Untitled - UFRJ
Untitled - UFRJ Untitled - UFRJ
Adaptive Proxy Maximum Probability Estimation ofMultidimensional Poisson IntensitiesJose Carlos Simon de MirandaIME-USPWe propose a non parametric methodology of estimation of the intensity for Poisson point processeson R m . We assume the observation region, O, is a bounded R m interval. The space of positive functionsformed by composition of L 2 (O)-functions with the exponential is endowed with a probability inducedfrom another one defined on the set of wavelet coefficients. This is a convenient space for the intensityto belong to and we choose as our first estimate for the intensity a function that corresponds to themaximum posterior probability given a trajectory of the Poisson process on O. This choice is done bydetermining the wavelet coefficients of its logarithm. A second estimate is obtained by suitably writingthe posterior probability as a product of functions that are maximized separately giving raise to a proxymaximum posterior probability estimate.This approach presents the desired feature of furnishing everywhere non negative estimates of theintensity that exhibit not only a minimization of the energy, relative to the wavelet basis, but also amaximization of the entropy of the process on O conditional to the realization. An adaptive thresholdingprocedure based on jointly testing hypothesis, on the wavelet coefficients, and adjusting the priors’locations is given. As an example of the general estimating procedure above, we specialize to self affineand self similar probability prior Poisson processes.101
A Practical Approach to Elicit Multivariate Prior DistributionsFernando Antônio MoalaDMEC, UNESPAnthony O’HaganUniversity of SheffieldMoala and O’Hagan (2010) present a method to quantify beliefs in the form of a multivariate priordistribution based on marginal and joint probabilities elicited from expert. The method uses a nonparametricbayesian framework with a Gaussian process prior proposed by elicitor (an statistician). Themain focus in that paper is theoretical and the approach is illustrated with two simulated data examples.However, to run our proposed procedure we must to specify some requirements to be asked to theexpert. This work offers a detailed guideline to elicit the expert’s prior distribution for general practicalsituations. It also brings our practical experience gained when applying our elicitation approach in a realsituation and exposes the challenges involved to implement it. More specifically, we discuss an applicationin clinical trial by building a joint prior distribution f(·) that represents the expert’s beliefs aboutfracture risks in patients suffering from osteoporosis under two treatments. Various posterior inferencesare computed and fedback to the expert.102
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Adaptive Proxy Maximum Probability Estimation ofMultidimensional Poisson IntensitiesJose Carlos Simon de MirandaIME-USPWe propose a non parametric methodology of estimation of the intensity for Poisson point processeson R m . We assume the observation region, O, is a bounded R m interval. The space of positive functionsformed by composition of L 2 (O)-functions with the exponential is endowed with a probability inducedfrom another one defined on the set of wavelet coefficients. This is a convenient space for the intensityto belong to and we choose as our first estimate for the intensity a function that corresponds to themaximum posterior probability given a trajectory of the Poisson process on O. This choice is done bydetermining the wavelet coefficients of its logarithm. A second estimate is obtained by suitably writingthe posterior probability as a product of functions that are maximized separately giving raise to a proxymaximum posterior probability estimate.This approach presents the desired feature of furnishing everywhere non negative estimates of theintensity that exhibit not only a minimization of the energy, relative to the wavelet basis, but also amaximization of the entropy of the process on O conditional to the realization. An adaptive thresholdingprocedure based on jointly testing hypothesis, on the wavelet coefficients, and adjusting the priors’locations is given. As an example of the general estimating procedure above, we specialize to self affineand self similar probability prior Poisson processes.101
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