Degeneracy in Candecomp/Parafac and Indscal Explained For Several Three-Sliced Arrays With A Two-Valued Typical Rank

The Candecomp/Parafac (CP) method decomposes a three-way array into a prespecified number R of rank-1 arrays, by minimizing the sum of squares of the residual array. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate sequences of solutions, in which several rank-1 arrays become highly correlated in all three modes and some elements of the rank-1 arrays become arbitrarily large. We consider the real-valued CP decomposition of all known three-sliced arrays, i.e., of size p×q×3, with a two-valued typical rank. These are the 5×3×3 and 8×4×3 arrays, and the 3×3×4 and 3×3×5 arrays with symmetric 3×3 slices. In the latter two cases, CP is equivalent to the Indscal model. For a typical rank of {m,m+1}, we consider the CP decomposition with R=m of an array of rank m+1. We show that (in most cases) the CP objective function does not have a minimum but an infimum. Moreover, any sequence of feasible CP solutions in which the objective value approaches the infimum will become degenerate. We use the tools developed in Stegeman (2006), who considers p×p×2 arrays, and present a framework of analysis which is of use to the future study of CP degeneracy related to a two-valued typical rank. Moreover, our examples show that CP uniqueness is not necessary for degenerate solutions to occur. The author is supported by the Dutch Organisation for Scientific Research (NWO), VENI grant 451-04-102.     

Sense and Simplicity: Wittgenstein's Argument for Simple Objects

This paper puts forward an alternative interpretation of the argument for simple objects advanced in the 2.0s of the Tractatus. In my view, Wittgenstein derives the simplicity of objects directly from his account of possible states, complex objects and senseful propositions. The key to Wittgenstein's argument is the idea that, if there were no simple objects, possible states would not be necessarily possible. If this were the case, however, there would be no senseful language, in Wittgenstein's view. One of the subsidiary aims of this paper is to question the idea that Wittgenstein posits simples because, without them, language would be infinitely analysable. ¹ 1 I am grateful to Malcolm Budd, Jane Heal and Jim Hopkins for their help in the development of this paper.     

Divine Simplicity and the Holy Trinity

The doctrine of divine simplicity is largely ignored in modern continental theology and has been criticized by some analytic theists. However, it plays a central role in patristic and medieval trinitarian thought, and is a doctrinal affirmation of the Catholic Church. This article seeks to illustrate the significance of the teaching first by examining the contrasting modern trinitarian theologies of Karl Barth and Richard Swinburne, noting how each suffers from a deficit of reference to the doctrine of divine simplicity. The article then presents four aspects of Aquinas’ teaching on divine simplicity. From this a consideration of trinitarian persons ensues that illustrates why the distinction of persons in God can best be understood by making use of Aquinas’ theology of ‘subsistent relations’, while the unified nature of God can best be understood in terms of ‘personal modes of subsistence’. Based on this analysis, the contrasting insights of both Barth and Swinburne can be fully retained, without the contrasting inherent problems that the theology of each presents.     

Simplicity and probability in causal explanation

What makes some explanations better than others? This paper explores the roles of simplicity and probability in evaluating competing causal explanations. Four experiments investigate the hypothesis that simpler explanations are judged both better and more likely to be true. In all experiments, simplicity is quantified as the number of causes invoked in an explanation, with fewer causes corresponding to a simpler explanation. Experiment 1 confirms that all else being equal, both simpler and more probable explanations are preferred. Experiments 2 and 3 examine how explanations are evaluated when simplicity and probability compete. The data suggest that simpler explanations are assigned a higher prior probability, with the consequence that disproportionate probabilistic evidence is required before a complex explanation will be favored over a simpler alternative. Moreover, committing to a simple but unlikely explanation can lead to systematic overestimation of the prevalence of the cause invoked in the simple explanation. Finally, Experiment 4 finds that the preference for simpler explanations can be overcome when probability information unambiguously supports a complex explanation over a simpler alternative. Collectively, these findings suggest that simplicity is used as a basis for evaluating explanations and for assigning prior probabilities when unambiguous probability information is absent. More broadly, evaluating explanations may operate as a mechanism for generating estimates of subjective probability.     

Persons,Simplicity, and Substance

A novel argument has recently been advanced against materialism—the view that human persons are identical to composite, material objects. The argument claims that pairs of people are not conscious and that the only viable explanation for why they are not is because pairs of people are not simple. The argument concludes that only a simple thing can be the subject of conscious states. In this paper, I offer an alternative explanation for why pairs of people are not conscious: pairs of people are not substances. I provide two characterizations of substantiality. The first proposal claims that substances have irreducible causal powers, and the second claims that substances cannot have other substances as proper parts. The alternative explanation based on these characterizations of substantiality shows that being conscious is compatible with materialism.     

Simplicity,personhood, and divinity

International Journal for Philosophy of Religion -     

Properties of and Algorithms for Fitting Three-Way Component Models with Offset Terms

Prior to a three-way component analysis of a three-way data set, it is customary to preprocess the data by centering and/or rescaling them. Harshman and Lundy (1984) considered that three-way data actually consist of a three-way model part, which in fact pertains to ratio scale measurements, as well as additive “offset” terms that turn the ratio scale measurements into interval scale measurements. They mentioned that such offset terms might be estimated by incorporating additional components in the model, but discarded this idea in favor of an approach to remove such terms from the model by means of centering. Then estimates for the three-way component model parameters are obtained by analyzing the centered data. In the present paper, the possibility of actually estimating the offset terms is taken up again. First, it is mentioned in which cases such offset terms can be estimated uniquely. Next, procedures are offered for estimating model parameters and offset parameters simultaneously, as well as successively (i.e., providing offset term estimates after the three-way model parameters have been estimated in the traditional way on the basis of the centered data). These procedures are provided for both the CANDECOMP/PARAFAC model and the Tucker3 model extended with offset terms. The successive and the simultaneous approaches for estimating model and offset parameters have been compared on the basis of simulated data. It was found that both procedures perform well when the fitted model captures at least all offset terms actually underlying the data. The simultaneous procedures performed slightly better than the successive procedures. If fewer offset terms are fitted than there are underlying the model, the results are considerably poorer, but in these cases the successive procedures performed better than the simultaneous ones. All in all, it can be concluded that the traditional approach for estimating model parameters can hardly be improved upon, and that offset terms can sufficiently well be estimated by the proposed successive approach, which is a simple extension of the traditional approach. The author is obliged to Jos M.F. ten Berge and Marieke Timmerman for helpful comments on an earlier version of this paper. The author is obliged to Iven van Mechelen for making available the data set used in Section 6.     

Extracting the Variance Inflation Factor and Other Multicollinearity Diagnostics from Typical Regression Results

Multicollinearity is a potential problem in all regression analyses. However, the examination of multicollinearity is rarely reported in primary studies. In this article we discuss and show several post hoc methods for assessing multicollinearity. One such multicollinearity diagnostic is the variance inflation factor. We outline the post hoc variance inflation factor method, which computes the variance inflation factor from the standardized regression coefficient and semi-partial correlation, both of which can be calculated from commonly reported regression results. Three examples of computing multicollinearity diagnostics using data from published studies are shown. We conclude with a discussion and practical implications.     

Cartesian Modes and The Simplicity of Mind

Malebranche argues that we lack a clear idea of the mind because we cannot, even in principle, derive all the possible modes of mind solely from the idea of thought. But we can, in principle, derive all the possible modes of body from the idea of extension. Therefore, there is epistemic asymmetry between our ideas of mind and body. I offer a defense of Descartes whereby he can assert that we have a clear idea of mind despite this asymmetry. I argue that he can do this by distinguishing the simplicity of mind from the divisibility of body.     

Simplicity and Language: Bonaventure's Semiotic Asymptoticism

Luy engages in a close reading of Bonaventure's doctrine of divine simplicity. He offers this reading in light of a keen awareness of contemporary critiques of the doctrine, especially from philosophers of religion who suggest that divine simplicity either means that our human words really cannot say anything intelligible about God, or that divine properties that are surely distinct (such as justice and goodness) are in fact absolutely identical. In sum, Luy recognizes that the doctrine of simplicity challenges the intelligibility of religious language. He points out that medieval thinkers, too, recognized this challenge, but they regarded it as a salutary reminder that God is ultimately incomprehensible to finite minds, even though we can speak true things about God. In expositing divine simplicity according to Bonaventure, Luy shows that Bonaventure expects that creation itself is designed to reveal God's limitless self‐communication. Divine simplicity, then, serves to affirm divine perfection, in a manner limited by the effort of finite words to express the infinite; but divine simplicity also reflects the “semiotic universe” that allows for, and exalts in, the wondrous expression of the divine plenitude.     

Karl Barth and the Purification of Divine Simplicity

Johnson investigates Karl Barth's critical appropriation of the doctrine of divine simplicity. While Barth is critical of traditional formulations of the doctrine, he understands himself to be refining the doctrine rather than rejecting it. Barth notes that Scripture attributes a diverse set of perfections to God in describing his salvific actions. These diverse perfections, however, have a fundamental unity: God does not contradict himself, but rather his perfections describe his unified, trustworthy agency. For this reason, we can know that in God's inmost being, God is not self‐contradictory but utterly unified or simple in his self‐fidelity. Johnson points out that a key element of Barth's doctrine of God is that it can never be the mere deduction of an abstract, transcendent entity; rather, it must begin with the transcendent God's relationship to creation, and therefore must begin with Jesus Christ, who reveals the true being of God. Johnson identifies three guidelines for speaking of Barth's doctrine: each one of God's perfections must be seen as perfections of his one divine being; God's one being does not exist above and behind his revealed perfections; and God's revealed perfections are essential to his divine nature. On this basis, Johnson explores what Barth has to say about the relationship between God's freedom and his self‐fidelity, including as this regards his freedom to live his one eternal life for us.     

Exploring the Accuracy and Simplicity of Idiographic Feeling-Performance Models

This study compared 3 types of idiosyncratic feeling-performance profiles: recall profiles, probabilistic-pIZOF profiles, and simplified “trigger” profiles. Ten tennis players created individualized (“initial recall”) profiles before monitoring their feelings and performances during matches (to generate “pIZOF profiles”). Participants identified 4 triggers items from their 12- or 13-item profiles that they felt were critical to their performance. Probabilistic-pIZOF profiles predicted performance correctly 64% of the time. Trigger profiles were 66% accurate (the top 3 were 70% accurate). Initial recall profiles were accurate 40% of the time. The study provides practical information about the utility of simplified profiles, and information about the accuracy of feeling-performance profile predictions. The results are also discussed within the framework of third-wave behavioral approaches such as mindfulness-acceptance-commitment therapies.     

Hume's Reflections on the Identity and Simplicity of Mind1

The article presents a new interpretation of Hume's treatment of personal identity, and his later rejection of it in the “Appendix” to the Treatise. Hume's project, on this interpretation, is to explain beliefs about persons that arise primarily within philosophical projects, not in everyday life. the belief in the identity and simplicity of the mind as a bundle of perceptions is an abstruse belief, not one held by the “vulgar” who rarely turn their minds on themselves so as to think of their perceptions. the author suggests that it is this philosophical observation of the mind that creates the problems that Hume finally acknowledges in the “Appendix.” He is unable to explain why we believe that the perceptions by means of which we observe our minds while philosophizing are themselves part of our minds. This suggestion is then tested against seven criteria that any interpretation of the “Appendix” must meet.     

Otto Rank: Pioneering Ideas for Social Work Theory and Practice

This article examines Otto Rank's psychology and its profound implications for social work knowledge-building and practice. The paper begins with a brief biographical portrait, highlighting the significance of Rank's relationship and eventual break with Freud, and contextualizing the ideas that became the basis for the Functional Approach. We discuss Rank's conception of human functioning in the social environment, his influence on later theories, and a contemporary critique of Rank's ideas. The paper concludes with a discussion of Rank's contributions to the Functional Approach and to social work practice in general.