1,720,987 research outputs found
Due approcci alla costruzione del modello statistico: confronti ed osservazioni
No full textLauritzen, in his book "Statistical models as extremal families", criticizes the usual procedure of giving in advance the statistical model and then applying the various inference principles, and states that the statistical analysis and the statistical model ought to be jointly considered. The technique which he suggests for the construction of a statistical model, may be traced, although in a simpler context , also in a theorem by Diaconis and Freedman. From a predictive view point the same problem has been also dealt with by Cifarelli and Regazzini. We present a new, and practically more useful version, of the deacons and freedman's Theorem, and analyse in relationship with the predictive viewpoint. Finally we describe a way to combine the two above-mentioned approaches into a unified framework
Discussion of "What’s the H in H-likelihood: A Holy Grail or an Achilles’ Heel?" by Meng, X.-L.
No full textMeng raises an interesting issue concerning the relationship among pivotal predictive distribution, posterior predictive distribution and h-distribution. He emphasizes how it is important using the "appropriate" scale both for observations and parameters and presents an example in which the three distributions coincide. We show why this result is not completely surprising and how part of the explanation can be found by extending a result due to Lindley who explains the relationship between a fiducial distribution and a posterior distribution
Sul significato previsivo di una particolare distribuzione iniziale coniugata nel modello di regressione lineare normale
No full textIl lavoro utilizza un criterio previsivo, precedentemente proposto in letteratura, per identificare una distribuzione non informativa. L'idea è di scegliere quella prior che massimizza la dipendenza, misurata attraverso l'indice di correlazione di Pearson, fra una variabile di cui interessa una previsione e il campione. Il criterio è applicato ad un modello di regressione lineare univariato. Le prior identificate, nel caso di varianza del modello nota e no, pur appartenendo alla famiglia coniugata differiscono da quelle usualmente impiegate in quanto dipendono dai regressori
A note on coherent invariant distributions as non-informative priors for exponential and location-scale families
No full textThe problem of finding a non-informative prior distribution for a parameter is approached using the notion of context-invariance. This concept is revisited and discussed with the aim of applying it to finding context-invariant non-informative priors for the one-parameter exponential family (suitably redefined) and the location-scale family. Our approach, carried-out in a finitely-additive framework, generally leads to a class of non-informative priors with respect to any given problem. For most common statistical models such a class does not always contain the corresponding Jeffreys' prior, but does contain the so-called ALI prior by Hartigan
Distribuzioni iniziali improprie nell'inferenza bayesiana ed indici di divergenze e informazione
No full textIl lavoro studia le distribuzioni iniziali su un parametro continuo che soddisfano a due criteri di "non informatività". Il primo suggerisce di massimizzare, per ogni osservazione X=x, una misura di divergenza fra la distribuzione iniziale e quella finale del parametro; il secondo di massimizzare la mutua informazione fra X e il parametro. Al fine di precisare e rendere operativi tali criteri si utilizza una definizione di divergenza e informazione che richieda solo la conoscenza della legge di probabilità (finitamente additiva) rilevante. Si dimostra che, sotto condizioni tipicamente soddisfatte dalle distribuzioni iniziali, se queste sono improprie e le finali proprie, la misura di divergenza è infinita. Questo risultato non sempre vale invece nel caso della mutua informazione, come mostrano alcuni esempi
Coherent distributions and Lindley's paradox
No full textA Bayesian test of a simple null hypothesis H_0 versus a composite alternative H_1 is performed using finitely additive prior distributions in order to investigate the so called Lindley's paradox. In particular two priors for the parameter under H_1 are considered. The first represents a coherently non-informative distributions which is shown to correctly yield the "paradox" because of the overall induced distribution of the parameter. The second, through the use of adherent masses the point specify by H_0, does instead avoid Lindley's paradox
Approximation of Distribution Functions: a Constructive Scheme with Application in Bayesian Nonparametrics
No full textMany statistical nonparametric techniques are based on the possibility of approximating a curve of interest (for example, the distribution function generating the data) by basis-functions expansions. In this paper we discuss the problem of constructively
approximating a deterministic or random probability distribution function by means of a sequence of distribution functions. Our proposal is based on a general approximation scheme referred to Feller, which we show to have nice probabilistic properties
when its basic elements are chosen in the natural exponential family. Some new results for this family are proved, which might be of autonomous interest; exploiting these properties, we can show connections between the proposed scheme and pproximation
by mixtures.
When the distribution function to be approximated is random, our procedure can be used for prior elicitation in Bayesian nonparametric inference. On one hand, it provides a constructive smoothing of discrete prior processes (e.g. the Dirichlet process),
so allowing to elicit a prior more suitable for nonparametric inference with continuous data. Also, it selects distribution functions which have the rather natural form of mixtures.
We have thus a general framework which includes continuous, countable and finite mixtures (with an unknown number of components), and various choices of the kernel.
Therefore, properties can be studied in a unified setting for a fairly large class of mixture models; in particular, we give a result on weak consistency of the proposed mixture prior
Unbiased Bayes estimates and improper prior
No full textGiven two random variables X and Y , the condition of unbiasedness states that E(X|Y = y) = y and E(Y |X = x) = x both almost surely (a.s.). If the prior on Y is proper and has finite expectation or nonnegative support, unbiasedness implies X = Y a.s. This paper examines the implications of unbiasedness when the prior on Y is improper. Since the improper case can be meaningfully analysed in a finitely additive framework, we revisit the whole issue of unbiasedness from this perspective. First we argue that a notion weaker than equality a.s., named coincidence, is more appropriate in a finitely additive setting. Next we discuss the meaning of unbiasedness from a Bayesian and fiducial perspective.We then show that unbiasedness and finite expectation of Y imply coincidence between X and Y , while a weaker conclusion follows if the improper prior on Y is only assumed to have positive support. We illustrate our approach
throughout the paper by revisiting some examples discussed in the recent literature
Conditionally reducible natural exponential families and enriched conjugate priors
No full textConsider a standard conjugate family of prior distributions for a vectorparameter indexing an exponential family. Two distinct model parameterizations may well lead to standard conjugate families which are not consistent, i.e. one family cannot be derived from the other by the usual change-ofvariable technique. This raises the problem of finding suitable parameterizations that may lead to enriched conjugate families which are more flexible than the traditional ones.
The previous remark motivates the definition of a new property for an exponential family, named conditional reducibility. Features of conditionally reducible natural exponential families are investigated thoroughly. In particular, we relate this new property to the notion of cut, and show that conditionally-reducible families admit a reparameterization in terms of a vector having likelihood-independent components. A general methodology to obtain enriched conjugate distributions for conditionally-reducible families
is described in detail, generalizing previous works and more recent contributions in the area.
The theory is illustrated with reference to natural exponential families having simple quadratic variance function
Un Algoritmo MCMC per l’ananlisi Bayesiana di modelli gerarchici partizionali
No full textIl lavoro propone un algoritmo MCMC per determinare la distribuzione a posteriori delle possibili strutture parzialmente scambialbili (modelli) generate da un modello gerarchico partizionale.
La progedura suggerita opera solo sullo spazio discreto dei modelli poiché è possibile determinare in modo analitico la distribuzione a posteriori dei parametri per ciascun modello. La performance dell'algoritmo è discussa in dettaglio ed appare soddisfacente sia quando il numero di modelli è moderato sia quando non lo è
- …
