Regular $g$-measures are not always Gibbsian

Fernández, Roberto; Gallo, Sandro; Maillard, G.

doi:10.1214/ecp.v16-1681

Cited by 28 publications

(47 citation statements)

References 15 publications

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“…the past are not necessarily continuous functions of this past. It was shown before that there exist g-measures which are not Gibbs measures [25]; our result answers a question raised in [27] and shows that neither class of measures contains the other one. Although the question had been posed before, it seems to be the case that there were no precise conjectures whether these Dyson Gibbs measures actually were g-measures or not.…”

Section: Introductionsupporting

confidence: 81%

“…However, it is far from obvious if such a description is always easily possible (see e.g. [27,28,25]). In fact, the non-equivalence between one-sided and two-sided conditionings, which we will demonstrate in detail later, serves as a warning to a too easy identification.…”

Section: General Definitions and Main Resultsmentioning

confidence: 99%

“…In fact, this equivalence applies for a large class of interactions which satisfy a strong uniqueness condition [27,28]. However, if we require only continuity of the conditional probabilities, there exist g-measures which are not Gibbs measures [25].…”

Section: Definitionmentioning

confidence: 98%

See 2 more Smart Citations

Entropic Repulsion and Lack of the g-Measure Property for Dyson Models

Bissacot

Endo

Enter

et al. 2018

Commun. Math. Phys.

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Section: Introductionsupporting

confidence: 81%

Section: General Definitions and Main Resultsmentioning

confidence: 99%

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Entropic Repulsion and Lack of the g-Measure Property for Dyson Models

Bissacot

Endo

Enter

et al. 2018

Commun. Math. Phys.

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“…Indeed, slight differences exist between the approaches coming from probability theory Georgii [1988], mathematical statistical mechanics Dobrushin [1968] or dynamical systems/ergodic theory Bowen [2008]. See also Fernandez et al [2011] for a discussion on different notions.…”

Section: Transition Probabilitiesmentioning

confidence: 99%

On the Mathematical Consequences of Binning Spike Trains

Cessac

Löcherbach

2017

Neural Computation

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We initiate a mathematical analysis of hidden effects induced by binning spike trains of neurons. Assuming that the original spike train has been generated by a discrete Markov process, we show that binning generates a stochastic process which is not Markov any more, but is instead a Variable Length Markov Chain (VLMC) with unbounded memory. We also show that the law of the binned raster is a Gibbs measure in the DLR (Dobrushin-Lanford-Ruelle) sense coined in mathematical statistical mechanics. This allows the derivation of several important consequences on statistical properties of binned spike trains. In particular, we introduce the DLR framework as a natural setting to mathematically formalize anticipation, i.e. to tell "how good" our nervous system is at making predictions. In a probabilistic sense, this corresponds to condition a process by its future and we discuss how binning may affect our conclusions on this ability. We finally comment what could be the consequences of binning in the detection of spurious phase transitions or in the detection of wrong evidences of criticality.

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“…They were introduced in the thirties and repeatedly rediscovered (under different names) [16,50,37,45,44]. A few years ago, Fernández, Gallo and Maillard [24] constructed a g-measure -with one-sided continuous conditional probabilities-which is not a Gibbs measure, as its two-sided conditional probabilities are not continuous.…”

Section: Introductionmentioning

confidence: 99%

One-Sided Versus Two-Sided Stochastic Descriptions

Enter

Aernout²

2019

Statistical Mechanics of Classical and Disordered Systems

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It is well-known that discrete-time finite-state Markov Chains, which are described by onesided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearestneighbor Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.

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Regular $g$-measures are not always Gibbsian

Cited by 28 publications

References 15 publications

Entropic Repulsion and Lack of the g-Measure Property for Dyson Models

Entropic Repulsion and Lack of the g-Measure Property for Dyson Models

On the Mathematical Consequences of Binning Spike Trains

One-Sided Versus Two-Sided Stochastic Descriptions

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