Prior Distributions on Spaces of Probability Measures

Ferguson, Thomas S.

doi:10.1214/aos/1176342752

Cited by 630 publications

(461 citation statements)

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“…Foundational work on Polya trees was presented by Ferguson (1974), Lavine (1992), Lavine (1994), andMauldin et al (1992). Mixtures of Polya trees were developed by Berger and Guglielmi (2001), Hanson and Johnson (2002), and Hanson (2006).…”

Section: Prior For Baseline Survivalmentioning

confidence: 99%

Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

2010

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The joint modeling of longitudinal and survival data has received extraordinary attention in the statistics literature recently, with models and methods becoming increasingly more complex. Most of these approaches pair a proportional hazards survival with longitudinal trajectory modeling through parametric or nonparametric specifications. In this paper we closely examine one data set previously analyzed using a two parameter parametric model for Mediterranean fruit fly (medfly) egg-laying trajectories paired with accelerated failure time and proportional hazards survival models. We consider parametric and nonparametric versions of these two models, as well as a proportional odds rate model paired with a wide variety of longitudinal trajectory assumptions reflecting the types of analyses seen in the literature. In addition to developing novel nonparametric Bayesian methods for joint models, we emphasize the importance of model selection from among joint and non joint models. The default in the literature is to omit at the outset non joint models from consideration. For the medfly data, a predictive diagnostic criterion suggests that both the choice of survival model and longitudinal assumptions can grossly affect model adequacy and prediction. Specifically for these data, the simple joint model used in by Tseng et al. (Biometrika 92:587-603, 2005) and models with much more flexibility in their longitudinal components are predictively outperformed by simpler analyses. This case study underscores the need for data analysts to compare on the basis of predictive performance different joint models and to include non joint models in the pool of candidates under consideration.

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Section: Prior For Baseline Survivalmentioning

confidence: 99%

Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

2010

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“…Under weak conditions on the so-called bandwidth h n ,k n (x) does converge (in L 2 norm and pointwise) to the true density f (x) (Silverman 1986). 4 The most common approach in Bayesian non-parametric Statistics is to use the so-called Dirichlet process distribution, D(F 0 , α), where F 0 is a cdf and α is a precision parameter (Ferguson 1974). This prior distribution enjoys the coherency property that, if…”

Section: Nonparametric Approachmentioning

confidence: 99%

Bayesian Modelling and Inference on Mixtures of Distributions

Marin¹,

Mengersen²,

Robert³

2005

Handbook of Statistics

328

364

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'But, as you have already pointed out, we do not need any more disjointed clues,' said Bartholomew. 'That has been our problem all along: we have a mass of small facts and small scraps of information, but we are unable to make any sense out of them. The last thing we need is more.' Susanna Gregory, A Summer of Discontent IntroductionToday's data analysts and modellers are in the luxurious position of being able to more closely describe, estimate, predict and infer about complex systems of interest, thanks to ever more powerful computational methods but also wider ranges of modelling distributions. Mixture models constitute a fascinating illustration of these aspects: while within a parametric family, they offer malleable approximations in non-parametric settings; although based on standard distributions, they pose highly complex computational challenges; and they are both easy to constrain to meet identifiability requirements and fall within the class of ill-posed problems. They also provide an endless benchmark for assessing new techniques, from the EM algorithm to reversible jump methodology. In particular, they exemplify the

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“…In this paper we show how a Bayesian nonparametric approach can be implemented in a simple way, using either Pólya tree prior distributions (Ferguson, 1974), or Bernstein polynomial prior distributions (Petrone, 1999a(Petrone, , 1999b. It is also well known that both types support to absolutely-continuous distributions.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

“…So let Y 0 , where is a sequence from f0; 1g, generate F 1 and Z 0 generate F 2 according to the description given for single Pólya tree prior distributions. See Ferguson (1974) and Lavine (1992) for further details. So, for example,…”

Section: Model Based On the Pólya Tree Priormentioning

confidence: 99%