640 research outputs found
A moment-matching Ferguson and Klass algorithm
Completely random measures (CRM) represent the key building block of a wide
variety of popular stochastic models and play a pivotal role in modern Bayesian
Nonparametrics. A popular representation of CRMs as a random series with
decreasing jumps is due to Ferguson and Klass (1972). This can immediately be
turned into an algorithm for sampling realizations of CRMs or more elaborate
models involving transformed CRMs. However, concrete implementation requires to
truncate the random series at some threshold resulting in an approximation
error. The goal of this paper is to quantify the quality of the approximation
by a moment-matching criterion, which consists in evaluating a measure of
discrepancy between actual moments and moments based on the simulation output.
Seen as a function of the truncation level, the methodology can be used to
determine the truncation level needed to reach a certain level of precision.
The resulting moment-matching \FK algorithm is then implemented and illustrated
on several popular Bayesian nonparametric models.Comment: 24 pages, 6 figures, 5 table
Nonlinear stiffness, Lyapunov exponents, and attractor dimension
I propose that stiffness may be defined and quantified for nonlinear systems
using Lyapunov exponents, and demonstrate the relationship that exists between
stiffness and the fractal dimension of a strange attractor: that stiff chaos is
thin chaos.Comment: See home page http://lec.ugr.es/~julya
On the sub-Gaussianity of the Beta and Dirichlet distributions
We obtain the optimal proxy variance for the sub-Gaussianity of Beta
distribution, thus proving upper bounds recently conjectured by Elder (2016).
We provide different proof techniques for the symmetrical (around its mean)
case and the non-symmetrical case. The technique in the latter case relies on
studying the ordinary differential equation satisfied by the Beta
moment-generating function known as the confluent hypergeometric function. As a
consequence, we derive the optimal proxy variance for the Dirichlet
distribution, which is apparently a novel result. We also provide a new proof
of the optimal proxy variance for the Bernoulli distribution, and discuss in
this context the proxy variance relation to log-Sobolev inequalities and
transport inequalities.Comment: 13 pages, 2 figure
Approximating predictive probabilities of Gibbs-type priors
Gibbs-type random probability measures, or Gibbs-type priors, are arguably
the most "natural" generalization of the celebrated Dirichlet prior. Among them
the two parameter Poisson-Dirichlet prior certainly stands out for the
mathematical tractability and interpretability of its predictive probabilities,
which made it the natural candidate in several applications. Given a sample of
size , in this paper we show that the predictive probabilities of any
Gibbs-type prior admit a large approximation, with an error term vanishing
as , which maintains the same desirable features as the predictive
probabilities of the two parameter Poisson-Dirichlet prior.Comment: 22 pages, 6 figures. Added posterior simulation study, corrected
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Bayesian nonparametric dependent model for partially replicated data: the influence of fuel spills on species diversity
We introduce a dependent Bayesian nonparametric model for the probabilistic
modeling of membership of subgroups in a community based on partially
replicated data. The focus here is on species-by-site data, i.e. community data
where observations at different sites are classified in distinct species. Our
aim is to study the impact of additional covariates, for instance environmental
variables, on the data structure, and in particular on the community diversity.
To that purpose, we introduce dependence a priori across the covariates, and
show that it improves posterior inference. We use a dependent version of the
Griffiths-Engen-McCloskey distribution defined via the stick-breaking
construction. This distribution is obtained by transforming a Gaussian process
whose covariance function controls the desired dependence. The resulting
posterior distribution is sampled by Markov chain Monte Carlo. We illustrate
the application of our model to a soil microbial dataset acquired across a
hydrocarbon contamination gradient at the site of a fuel spill in Antarctica.
This method allows for inference on a number of quantities of interest in
ecotoxicology, such as diversity or effective concentrations, and is broadly
applicable to the general problem of communities response to environmental
variables.Comment: Main Paper: 22 pages, 6 figures. Supplementary Material: 11 pages, 1
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Bayesian optimal adaptive estimation using a sieve prior
We derive rates of contraction of posterior distributions on nonparametric
models resulting from sieve priors. The aim of the paper is to provide general
conditions to get posterior rates when the parameter space has a general
structure, and rate adaptation when the parameter space is, e.g., a Sobolev
class. The conditions employed, although standard in the literature, are
combined in a different way. The results are applied to density, regression,
nonlinear autoregression and Gaussian white noise models. In the latter we have
also considered a loss function which is different from the usual l2 norm,
namely the pointwise loss. In this case it is possible to prove that the
adaptive Bayesian approach for the l2 loss is strongly suboptimal and we
provide a lower bound on the rate.Comment: 33 pages, 2 figure
Fuzzy Control of Chaos
We introduce the idea of the fuzzy control of chaos: we show how fuzzy logic
can be applied to the control of chaos, and provide an example of fuzzy control
used to control chaos in Chua's circuit
Dirichlet process mixtures under affine transformations of the data
Location-scale Dirichlet process mixtures of Gaussians (DPM-G) have proved
extremely useful in dealing with density estimation and clustering problems in
a wide range of domains. Motivated by an astronomical application, in this work
we address the robustness of DPM-G models to affine transformations of the
data, a natural requirement for any sensible statistical method for density
estimation and clustering. First, we devise a coherent prior specification of
the model which makes posterior inference invariant with respect to affine
transformations of the data. Second, we formalise the notion of asymptotic
robustness under data transformation and show that mild assumptions on the true
data generating process are sufficient to ensure that DPM-G models feature such
a property. Our investigation is supported by an extensive simulation study and
illustrated by the analysis of an astronomical dataset consisting of physical
measurements of stars in the field of the globular cluster NGC 2419.Comment: 36 pages, 7 Figure
Comparison of three different instruments for orthodontic study model analysis
INTRODUCTION: A proper model analysis forms a vital part of the orthodontic diagnosis process, but it remains a time-consuming procedure. In day-to-day practice, many orthodontists assess the models subjectively, without applying analytical tests, due to the time it takes to do proper model analysis.1,2 Plaster dental models have long been the gold standard for orthodontic study model analysis and to calculate the Bolton index for tooth size disproportions, as well as intra-arch space discrepancies.3,4 Vernier callipers or needle pointed dividers are traditionally used to perform measurements on dental models.5 More recently digital orthodontic study models that are computer-based have been developed and have the potential to replace the traditional plaster orthodontic models.6 AIMS AND OBJECTIVES: The aim of this study was to do model analysis on one hundred orthodontic cases by making use of three different measuring tools. The objective was to see if a difference exists with regards to the measurements produced by the three different instruments and to compare the instruments with each other. MATERIAL AND METHODS: Three different instruments were used to measure Ave values on one hundred orthodontic study models. The three instruments included a Boley Gauge, Digital Vernier Calliper and Carestream 3600 scanner with accompanying software. The five values measured on the study models were: maxillary intercanine width, maxillary intermolar width, mesio-distal width of tooth 11, mesio-distal width of tooth 46 and mesio-distal width of tooth 41. RESULTS: The statistical analysis performed showed that the difference in measurements produced by the three instruments were not statistically significant for the inter-molar width (p = 0.849), intercanine width (p = 0.657), mesio-distal width of tooth 11 (p = 0.178) and mesio-distal width of tooth 41 (p = 0.240 The difference in measurements for the mesio-distal width of tooth 46 were statistically significant (p<0.01). However no clinically significant difference was found when the measurements produced by the three instruments were compared. CONCLUSIONS: All three of the instruments produced accurate measurements and can be used confidently when doing a comprehensive study model analysis for orthodontic diagnosis and treatment planning. The values produced were similar for all three instruments with insignificant differences between the three
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