1,721,095 research outputs found
Compatibility of Prior Specifications across Linear Model
No full textModel comparison within a collection of candidate models is an important and active area of statistical methodology and practice, especially from the Bayesian perspective. An essential requirement for applying the Bayesian paradigm is the specification of a prior distribution on the parameter space of each candidate model: clearly this task becomes prohibitive as soon as the number of models is only moderately large. However, when models are nested within an encompassing (or full) model, one can try to relate priors across models. In this way, only one prior
on the parameter space of the full model need be assigned, while the prior under each submodel is derived from it. This would solve not only the elicitation problem, but also achieve some sort of prior compatibility across models.
In this paper we provide a unified framework for the assignment of priors under a collection of submodels, given a prior on the full model. We introduce two interpretations of nested models, carefully describing the allied notation, and describe some procedures to derive priors under each submodel, namely: marginalization, conditioning, and Kullback-Leibler (KL) projection.
We motivate and illustrate the general methodology through the variable selection problem of linear regression, and illustrate the methods with three examples. In the light of our findings, we conclude that the procedure based on conditioning is not particularly advisable, while KL projection priors together with a default improper prior may jointly contribute to identify a collection of plausible models for Bayesian variable selection
A Bayesian method for combining results from several binomial experiments
No full textThe problem of combining information related to I binomial experiments, each having a distinct probability of success θi, is considered. Instead of using a standard exchangeable prior for θ; = (θ1, ..., θI), we propose a more flexible distribution that takes into account various degrees of similarity among the θi’s. Using ideas developed by Malec and Sedransk, we consider a partition g of the experiments and take the θi’s belonging to the same partition subset to be exchangeable and the θi’s belonging to distinct subsets to be independent. Next we perform Bayesian inference on θ; conditional on g. of course, one is typically uncertain about which partition to use, and so a prior distribution is assigned on a set of plausible partitions g. The final inference on θ; is obtained by combining the conditional inferences according to the posterior distribution of g. The methodology adopted in this article offers a wide flexibility in structuring the dependence among the θi’s. This allows the estimate of θi to borrow strength from all other experiments according to an adaptive process governed by the data themselves. The method may be usefully applied to the analysis of binary response variables in the presence of categorical covariates. The latter are used to identify a collection of suitable partitions g, representing factor main effects and interactions, whose relevance will be summarized in the posterior distribution of g. Besides providing novel interpretations on the role played by the various factors, the procedure will also produce parameter estimates that may differ, sometimes in an appreciable manner, from those obtained using more traditional techniques. Finally, three real data sets are used to illustrate the methodology and compare it with other approaches, such as empirical Bayes (both parametric and nonparametric), logistic regression, and multiple shrinkage estimators. © 1995 Taylor & Francis Group, LLC
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
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
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
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
Invariance and Bayesian predictive analysis with an application to the diallel cross design
No full textThe predictive approach to Bayesian inference is enriched with invariance considerations which naturally occur in experimental designs. Invariance is used extensively both to compute predictive distributions and to derive predictive models together with the posterior parameter distribution. Under the usual assumption of normality, a simple and direct relationship between predictive and posterior distribution is shown to exist. General results are illustrated in detail with reference to the diallel cross experimental design
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