1,721,087 research outputs found
A Berry–Esseen theorem for Pitman’s -diversity
Full text linkThis paper contributes to the study of the random number K_n of blocks in the random partition of {1,...,n} induced by random sampling from
the celebrated two-parameter Poisson–Dirichlet process. For any α ∈ (0, 1) and θ > −α, Pitman (Combinatorial Stochastic Processes (2006) Springer, Berlin) showed that (K_n)/(n^α) → S as n → +∞, where the limiting random variable, referred to as Pitman’s α-diversity, is distributed according to a polynomially scaled Mittag–Leffler distribution function. Our main result is a Berry–Esseen theorem for Pitman’s α-diversity S. The proof relies on three intermediate novel results which are of independent interest: (i) a probabilistic representation of the distribution of K_n in terms of a compound distribution; (ii) a quantitative version of the Laplace’s approximation method for integrals; (iii) a refined quantitative bound for Poisson approximation. An application of our Berry–Esseen theorem is presented in the context of Bayesian nonparametric inference for species sampling problems, quantifying the error of a posterior approximation that has been extensively applied to infer the number of unseen species in a population
Salvatore Pugliatti e il suo tempo: per un diritto tra sistema e storia
Full text linkThe dissertation entitled “Salvatore Pugliatti and his own time: for a Law between system and History” examines carefully, in a reconstructive and declarative way, the legal and methodological thought of Salvatore Pugliatti. He was a Sicilian jurist, who died in 1976. His ideas grew and developed in the decades around the half of the last century during which he founded the School of Civil Law at the University of Messina.
In order to elucidate his thought, this dissertation faces the study either of Pugliatti’s works directly addressed to the analysis of theoretic and methodological matters, and of those, among his works of positive law, in which it is possible to evict, through his exposition, the fulfilment of his speculative goal somewhere else expressed.
For this reason, this work is divided into three chapters.
The first one has the aim to frame the scientific figure of Salvatore Pugliatti in his own times, between 1927 and the first half of the seventies, during which he operated, so it reconstructs either the historical elements and the juridical-philosophical and methodological ones which influenced Pugliatti’s thought.
From the historical point of view many elements emerge and are considered: the ascent and the fall of Fascist regime, the ruin of the Second World War, the slow rebuilding of the Italian institutional order consequent upon it.
From a juridical point of view the most important elements which are examined are: the special legislation of Mussolini’s regime, the fascist re-codification, the Civil Code that came into force in 1942 and, above all, the Constitution, which came forth in 1948, whose formal importance and substantial content obliged the Italian civil law doctrine, and so Pugliatti either, to redefine at least a part of the dogmatic certitudes achieved by juridical science till that moment.
Moreover, concerning the philosophical and methodological context, this study inspects the main cultural trends which could have affected Pugliatti’s thought.
Firstly, the direction of the juridical-formalistic Positivism which was dogmatic and post-pandettistic: it was inclined to reduce the Philosophy of Law within the borders of juridical science structured in conformity with experimental science. Secondly, the direction graviting in the orbit of Italian juridical Neo-Idealism of Gentile’s actualism and Croce’s historicism, which was inclined to exclude the chance to provide the scientific characteristic both to the Philosophy of Law and to the Juridical Science. Thirdly the direction of the Juridical Empiric Historical Idealism post- Croce which was inclined to make up, starting from Croce’s and Gentile’s idealism, the historical value of juridical character as the point of departure of any form of reflection about Law. In particular Angelo Ermanno Cammarata and Widar Cesarini Sforza embodied this idea and they influenced Pugliatti’s thought.
Then the second chapter faces, in light of this context, the real nucleus of Pugliatti’s theoretic reflection, following analytically, through a careful exam of Pugliatti’s contributions dedicated in particular to methodological topics, the lines, even chronological, of his progressive evolution.
First of all, this study analyses the first juvenile period of Pugliatti’s scientific production: it appears, considering his theoretic-juridical convincement, as strongly hinged on a methodological paradigm of a strict systematic rigour based on a Positivistic logical juridical nature and on an inevitable dualism between the formal abstractness of the conceptual system created by the jurist and the concreteness of historical interests that originates it.
The second, more mature period of Pugliatti’s speculative iter, in the Forties, allows to single out an evolution in his methodological creed: he gathered solicitations coming in particular from his active participation to the well known polemic “about the nature of juridical concepts” and to the debate about the crisis of Law and Juridical Science, but also from his clear interest for Linguistics and from the reading and knowing of eminent figures like Santi Romano and Capograssi. In these years Pugliatti arrived to consolidate his own convincement in a new idea of Jurisprudence, that is qualified as a practical science: in fact it is undoubtedly bond to the real data and functional to the construction of a perfect, abstract, logical and conceptual juridical system, nevertheless it has also to root even in the concrete interest of consociate, in the magmatic flowing of History and in the changeable becoming of rules, so that it remains science, but qualified as a practical one, and it finds in itself the chance of a dialectic composition of the dualism form-substance, abstract-concrete, that, in other way, would remain inerasable.
Finally the analysis of the third period of Pugliatti’s thought underlines a gradual perfecting of his conviction previously achieved through a deepening of the problem related to the connection between continuous and discontinuous in Law. The dialectic synthesis of the dualism between form and substance, realised through the conception of Jurisprudence as a practical science, is crystallized in the idea- conclusive in Pugliatti’s methodological course- that the complexity of juridical experience, in which coexist either the juridical synchrony of abstract discontinuous of the Law system or the historical diachrony of mobile continuous of the juridical order tout court (which couldn’t resolve in rule because it is other to them). His final idea is realised in the formula that qualifies Law as a “ synthesis of manifold in one”.
At last the third chapter analyses some of the main works of Pugliatti related to some topic of Positive Law in which the author uses his own methodological tool when he faced technical problems of pure civil law. The analysis passes from the representation, whose importance is emphasized by Pugliatti, in the description of the institutes, the concrete and historical relationship between dominus and representer, to the phenomenon of coercive transfer; from the every important studies about properties (where the owner is considered plural, no more singular) to the brief exposition of the dialectic connection existing, according to Pugliatti, between the public and private Law.
From this analysis, as a synthesis of the figure of Pugliatti, emerges an image of a complete jurist, gifted with a great culture and a great juridical background, constantly researching for a clarification of methodological criteria of his own action; moreover, even though his thought never manages to emancipate completely from the persistence of the dualism between abstract and concrete, between form and History, which juridical science seems condemned to, refalling into a juridical Positivism only partly, critically and consciously subdue, Pugliatti’s thought anyway leaves, as a methodological inheritance for the jurist of today, a deep and bright breath of his own solid complexity, based on the intuition of the constant necessity of exegesis and of multidisciplinary, of the strong need of the reference to a positive data and of the fundamental necessity to calibrate any philosophical, methodological or legal reflection, on and with Law, within an historical and concrete frame of reference.L’elaborato, dal titolo “Salvatore Pugliatti e il suo tempo: per un diritto tra sistema e storia”, effettua una disamina ricostruttiva ed espositiva del pensiero giuridico e metodologico di Salvatore Pugliatti, giurista siciliano dei decenni posti a cavallo della metà del secolo scorso, morto nel 1976, fondatore della Scuola di diritto civile dell’Università di Messina, affrontando lo studio sia delle opere dell’Autore direttamente indirizzate all’analisi di questioni di indole teorico-metodologica, sia di quelle, tra le sue opere di diritto positivo, in cui è dato evincere, nelle pieghe dell’esposizione, l’attuazione dei traguardi speculativi di fondo altrove espressi.
A tale scopo, il lavoro è ripartito in tre capitoli.
Il primo di essi si prefigge l’obiettivo di inquadrare la figura scientifica di Salvatore Pugliatti entro il contesto temporale, compreso tra il 1927 e la prima metà degli anni Settanta, in cui l’Autore ha operato, ed è dedicato alla ricostruzione sia delle componenti storiche, sia di quelle più specificamente giuridico-normative e filosofico-metodologiche, che ne hanno maggiormente influenzato il pensiero. Se, dal punto di vista storico, emergono, e vengono analizzati, l’ascesa e la caduta del regime fascista, lo sfacelo della seconda guerra mondiale, e la lenta ricostruzione dell’assetto istituzionale italiano ad essa susseguente, dal punto di vista normativo assumono rilievo, e vengono via via analizzati, la legislazione speciale del regime mussoliniano, la ricodificazione fascista, l’entrata in vigore del Codice Civile del 1942 e, soprattutto, l’irrompere della Costituzione del 1948, che, per la sua portata formale e per il suo contenuto sostanziale, ha costretto la dottrina civilistica italiana, e dunque anche Pugliatti, a ridefinire quantomeno parte delle certezze dogmatiche sino ad allora raggiunte dalla scienza giuridica. Per quanto riguarda, inoltre, il contesto più marcatamente giusfilosofico e metodologico di riferimento, l’elaborato passa in rassegna le principali tendenze culturali che si sono manifestate suscettibili di influire sul pensiero di Pugliatti, ossia, più specificamente: in primo luogo, l’indirizzo, normativista ed astrattizzante, proprio del positivismo giuridico formalistico di matrice dogmatica e postpandettistica, tendente a ridurre la filosofia del diritto entro i confini della scienza giuridica strutturata secondo il metodo delle scienze sperimentali; in secondo luogo, l’indirizzo, gravitante nell’orbita del neoidealismo giuridico italiano, proprio dell’attualismo gentiliano e dello storicismo crociano, tendente ad escludere, seppure per vie diverse, la possibilità di munire del carattere della scientificità tanto la filosofia del diritto quanto la scienza giuridica; in terzo luogo, l’indirizzo, proprio del cosiddetto idealismo giuridico post-crociano empirico-storicistico, in cui spiccano, per l’influenza che hanno avuto sulla riflessione pugliattiana, le figure di Angelo Ermanno Cammarata e Widar Cesarini Sforza, e tendente a recuperare, a partire dall’idealismo stesso di Croce e Gentile, il valore storicamente concreto della giuridicità da cui promana qualsiasi forma di riflessione sul diritto.
Il secondo capitolo affronta invece, alla luce di un simile contesto, il nucleo vero e proprio della riflessione teorica di Pugliatti, seguendo analiticamente, attraverso la disamina dei contributi dell’Autore dedicati a tematiche di indole specificamente metodologica, le linee anche cronologiche della sua progressiva evoluzione. L’elaborato analizza, innanzitutto, la prima fase, giovanile, della produzione scientifica pugliattiana, che appare ancora fortemente incardinata, sotto il profilo dei convincimenti teorico-giuridici, sui paradigmi metodologici di un deciso rigore sistematico di natura logico-normativista e di matrice giuspositivistica, e di un inevitabile dualismo tra le astrattezze formali del sistema concettuale elaborato dal giurista e la concretezza degli interessi storici ad essi sottesi ma ad essi, necessariamente, anche rigorosamente estranei. La seconda fase, più matura, dell’iter speculativo di Pugliatti, relativa agli anni Quaranta, consente di individuare un’evoluzione nel credo metodologico dell’Autore: raccogliendo le sollecitazioni provenienti in modo particolare dalla sua partecipazione attiva alla nota “polemica sulla natura dei concetti giuridici” e al dibattito sulla “crisi” del diritto e della
scienza giuridica, ma provenienti anche dal manifestato interesse per la linguistica e dalla lettura e conoscenza di figure eminenti come Santi Romano e Capograssi, Pugliatti giunge in questi anni a consolidare i propri convincimenti in una concezione rinnovata di giurisprudenza, che viene qualificata come scienza pratica, cioè come scienza che, senza dubbio ancorata al dato positivo e funzionale alla costruzione di perfetti sistemi giuridici astratti, logici e concettuali, non può tuttavia non radicarsi, traendone origine, anche negli interessi concreti dei consociati, nel fluire magmatico della storia, e, con esso, nel divenire mutevole delle norme, così che essa, rimanendo scienza, ma qualificandosi come scienza pratica, trova dunque in sé stessa la possibilità di una composizione dialettica del dualismo forma-sostanza, astratto-concreto, che altrimenti rimarrebbe insopprimibile. L’analisi della terza fase del pensiero di Pugliatti, infine, evidenzia nell’Autore il graduale perfezionamento, operato attraverso l’approfondimento della problematica relativa al rapporto tra il “continuo” e il “discontinuo” nel diritto, dei convincimenti precedentemente acquisiti: la sintesi dialettica del dualismo tra forma e sostanza, operata attraverso la concezione della giurisprudenza come scienza pratica, si cristallizza così nell’idea – conclusiva del percorso metodologico di Pugliatti – secondo cui la complessità dell’esperienza giuridica, nella quale convivono la sincronia giuridica del discontinuo astratto del sistema normativo e la diacronia storica del continuo mobile dell’ordinamento giuridico tout court (che non si risolve nelle norme poiché è anche altro rispetto ad esse), trova compimento nella formula che qualifica il diritto come “sintesi del molteplice nell’uno”.
Il terzo capitolo analizza, infine, alcune tra le principali opere di Pugliatti relative a tematiche di diritto positivo in cui l’Autore impiega, nel momento in cui affronta questioni tecniche di puro diritto civile, lo strumentario metodologico di cui si è dotato: l’analisi passa dalla rappresentanza, di cui è valorizzato da Pugliatti il rilievo che assume, nella descrizione dei caratteri dell’istituto, il concreto e storico rapporto sottostante di gestione tra il dominus e il rappresentante, al fenomeno dei trasferimenti coattivi, dagli importantissimi studi pugliattiani sulle proprietà (in cui l’istituto proprietario è considerato al plurale, non più al singolare, stante la polverizzazione delle tipologie di diritti dominicali rinvenibili nell’ordinamento a fronte della diversificazione dei soggetti titolari del diritto e delle funzioni proprie dei singoli beni che ne sono oggetto), alla breve esposizione del rapporto dialettico intercorrente, per Pugliatti, tra il diritto pubblico e il diritto privato.
Dall’analisi effettuata emerge, quale sintesi della figura di Pugliatti, l’immagine di un giurista completo, dotato di un forte bagaglio culturale e giuridico, costantemente proteso verso la ricerca di una chiarificazione dei criteri metodologici del proprio operare, il cui pensiero, se non riesce invero mai ad emanciparsi completamente dalla persistenza del dualismo tra astratto e concreto, tra forma e storia, cui la scienza giuridica sembra essere condannata, ricadendo tendenzialmente entro le spire di un normativismo positivista solo parzialmente, criticamente e consapevolmente attenuato, lascia comunque, come eredità metodologica per il giurista di oggi, il respiro lucido e profondo della propria solida complessità, fondata sull’intuizione della necessità costante dell’esegesi e della multidisciplinarietà, dell’altrettanto forte necessità del riferimento al dato positivo, e dell’insopprimibile necessità di calibrare qualsiasi riflessione, sia essa filosofica, metodologica o di diritto positivo, sul e col diritto, all’interno della cornice storica e concreta di riferimento
Contributions to the Dirichlet process and related classes of random probability measures
Full text linkRates of convergence in de Finetti’s representation theorem, and Hausdorff moment problem
Full text linkGiven a sequence {X_n} of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that
(X_1+...+X_n)/n → Y a.s. for a suitable random variable Y : Omega → [0, 1] satisfying
P[X_1 = x_1,..., X_n = x_n |Y]=Y^(X_1+...+X_n) * (1−Y)^(n - X_1-...-X_n).
In this paper, we study the rate of convergence in law of (X_1+...+X_n)/n to Y under the Kolmogorov distance. After showing that a rate of the type of (1/n)^α can be obtained for any index α ∈ (0, 1], we find a sufficient condition on the distribution of Y for the
achievement of the optimal rate of convergence, that is 1/n. Besides extending and strengthening recent results under the weaker Wasserstein distance, our main result weakens the regularity hypotheses on Y in the context of the Hausdorff moment problem
Learning-augmented count-min sketches via Bayesian nonparametrics
No full textThe count-min sketch (CMS) is a time and memory efficient randomized data
structure that provides estimates of tokens' frequencies in a data stream of
tokens, i.e. point queries, based on random hashed data. A learning-augmented
version of the CMS, referred to as CMS-DP, has been proposed by Cai,
Mitzenmacher and Adams (\textit{NeurIPS} 2018), and it relies on Bayesian
nonparametric (BNP) modeling of the data stream of tokens via a Dirichlet
process (DP) prior, with estimates of a point query being obtained as suitable
mean functionals of the posterior distribution of the point query, given the
hashed data. While the CMS-DP has proved to improve on some aspects of CMS, it
has the major drawback of arising from a ``constructive" proof that builds upon
arguments tailored to the DP prior, namely arguments that are not usable for
other nonparametric priors. In this paper, we present a ``Bayesian" proof of
the CMS-DP that has the main advantage of building upon arguments that are
usable, in principle, within a broad class of nonparametric priors arising from
normalized completely random measures. This result leads to develop a novel
learning-augmented CMS under power-law data streams, referred to as CMS-PYP,
which relies on BNP modeling of the data stream of tokens via a Pitman-Yor
process (PYP) prior. Under this more general framework, we apply the arguments
of the ``Bayesian" proof of the CMS-DP, suitably adapted to the PYP prior, in
order to compute the posterior distribution of a point query, given the hashed
data. Applications to synthetic data and real textual data show that the
CMS-PYP outperforms the CMS and the CMS-DP in estimating low-frequency tokens,
which are known to be of critical interest in textual data, and it is
competitive with respect to a variation of the CMS designed for low-frequency
tokens. An extension of our BNP approach to more general queries is also
discussed.Comment: 47 page
Deep Stable neural networks: large-width asymptotics and convergence rates
Full text linkIn modern deep learning, there is a recent and growing literature on the interplay between large-width asymptotic
properties of deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed weights, and Gaussian stochastic processes (SPs). Motivated by empirical analyses that show the potential of replacing Gaussian
distributions with Stable distributions for the NN’s weights, in this paper we present a rigorous analysis of the
large-width asymptotic behaviour of (fully connected) feed-forward deep Stable NNs, i.e. deep NNs with Stabledistributed
weights. We show that as the width goes to infinity jointly over the NN’s layers, i.e. the “joint growth”
setting, a rescaled deep Stable NN converges weakly to a Stable SP whose distribution is characterized recursively
through the NN’s layers. Because of the non-triangular structure of the NN, this is a non-standard asymptotic
problem, to which we propose an inductive approach of independent interest. Then, we establish sup-norm convergence rates of the rescaled deep Stable NN to the Stable SP, under the “joint growth” and a “sequential growth” of the width over the NN’s layers. Such a result provides the difference between the “joint growth” and the “sequential growth” settings, showing that the former leads to a slower rate than the latter, depending on the depth of the layer and the number of inputs of the NN. Our work extends some recent results on infinitely wide limits for deep Gaussian NNs to the more general deep Stable NNs, providing the first result on convergence rates in the “joint growth” setting
Asymptotics for a Bayesian nonparametric estimator of species richness
No full textIn Bayesian nonparametric inference, random discrete probability measures are commonly used as priors within hierarchical mixture models for density estimation and for inference on the clustering of the data. Recently it has been shown that they can also be exploited in species sampling problems: indeed they are natural tools for modeling the random proportions of species within a population thus allowing for inference on various quantities of statistical interest. For applications that involve large samples, the exact evaluation of the corresponding estimators becomes impracticable and, therefore, asymptotic approximations are sought. In the present paper we study the limiting behaviour of the number of new species to be observed from further sampling, conditional on observed data, assuming the observations are exchangeable and directed by a normalized generalized gamma process prior. Such an asymptotic study highlights a connection between the normalized generalized gamma process and the two-parameter Poisson-Dirichlet process that was previously known only in the unconditional case
Bayesian nonparametric disclosure risk assessment
Full text linkAny decision about the release of microdata for public use is supported by the estimation of measures of disclosure risk, the most popular being the number τ1 of sample uniques that are also population uniques. In such a context, parametric and nonparametric partition-based models have been shown to have: i) the strength of leading to estimators of τ1 with de- sirable features, including ease of implementation, computational efficiency and scalability to massive data; ii) the weakness of producing underesti- mates of τ1 in realistic scenarios, with the underestimation getting worse as the tail behaviour of the empirical distribution of microdata gets heavier. To fix this underestimation phenomenon, we propose a Bayesian nonpara- metric partition-based model that can be tuned to the tail behaviour of the empirical distribution of microdata. Our model relies on the Pitman–Yor process prior, and it leads to a novel estimator of τ1 with all the desir- able features of partition-based estimators and that, in addition, allows to reduce underestimation by tuning a “discount” parameter. We show the effectiveness of our estimator through its application to synthetic data and real data
Infinitely wide limits for deep Stable neural networks: sub-linear, linear and super-linear activation functions
Full text linkThere is a growing literature on the study of large-width properties of deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed parameters or weights, and Gaussian stochastic processes. Motivated by some empirical and theoretical studies showing the potential of replacing Gaussian distributions with Stable distributions, namely distributions with heavy tails, in this paper we investigate large-width properties of deep Stable NNs, i.e. deep NNs with Stable-distributed parameters. For sub-linear activation functions, a recent work has characterized the infinitely wide limit of a suitable rescaled deep Stable NN in terms of a Stable stochastic process, both under the assumption of a ``joint growth" and under the assumption of a ``sequential growth" of the width over the NN's layers. Here, assuming a ``sequential growth" of the width, we extend such a characterization to a general class of activation functions, which includes sub-linear, asymptotically linear and super-linear functions. As a novelty with respect to previous works, our results rely on the use of a generalized central limit theorem for heavy tails distributions, which allows for an interesting unified treatment of infinitely wide limits for deep Stable NNs. Our study shows that the scaling of Stable NNs and the stability of their infinitely wide limits may depend on the choice of the activation function, bringing out a critical difference with respect to the Gaussian setting
Conditional formulae for Gibbs-type exchangeable random partitions
No full textGibbs–type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs–type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviours. Such expressions can be interpreted as Bayesian nonparametric estimators of the rare species variety and their performance is tested on some real genomic data
- …
