Donato Michele Cifarelli

Priors for Exponential Families Which Maximize the Association between Past and Future Observations

Throughout the present paper, {Xn} denotes a sequence of random quantities which are regarded as exchangeable, and which are assessed with a probability measure P(•) which is a member of the mixture-exponential family. To be precise, it will be presumed that the assessment P(•) for any finite subsequence (X1,…,Xn) can be represented using the product of an identical non-degenerate parametric measure for each Xi, Pθ (•)= P(•θ=θ), determined by

Frequentistic approximations to Bayesian prevision of exchangeable random elements

{\rm{d}}{{\rm{P}}_{\rm{\theta }}} = \exp \{ {\rm{\theta x}} - {\rm{M}}({\rm{\theta }})\} {\rm{d\mu }}

Estimates of regression coefficient based on gini’s rank association coefficient

(1.1) μ being a σ-finite measure on the class B of Borel sets of IR. It will always be assumed that the interior X° of the convex hull X of the support of μ (in symbols:suρp(μ))is a nonempty open set (interval) in IR and that {Pθ;θeΘ} is a regular exponential family (cf. Barndorff - Nielsen 1978, p.116). The latter condition implies that Θ = {θ:M(θ)<∞ } is an open interval in IR. Moreover, we will suppose that the set of the logically possible values of θ coincides with θ. Given such a particular frame, the present paper deals with the choice of a prior for (1.1); an excellent treatment of the same topic is included in Diaconis and Ylvisaker (1979, 1985). Our approach bases itself on the obvious remark that the choice of a prior establishes the strength of the dependence among the elements of the sequence {Xn} and, consequently, the strength of the influence exercised by experience on our future predictions. This subjective standpoint is skilfully expounded in de Finetti (1937).

Distribution Functions of Means of a Dirichlet Process

Si fanno alcune considerazioni di natura generale sui problemi statistici non parametrici e sulla loro risoluzione dal punto di vista bayesiano. Dopo aver posto in evidenza l’importanza delle medie associative come strumenti di analisi statistica dei suddetti problemi, ci si propone di determinare la distribuzione iniziale e finale di una generica media associativa nell’ipotesi che le misure di probabilità assegnabili al fenomeno in esame siano rette da un processo di Dirichlet. Ricordate le proprietà fondamentali di tale processo, si rinvia la risoluzione del problema posto alla seconda parte del lavoro.AbstractThis paper presents some aspects of Bayesian analysis of nonparametric problems and it is splitted up in two parts.It is the purpose of the first one to emphasize the role of the associative mean valuesϕ(x) is a continuous and strictly increasing function andF(x)=P((− ∞,x]), whereP is a Dirichlet process (see T. S. Ferguson (1973)).We review the basic properties of Ferguson’s Dirichlet process and we characterize such a processvia exchangeability of the sequence of the observable random variables. We take the opportunity to remind that an elementary version of Dirichlet process was given by C. Gini (1911) and by G. Pompilj (1951).It is the purpose of the second part, which will appear in the next number of this Journal, to exhibit the initial and final distributions ofMϕ.

De Finetti's contribution to probability and statistics

Given a sequence \xi_1, \xi_2,... of X-valued, exchangeable random elements, let q(\xi^(n)) and p_m(\xi^(n)) stand for posterior and predictive distribution, respectively, given \xi^(n) = (\xi_1,..., \xi_n). We provide an upper bound for limsup b_n d_[[X]](q(\xi^(n)), \delta_\empiricn) and limsup b_n d_[X^m](p_m(\xi^(n)), \empiricn^m), where \empiricn is the empirical measure, b_n is a suitable sequence of positive numbers increasing to +\infty, d_[[X]] and d_[X^m] denote distinguished weak probability distances on [[X]] and [X^m], respectively, with the proviso that [S] denotes the space of all probability measures on S. A characteristic feature of our work is that the aforesaid bounds are established under the law of the \xi_ns, unlike the more common literature on Bayesian consistency, where they are studied with respect to product measures (p_0)^\infty, as p_0 varies among the admissible determinations of a random probability measure.

On the asymptotic distribution of a general measure of monotone dependence

Abstract This note points some ambiguities in the notation adopted in “Frequentistic approximations to Bayesian prevision of exchangeable random elements” [Int. J. Approx. Reason. 78 (2016) 138–152] and provides the correct way to read those statements and proofs which are affected by the aforesaid ambiguities.

A general approach to Bayesian analysis of nonparametric problems: The associative mean values within the framework of the Dirichlet process

AbstractSi considera il modello lineare semplice: conxi costanti assegnate edɛ1,ɛ2, ...,ɛN v.c. mutuamente indipendenti e somiglianti con funzioni di ripartizioneF continua e si propone uno stimatore diβ, costruito a partire dal processo stocastico definito dalla (2), dato da