5,339 research outputs found
Approximating Probability Densities by Iterated Laplace Approximations
The Laplace approximation is an old, but frequently used method to
approximate integrals for Bayesian calculations. In this paper we develop an
extension of the Laplace approximation, by applying it iteratively to the
residual, i.e., the difference between the current approximation and the true
function. The final approximation is thus a linear combination of multivariate
normal densities, where the coefficients are chosen to achieve a good fit to
the target distribution. We illustrate on real and artificial examples that the
proposed procedure is a computationally efficient alternative to current
approaches for approximation of multivariate probability densities. The
R-package iterLap implementing the methods described in this article is
available from the CRAN servers.Comment: to appear in Journal of Computational and Graphical Statistics,
http://pubs.amstat.org/loi/jcg
Conjugate Bayes for probit regression via unified skew-normal distributions
Regression models for dichotomous data are ubiquitous in statistics. Besides
being useful for inference on binary responses, these methods serve also as
building blocks in more complex formulations, such as density regression,
nonparametric classification and graphical models. Within the Bayesian
framework, inference proceeds by updating the priors for the coefficients,
typically set to be Gaussians, with the likelihood induced by probit or logit
regressions for the responses. In this updating, the apparent absence of a
tractable posterior has motivated a variety of computational methods, including
Markov Chain Monte Carlo routines and algorithms which approximate the
posterior. Despite being routinely implemented, Markov Chain Monte Carlo
strategies face mixing or time-inefficiency issues in large p and small n
studies, whereas approximate routines fail to capture the skewness typically
observed in the posterior. This article proves that the posterior distribution
for the probit coefficients has a unified skew-normal kernel, under Gaussian
priors. Such a novel result allows efficient Bayesian inference for a wide
class of applications, especially in large p and small-to-moderate n studies
where state-of-the-art computational methods face notable issues. These
advances are outlined in a genetic study, and further motivate the development
of a wider class of conjugate priors for probit models along with methods to
obtain independent and identically distributed samples from the unified
skew-normal posterior
The joint projected normal and skew-normal: a distribution for poly-cylindrical data
The contribution of this work is the introduction of a multivariate
circular-linear (or poly- cylindrical) distribution obtained by combining the
projected and the skew-normal. We show the flexibility of our proposal, its
property of closure under marginalization and how to quantify multivariate
dependence. Due to a non-identifiability issue that our proposal inherits from
the projected normal, a compu- tational problem arises. We overcome it in a
Bayesian framework, adding suitable latent variables and showing that posterior
samples can be obtained with a post-processing of the estimation algo- rithm
output. Under specific prior choices, this approach enables us to implement a
Markov chain Monte Carlo algorithm relying only on Gibbs steps, where the
updates of the parameters are done as if we were working with a multivariate
normal likelihood. The proposed approach can be also used with the projected
normal. As a proof of concept, on simulated examples we show the ability of our
algorithm in recovering the parameters values and to solve the identification
problem. Then the proposal is used in a real data example, where the
turning-angles (circular variables) and the logarithm of the step-lengths
(linear variables) of four zebras are jointly modelled
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