46,674 research outputs found
Gaussianisation for fast and accurate inference from cosmological data
We present a method to transform multivariate unimodal non-Gaussian posterior
probability densities into approximately Gaussian ones via non-linear mappings,
such as Box--Cox transformations and generalisations thereof. This permits an
analytical reconstruction of the posterior from a point sample, like a Markov
chain, and simplifies the subsequent joint analysis with other experiments.
This way, a multivariate posterior density can be reported efficiently, by
compressing the information contained in MCMC samples. Further, the model
evidence integral (i.e. the marginal likelihood) can be computed analytically.
This method is analogous to the search for normal parameters in the cosmic
microwave background, but is more general. The search for the optimally
Gaussianising transformation is performed computationally through a
maximum-likelihood formalism; its quality can be judged by how well the
credible regions of the posterior are reproduced. We demonstrate that our
method outperforms kernel density estimates in this objective. Further, we
select marginal posterior samples from Planck data with several distinct
strongly non-Gaussian features, and verify the reproduction of the marginal
contours. To demonstrate evidence computation, we Gaussianise the joint
distribution of data from weak lensing and baryon acoustic oscillations (BAO),
for different cosmological models, and find a preference for flat CDM.
Comparing to values computed with the Savage-Dickey density ratio, and
Population Monte Carlo, we find good agreement of our method within the spread
of the other two.Comment: 14 pages, 9 figure
The Bivariate Normal Copula
We collect well known and less known facts about the bivariate normal
distribution and translate them into copula language. In addition, we prove a
very general formula for the bivariate normal copula, we compute Gini's gamma,
and we provide improved bounds and approximations on the diagonal.Comment: 24 page
On the estimation of normal copula discrete regression models using the continuous extension and simulated likelihood
The continuous extension of a discrete random variable is amongst the
computational methods used for estimation of multivariate normal copula-based
models with discrete margins. Its advantage is that the likelihood can be
derived conveniently under the theory for copula models with continuous
margins, but there has not been a clear analysis of the adequacy of this
method. We investigate the asymptotic and small-sample efficiency of two
variants of the method for estimating the multivariate normal copula with
univariate binary, Poisson, and negative binomial regressions, and show that
they lead to biased estimates for the latent correlations, and the univariate
marginal parameters that are not regression coefficients. We implement a
maximum simulated likelihood method, which is based on evaluating the
multidimensional integrals of the likelihood with randomized quasi Monte Carlo
methods. Asymptotic and small-sample efficiency calculations show that our
method is nearly as efficient as maximum likelihood for fully specified
multivariate normal copula-based models. An illustrative example is given to
show the use of our simulated likelihood method
Mixed models for longitudinal left-censored repeated measures
Longitudinal studies could be complicated by left-censored repeated measures.
For example, in Human Immunodeficiency Virus infection, there is a detection
limit of the assay used to quantify the plasma viral load. Simple imputation of
the limit of the detection or of half of this limit for left-censored measures
biases estimations and their standard errors. In this paper, we review two
likelihood-based methods proposed to handle left-censoring of the outcome in
linear mixed model. We show how to fit these models using SAS Proc NLMIXED and
we compare this tool with other programs. Indications and limitations of the
programs are discussed and an example in the field of HIV infection is shown
A New Monte Carlo Based Algorithm for the Gaussian Process Classification Problem
Gaussian process is a very promising novel technology that has been applied
to both the regression problem and the classification problem. While for the
regression problem it yields simple exact solutions, this is not the case for
the classification problem, because we encounter intractable integrals. In this
paper we develop a new derivation that transforms the problem into that of
evaluating the ratio of multivariate Gaussian orthant integrals. Moreover, we
develop a new Monte Carlo procedure that evaluates these integrals. It is based
on some aspects of bootstrap sampling and acceptancerejection. The proposed
approach has beneficial properties compared to the existing Markov Chain Monte
Carlo approach, such as simplicity, reliability, and speed
Optimized recentered confidence spheres for the multivariate normal mean
Casella and Hwang, 1983, JASA, introduced a broad class of recentered
confidence spheres for the mean of a multivariate normal
distribution with covariance matrix , for
known. Both the center and radius functions of these confidence spheres are
flexible functions of the data. For the particular case of confidence spheres
centered on the positive-part James-Stein estimator and with radius determined
by empirical Bayes considerations, they show numerically that these confidence
spheres have the desired minimum coverage probability and dominate
the usual confidence sphere in terms of scaled volume. We shift the focus from
the scaled volume to the scaled expected volume of the recentered confidence
sphere. Since both the coverage probability and the scaled expected volume are
functions of the Euclidean norm of , it is feasible to
optimize the performance of the recentered confidence sphere by numerically
computing both the center and radius functions so as to optimize some clearly
specified criterion. We suppose that we have uncertain prior information that
. This motivates us to determine the
center and radius functions of the confidence sphere by numerical minimization
of the scaled expected volume of the confidence sphere at , subject to the constraints that (a) the coverage probability
never falls below and (b) the radius never exceeds the radius of the
standard confidence sphere. Our results show that, by focusing on
this clearly specified criterion, significant gains in performance (in terms of
this criterion) can be achieved. We also present analogous results for the much
more difficult case that is unknown.Comment: arXiv admin note: text overlap with arXiv:1306.241
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