scholarly journals Probabilities

2021 ◽  
pp. 158-220
Author(s):  
Nuel Belnap ◽  
Thomas MÜller ◽  
Tomasz Placek
Keyword(s):  
Single Case ◽  
Causal Analysis ◽  
Probability Measures ◽  
Probability Spaces ◽  
Indeterministic Causation

This chapter offers a BST theory of propensities (i.e., of objective single-case probabilities), which builds on the account of indeterministic causation developed in Chapter 6. Propensities are shown to deliver classical (Kolmogorovian) probability spaces. The chapter draws a distinction between propensities and probability measures. The former are assigned to sets of BST transitions, in particular to sets of causae causantes of transitions, and are interpreted as degrees of possibility of these transitions. The latter are defined in terms of propensities and are measures of Komogorovian probability spaces. Features of propensities are derived from a logico-causal analysis. Finally, the chapter discusses how the theory developed here handles well-known objections to propensities due to Humphreys and to Salmon, especially Humphreys’s paradox.

Statistical maps and generalized random walks

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Roman Frič ◽  
Martin Papčo
Keyword(s):  
Probability Measure ◽  
Random Walks ◽  
Measurable Space ◽  
Probability Measures ◽  
Statistical Map ◽  
Probability Spaces ◽  
Special Matrix ◽  
Fuzzy Probability Theory ◽  
Fuzzy Random ◽  
Joint Probabilities

AbstractWe continue our study of statistical maps (equivalently, fuzzy random variables in the sense of Gudder and Bugajski). In the realm of fuzzy probability theory, statistical maps describe the transportation of probability measures on one measurable space into probability measures on another measurable space. We show that for discrete probability spaces each statistical map can be represented via a special matrix the rows of which are probability functions related to conditional probabilities and the columns are related to fuzzy n-partitions of the domain. Discrete statistical maps sending a probability measure p to a probability measure q can be represented via conditional distributions and correspond to joint probabilities on the product. The composition of statistical maps provide a tool to describe and to study generalized random walks and Markov chains.

scholarly journals Relative and Discrete Utility Maximising Entropy

2008 ◽  
Vol 15 (04) ◽  
pp. 303-327
Author(s):  
Grzegorz Harańczyk ◽  
Wojciech Słomczyński ◽  
Tomasz Zastawniak
Keyword(s):  
Probability Measure ◽  
Utility Function ◽  
Expected Utility ◽  
Mathematical Finance ◽  
Second Law ◽  
Probability Measures ◽  
Discrete Probability ◽  
Terminal Wealth ◽  
Probability Spaces ◽  
Rényi Relative Entropy

The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied in [37], is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based on the idea, borrowed from mathematical finance, of maximising the expected utility of the terminal wealth of an investor. Moreover, u-entropy is also relevant in thermodynamics, as it can replace the standard Boltzmann-Shannon entropy in the Second Law. If the utility function is logarithmic or isoelastic (a power function), then the well-known notions of Boltzmann-Shannon and Rényi relative entropy are recovered. We establish the principal properties of relative and discrete u-entropy and discuss the links with several related approaches in the literature.

scholarly journals A Unifying Framework for Causal Analysis in Set-Theoretic Multimethod Research

2016 ◽  
Vol 47 (1) ◽  
pp. 37-63 ◽  
Author(s):  
Ingo Rohlfing ◽  
Carsten Q. Schneider
Keyword(s):  
Comparative Analysis ◽  
Causal Inference ◽  
Empirical Research ◽  
Set Theory ◽  
Single Case ◽  
Qualitative Comparative Analysis ◽  
Causal Analysis ◽  
Process Tracing ◽  
Case Method ◽  
Multimethod Research

The combination of Qualitative Comparative Analysis (QCA) with process tracing, which we call set-theoretic multimethod research (MMR), is steadily becoming more popular in empirical research. Despite the fact that both methods have an elected affinity based on set theory, it is not obvious how a within-case method operating in a single case and a cross-case method operating on a population of cases are compatible and can be combined in empirical research. There is a need to reflect on whether and how set-theoretic MMR is internally coherent and how QCA and process tracing can be integrated in causal analysis. We develop a unifying foundation for causal analysis in set-theoretic MMR that highlights the roles and interplay of QCA and process tracing. We argue that causal inference via counterfactuals on the level of single cases integrates QCA and process tracing and assigns proper and equally valuable roles to both methods.

Probability Spaces

2021 ◽  
pp. 147-153
Author(s):  
James Davidson
Keyword(s):  
Conditional Probability ◽  
General Context ◽  
Probability Measures ◽  
Product Spaces ◽  
Probability Spaces ◽  
Product Measures

This chapter defines probability measures and probability spaces in a general context, as a case of the concepts introduced in Chapter 3. The axioms of probability are explained, and the important concepts of conditional probability and independence are introduced and linked to the role of product spaces and product measures.

On disjointness of dynamical systems

1979 ◽  
Vol 85 (3) ◽  
pp. 477-491 ◽  
Author(s):  
J. Auslander ◽  
Y. N. Dowker
Keyword(s):  
Dynamical System ◽  
Metric Spaces ◽  
Probability Measures ◽  
Compact Metric Spaces ◽  
Probability Spaces ◽  
Dynamical Properties ◽  
Borel Probability ◽  
The Given ◽  
Special Case ◽  
Measure Preserving Transformations

By a dynamical system we mean one of several related objects: measure preserving transformations on probability spaces (processes), self homeomorphisms of compact metric spaces (compact systems), or a combination of these, namely compact systems provided with invariant Borel probability measures. It is the latter, which we call compact processes, which will be of most interest in this paper. In particular, we will study the dynamical properties of the product of two processes with respect to compatible measures – those measures which project to the given measures on the component spaces. This leads to the notion of disjointness of two processes – the only compatible measure is the product measure. As an application we obtain a theorem, a special case of which gives rise to a class of transformations which preserve normal sequences. Finally, we study a topological analog (topological disjointness) and briefly consider the relation between the two notions of disjointness.

scholarly journals Reconciliation of probability measures

2018 ◽  
Vol 30 (6) ◽  
pp. 1300-1310
Author(s):  
JEAN BÉRARD ◽  
NICOLAS JUILLET
Keyword(s):  
Probability Measure ◽  
Recent Work ◽  
Sample Space ◽  
Probability Measures ◽  
Probability Spaces

We discuss the reconciliation problem between probability measures: given n⩾2 probability spaces $(\Omega,{\mathcal{F}}_1,{\mathbb{P}}_1),\ldots,(\Omega,{\mathcal{F}}_n,{\mathbb{P}}_n)$ with a common sample space, does there exist an overall probability measure ${\mathbb{P}} \ \text{on} \ {\mathcal{F}} = \sigma({\mathcal{F}}_1,\ldots,{\mathcal{F}}_n)$ such that, for all i, the restriction of ${\mathbb{P}} \ \text{to} \ {\mathcal{F}}_i$ coincides with ${\mathbb{P}}_i$? General criteria for the existence of a reconciliation are stated, along with some counterexamples that highlight some delicate issues. Connections to earlier (recent and far less recent) work are discussed, and elementary self-contained proofs for the various results are given.

Moving Forward Four Words at a Time: Effects of a Supplemental Preschool Vocabulary Intervention

2020 ◽  
Vol 51 (1) ◽  
pp. 165-175 ◽  
Author(s):  
Lindsey A. Peters-Sanders ◽  
Elizabeth S. Kelley ◽  
Christa Haring Biel ◽  
Keri Madsen ◽  
Xigrid Soto ◽  
...  
Keyword(s):  
Group Intervention ◽  
Single Case ◽  
Single Case Design ◽  
Vocabulary Intervention ◽  
Iterative Development ◽  
At Risk Children ◽  
Before And After ◽  
Child Friendly ◽  
Interactive Lessons ◽  
Case Design

Purpose This study evaluated the effects of an automated, small-group intervention designed to teach preschoolers challenging vocabulary words. Previous studies have provided evidence of efficacy. In this study, we evaluated the effects of the program after doubling the number of words taught from 2 to 4 words per book. Method Seventeen preschool children listened to 1 prerecorded book per week for 9 weeks. Each storybook had embedded, interactive lessons for 4 target vocabulary words. Each lesson provided repeated exposures to words and their definitions, child-friendly contexts, and multiple opportunities for children to respond verbally to instructional prompts. Participants were asked to define the weekly targeted vocabulary before and after intervention. A repeated acquisition single-case design was used to examine the effects of the books and embedded lessons on learning of target vocabulary words. Results Treatment effects were observed for all children across many of the books. Learning of at least 2 points (i.e., 1 word) was replicated for 74.5% of 149 books tested across the 17 participants. On average, children learned to define 47% of the target vocabulary words (17 out of 36). Conclusions Results support including 4 challenging words per book, as children learned substantially more words when 4 words were taught, in comparison to previous studies. Within an iterative development process, results of the current study take us 1 step closer to creating an optimal vocabulary intervention that supports the language development of at-risk children.

A Pilot Study of Early Storybook Reading With Babies With Hearing Loss

2019 ◽  
Vol 62 (9) ◽  
pp. 3397-3412
Author(s):  
Michelle I. Brown ◽  
David Trembath ◽  
Marleen F. Westerveld ◽  
Gail T. Gillon
Keyword(s):  
Hearing Loss ◽  
Pilot Study ◽  
Visual Analysis ◽  
Single Case ◽  
Eye Contact ◽  
Preliminary Evidence ◽  
Storybook Reading ◽  
Book Selection ◽  
Turn Taking ◽  
Parent Child

Purpose This pilot study explored the effectiveness of an early storybook reading (ESR) intervention for parents with babies with hearing loss (HL) for improving (a) parents' book selection skills, (b) parent–child eye contact, and (c) parent–child turn-taking. Advancing research into ESR, this study examined whether the benefits from an ESR intervention reported for babies without HL were also observed in babies with HL. Method Four mother–baby dyads participated in a multiple baseline single-case experimental design across behaviors. Treatment effects for parents' book selection skills, parent–child eye contact, and parent–child turn-taking were examined using visual analysis and Tau-U analysis. Results Statistically significant increases, with large to very large effect sizes, were observed for all 4 participants for parent–child eye contact and parent–child turn-taking. Limited improvements with ceiling effects were observed for parents' book selection skills. Conclusion The findings provide preliminary evidence for the effectiveness of an ESR intervention for babies with HL for promoting parent–child interactions through eye contact and turn-taking.

Successful Implicit Vocabulary Intervention for Three Cantonese-Speaking Toddlers: A Replicated Single-Case Design

2020 ◽  
Vol 63 (12) ◽  
pp. 4148-4161
Author(s):  
Christine S.-Y. Ng ◽  
Stephanie F. Stokes ◽  
Mary Alt
Keyword(s):  
Single Case ◽  
Cross Cultural ◽  
Neighborhood Density ◽  
Expressive Vocabulary ◽  
Single Case Design ◽  
Vocabulary Intervention ◽  
Phonological Neighborhood Density ◽  
Phonological Neighborhood ◽  
And Control ◽  
Case Design

Purpose We report on a replicated single-case design study that measured the feasibility of an expressive vocabulary intervention for three Cantonese-speaking toddlers with small expressive lexicons relative to their age. The aim was to assess the cross-cultural and cross-linguistic feasibility of an intervention method developed for English-speaking children. Method A nonconcurrent multiple-baseline design was used with four baseline data points and 16 intervention sessions per participant. The intervention design incorporated implicit learning principles, high treatment dosage, and control of the phonological neighborhood density of the stimuli. The children (24–39 months) attended 7–9 weeks of twice weekly input-based treatment in which no explicit verbal production was required from the child. Each target word was provided as input a minimum of 64 times in at least two intervention sessions. Treatment feasibility was measured by comparison of how many of the target and control words the child produced across the intervention period, and parent-reported expressive vocabulary checklists were completed for comparison of pre- and postintervention child spoken vocabulary size. An omnibus effect size for the treatment effect of the number of target and control words produced across time was calculated using Kendall's Tau. Results There was a significant treatment effect for target words learned in intervention relative to baselines, and all children produced significantly more target than control words across the intervention period. The effect of phonological neighborhood density on expressive word production could not be evaluated because two of the three children learned all target words. Conclusion The results provide cross-cultural evidence of the feasibility of a model of intervention that incorporated a high-dosage, cross-situational statistical learning paradigm to teach spoken word production to children with small expressive lexicons.

Bang for Your Buck: A Single-Case Experimental Design Study of Practice Amount and Distribution in Treatment for Childhood Apraxia of Speech

2019 ◽  
Vol 62 (9) ◽  
pp. 3160-3182 ◽  
Author(s):  
Edwin Maas ◽  
Christina Gildersleeve-Neumann ◽  
Kathy Jakielski ◽  
Nicolette Kovacs ◽  
Ruth Stoeckel ◽  
...  
Keyword(s):  
Single Case ◽  
Evidence Base ◽  
Distributed Practice ◽  
Single Subject ◽  
Single Subject Design ◽  
Apraxia Of Speech ◽  
Childhood Apraxia Of Speech ◽  
Opposite Pattern ◽  
Massed Practice ◽  
Treatment Conditions

Purpose The aim of this study was to examine 2 aspects of treatment intensity in treatment for childhood apraxia of speech (CAS): practice amount and practice distribution. Method Using an alternating-treatments single-subject design with multiple baselines, we compared high versus low amount of practice, and massed versus distributed practice, in 6 children with CAS. Conditions were manipulated in the context of integral stimulation treatment. Changes in perceptual accuracy, scored by blinded analysts, were quantified with effect sizes. Results Four children showed an advantage for high amount of practice, 1 showed an opposite effect, and 1 showed no condition difference. For distribution, 4 children showed a clear advantage for massed over distributed practice post treatment; 1 showed an opposite pattern, and 1 showed no clear difference. Follow-up revealed a similar pattern. All children demonstrated treatment effects (larger gains for treated than untreated items). Conclusions High practice amount and massed practice were associated with more robust speech motor learning in most children with CAS, compared to low amount and distributed practice, respectively. Variation in effects across children warrants further research to determine factors that predict optimal treatment conditions. Finally, this study adds to the evidence base supporting the efficacy of integral stimulation treatment for CAS. Supplemental Material https://doi.org/10.23641/asha.9630599

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