<ref>Achim Klenke, ''Probability Theory'' (2006) Springer-Verlag, ISBN 978-1-848000-047-6 doi:10.1007/978-1-848000-048-3</ref>
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| :<math>\mu = \{p_1,\ldots,p_N\}^\mathbb{Z}</math>
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| The [[sigma-algebra|σ-algebra]] <math>\mathcal{A}</math> 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
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| :<math>(X,\mathcal{A},\mu)</math>
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| is a [[measure space]]. A basis of <math>\mathcal{A}</math> is the [[cylinder set]]s. Given a cylinder set <math>[x_0, x_1,\ldots,x_n]</math>, its measure is
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| :<math>\mu\left([x_0, x_1,\ldots,x_n]\right)=
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| \prod_{i=0}^n p_{x_i}</math>
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| The equivalent expression, using the notation of probability theory, is
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| :<math>\mu\left([x_0, x_1,\ldots,x_n]\right)=
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| \mathrm{Pr}(X_0=x_0, X_1=x_1, \ldots, X_n=x_n)</math>
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| for the random variables <math>\{X_k\}</math>
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| The Bernoulli scheme, as any stochastic process, may be viewed as a [[dynamical system]] by endowing it with the [[shift operator]] ''T'' where
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| :<math>Tx_k = x_{k+1}.</math>
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| Since the outcomes are independent, the shift preserves the measure, and thus ''T'' is a [[measure-preserving transformation]]. The quadruplet
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| :<math>(X,\mathcal{A},\mu, T)</math>
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| is a [[measure-preserving dynamical system]], and is called a '''Bernoulli scheme''' or a '''Bernoulli shift'''. It is often denoted by | |
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| :<math>BS(p)=BS(p_1,\ldots,p_N).</math>
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| 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 (graph theory)|clique]].
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| ==Generalizations==
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| 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]] <math>(Y,\mathcal{B},\nu)</math>, and define the Bernoulli scheme as
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| :<math>(X, \mathcal{A}, \mu)=(Y,\mathcal{B},\nu)^\mathbb{Z}</math>
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| This works because the countable direct product of a standard probability space is again a standard probability space.
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| As a further generalization, one may replace the in integers <math>\mathbb{Z}</math> by a [[countable]] [[discrete group]] <math>G</math>, so that
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| :<math>(X, \mathcal{A}, \mu)=(Y,\mathcal{B},\nu)^G</math>
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| For this last case, the shift operator is replaced by the [[group action]]
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| :<math>gx(f)=x(g^{-1}f)</math>
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| for group elements <math>f,g\in G</math> and <math>x\in Y^G</math> understood as a function <math>x:G\to Y</math> (any direct product <math>Y^G</math> can be understood to be the set of functions <math>[G\to Y]</math>, as this is the [[exponential object]]). The measure <math>\mu</math> is taken as the [[Haar measure]], which is invariant under the group action:
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| :<math>\mu(gx)=\mu(x). \, </math>
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| These generalizations are also commonly called Bernoulli schemes, as they still share most properties with the finite case.
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| ==Properties==
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| [[Ya. Sinai]] demonstrated that the [[Kolmogorov entropy]] of a Bernoulli scheme is given by<ref>Ya.G. Sinai, (1959) "On the Notion of Entropy of a Dynamical System", ''Doklady of Russian Academy of Sciences'' '''124''', pp. 768–771.</ref><ref>Ya. G. Sinai, (2007) "[http://web.math.princeton.edu/facultypapers/Sinai/MetricEntropy2.pdf Metric Entropy of Dynamical System]"</ref>
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| :<math>H = -\sum_{i=1}^N p_i \log p_i .</math> | |
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| 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 <math>(Y, \mathcal{B}, \nu)</math> (''i.e.'' a base space which is not countable), one typically considers the [[relative entropy]]. So, for example, if one has a countable [[partition of a set|partition]] <math>Y'\subset Y</math> of the base ''Y'', such that <math>\nu(Y')=1</math>, one may define the entropy as
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| :<math>H_{Y'} = -\sum_{y'\in Y'} \nu(y') \log \nu(y') .</math>
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| In general, this entropy will depend on the partition; however, for many [[dynamical system]]s, 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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| The [[Ornstein isomorphism theorem]] states that two Bernoulli schemes with the same entropy are [[isomorphism of dynamical systems|isomorphic]].<ref>Donald Ornstein, "Bernoulli shifts with the same entropy are isomorphic", ''Advances in Math.'' '''4''' (1970), pp.337–352</ref> The result is sharp,<ref>Christopher Hoffman, "[http://www.ams.org/journals/tran/1999-351-10/S0002-9947-99-02446-0/ A K counterexample machine]", ''Trans. Amer. Math. Soc.'' '''351''' (1999), pp 4263–4280</ref> in that very similar, non-scheme systems, such as [[Kolmogorov automorphism]]s, do not have this property.
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| The Ornstein isomorphism theorem is in fact considerably deeper: it provides a simple criterion by which many different [[measure-preserving dynamical system]]s 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{{clarify|date=November 2010}} [[stationary stochastic process]]es, [[subshifts of finite type]], finite [[Markov chain]]s, [[Anosov flow]]s, and [[Sinai's billiards]]: these are all isomorphic to Bernoulli schemes.
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| For the generalized case, the Ornstein isomorpism theorem still holds if the group ''G'' is a countably infinite [[amenable group]].
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| <ref>D. Ornstein and B. Weiss. "Entropy and isomorphism theorems for actions of amenable groups." ''J. Analyse Math.'' '''48''' (1987), pp. 1–141.</ref><ref>Lewis Bowen (2011), "[http://arxiv.org/abs/1103.4424 Every countably infinite group is almost Ornstein]", ArXiv abs/1103.4424</ref>
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| ==Bernoulli automorphism==
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| An invertible, [[measure-preserving transformation]] of a [[standard probability space]] (Lebesgue space) is called a '''Bernoulli automorphism''' if it [[isomorphism of dynamical systems|isomorphic]] to a [[Bernoulli shift]].<ref>Peter Walters (1982) ''An Introduction to Ergodic Theory'', Springer-Verlag, ISBN 0-387-90599-5</ref>
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| ==See also==
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| * [[Shift of finite type]]
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| * [[Markov chain]]
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| * [[Hidden Bernoulli model]]
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Bernoulli Scheme}}
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| [[Category:Markov models]]
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| [[Category:Ergodic theory]]
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| [[Category:Stochastic processes]]
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| [[Category:Symbolic dynamics]]
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