École d'Été de Probabilités de Saint-Flour XIII — 1983

Aldous, David; Ибрагимов, И. А.; Jacod, Jean; Saint-Flour, École de probabilités de d'été; Hennequin, Paul Louis

doi:10.1007/bfb0099420

Cited by 436 publications

(649 citation statements)

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“…thereby make it possible to predict the properties of future observations. A statistically optimal cluster-learning model, which has been increasingly popular in cognitive science, is the Chinese restaurant process (CRP) mixture model (15). The intuition behind the CRP mixture model is that discovering latent clusters underlying stimuli bears similarity to the task of seating customers at a restaurant.…”

Section: Significancementioning

confidence: 99%

Learning bundles of stimuli renders stimulus order as a cue, not a confound

Qian

Aslin

2014

Proc. Natl. Acad. Sci. U.S.A.

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The order in which stimuli are presented in an experiment has long been recognized to influence behavior. Previous accounts have often attributed the effect of stimulus order to the mechanisms with which people process information. We propose that stimulus order influences cognition because it is an important cue for learning the underlying structure of a task environment. In particular, stimulus order can be used to infer a "stimulus bundle"-a sequence of consecutive stimuli that share the same underlying latent cluster. We describe a clustering model that successfully explains the perception of streak shooting in basketball games, along with two other cognitive phenomena, as the outcome of finding the statistically optimal bundle representation. We argue that the perspective of viewing stimulus order as a cue may hold the key to explaining behaviors that seemingly deviate from normative theories of cognition and that in task domains where the assumption of stimulus bundles is intuitively appropriate, it can improve the explanatory power of existing models.H uman cognition typically takes place in an environment where information becomes available over time. The sequential nature of information availability implies that stimulus order-the order in which a sequence of stimuli is presentedmay strongly influence the interpretation of a task environment. Consider the weather patterns in Fig. 1 as an example. If we were to record the dominant weather phenomenon in each month of a year, order 1, as shown in Fig. 1, would likely represent a typical city in upstate New York, such as Rochester. Given this sequence of stimuli, it is relatively easy for an observer to infer the existence of "seasons" underlying the weather pattern-that is, winter seems to be in effect from November to April, followed by spring/summer, and then by fall. However, if we reorder this set of stimuli from order 1 to order 2 as shown in Fig. 1, the inference of seasons becomes much less apparent. Even when the exact same set of stimuli is presented in both sequences, the difference in stimulus order has led to a dramatically different interpretation about its underlying structure. Why does stimulus order affect human cognition in this interesting way?The effect of stimulus order on cognition has often been viewed as a consequence of the way that the human mind processes information (1-8). For example, Elliott and Anderson (5) accounted for the effect of stimulus order on category learning by incorporating the mechanisms of memory decay into a categorization model. Memory decay intuitively captures the assumption that the human mind tends to be influenced by nearby stimuli more than by distant ones. It has also been hypothesized that when a sequence of stimuli is ordered in such a way that the similarity between successive stimuli is maximized, it offers a processing benefit because the stimuli being kept in working memory will be similar to each other and thus allow for easier abstraction of concepts (4). In addition, the effects of stimulus ord...

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Section: Significancementioning

confidence: 99%

Learning bundles of stimuli renders stimulus order as a cue, not a confound

Qian

Aslin

2014

Proc. Natl. Acad. Sci. U.S.A.

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“…So we identify V with this set of Markovian laws. By some well known general theory [1,5,6,11,15]), each extreme law in V can be represented as a weak limit of the conditional laws for (K n ) given K ν = κ ν for ν → ∞ and some sequence (κ ν , ν = 1, 2, . .…”

Section: The Boundary Problemmentioning

confidence: 99%

“…is regular if and only if there is a limit κ ν /ν → s for some s ∈ [0, 1], which corresponds to a boundary element V (s) with V n,k (s) = s k (1−s) n−k , hence the boundary is homeomorphic to [0,1]. This is de Finetti's representation of infinite exchangeable sequences of zeros and ones.…”

Section: The Boundary Problemmentioning

confidence: 99%

Exchangeable Gibbs partitions and Stirling triangles

2006

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For two collections of nonnegative and suitably normalised weights W = (W j ) and V = (V n,k ), a probability distribution on the set of partitions of the set {1, . . . , n} is defined by assigning to a generic partition {A j , j ≤ k} the probability V n,k W |A1| · · · W |A k | , where |A j | is the number of elements of A j . We impose constraints on the weights by assuming that the resulting random partitions Π n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [−∞, 1]. The case α = 1 is trivial, and for each value of α = 1 the set of possible V -weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalised Stirling triangle. In particular, we show that the boundary is discrete for −∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α, θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α, θ)-partition on the asymptotics of the number of blocks of Π n as n tends to infinity.

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“…y ∈ Y . The reverse martingale theorem and Lemma 2.4 imply that (2) ) is the equivalence relation on Σ ). Proof.…”

Section: The Main Examplesmentioning

confidence: 97%

“…It took some time to sort out the relationships among the various tail fields [16,21,22,32,36,37,39,38], to understand the representation of measures as mixtures in terms of the theory of Choquet simplices and ergodic decompositions [8,7,9,42,44,56], and to investigate exchangeability in more general contexts [8,7,13,20,30,42,51]. For a survey of exchangeability, see [2]. Diaconis and Freedman [7] gave a necessary and sufficient condition for a measure to be a mixture of Markov measures, in terms of 'partial exchangeability'-invariance under the subgroup of Π that preserves transition counts (as well as symbol counts).…”

Section: Introductionmentioning

confidence: 99%

Symmetric Gibbs measures

Petersen

Schmidt

1997

Trans. Amer. Math. Soc.

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Abstract. We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, 'canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

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École d'Été de Probabilités de Saint-Flour XIII — 1983

Cited by 436 publications

References 0 publications

Learning bundles of stimuli renders stimulus order as a cue, not a confound

Learning bundles of stimuli renders stimulus order as a cue, not a confound

Exchangeable Gibbs partitions and Stirling triangles

Symmetric Gibbs measures

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