124,927 research outputs found
Computing the Gamma function using contour integrals and rational approximations
Some of the best methods for computing the gamma function are based on numerical evaluation of Hankel's contour integral. For example, Temme evaluates this integral based on steepest-decent contours by the trapezoid rule. Here we investigate a different approach to the integral: the application of the trapezoid rule on Talbot-type contours using optimal parameters recently derived by Weideman for computing inverse Laplace transforms. Relatedly, we also investigate quadrature formulas derived from best approximations to exp(z) on the negative real axis, following Cody, Meinardus and Varga. The two methods are closely related and both converge geometrically. We find that the new methods are competitive with existing ones, even though they are based on generic tools rather than on specific analysis of the gamma function
Black Box Variational Inference
Variational inference has become a widely used method to approximate
posteriors in complex latent variables models. However, deriving a variational
inference algorithm generally requires significant model-specific analysis, and
these efforts can hinder and deter us from quickly developing and exploring a
variety of models for a problem at hand. In this paper, we present a "black
box" variational inference algorithm, one that can be quickly applied to many
models with little additional derivation. Our method is based on a stochastic
optimization of the variational objective where the noisy gradient is computed
from Monte Carlo samples from the variational distribution. We develop a number
of methods to reduce the variance of the gradient, always maintaining the
criterion that we want to avoid difficult model-based derivations. We evaluate
our method against the corresponding black box sampling based methods. We find
that our method reaches better predictive likelihoods much faster than sampling
methods. Finally, we demonstrate that Black Box Variational Inference lets us
easily explore a wide space of models by quickly constructing and evaluating
several models of longitudinal healthcare data
parallelMCMCcombine: An R Package for Bayesian Methods for Big Data and Analytics
Recent advances in big data and analytics research have provided a wealth of
large data sets that are too big to be analyzed in their entirety, due to
restrictions on computer memory or storage size. New Bayesian methods have been
developed for large data sets that are only large due to large sample sizes;
these methods partition big data sets into subsets, and perform independent
Bayesian Markov chain Monte Carlo analyses on the subsets. The methods then
combine the independent subset posterior samples to estimate a posterior
density given the full data set. These approaches were shown to be effective
for Bayesian models including logistic regression models, Gaussian mixture
models and hierarchical models. Here, we introduce the R package
parallelMCMCcombine which carries out four of these techniques for combining
independent subset posterior samples. We illustrate each of the methods using a
Bayesian logistic regression model for simulation data and a Bayesian Gamma
model for real data; we also demonstrate features and capabilities of the R
package. The package assumes the user has carried out the Bayesian analysis and
has produced the independent subposterior samples outside of the package. The
methods are primarily suited to models with unknown parameters of fixed
dimension that exist in continuous parameter spaces. We envision this tool will
allow researchers to explore the various methods for their specific
applications, and will assist future progress in this rapidly developing field.Comment: for published version see:
http://www.plosone.org/article/fetchObject.action?uri=info%3Adoi%2F10.1371%2Fjournal.pone.0108425&representation=PD
Fast and scalable non-parametric Bayesian inference for Poisson point processes
We study the problem of non-parametric Bayesian estimation of the intensity
function of a Poisson point process. The observations are independent
realisations of a Poisson point process on the interval . We propose two
related approaches. In both approaches we model the intensity function as
piecewise constant on bins forming a partition of the interval . In
the first approach the coefficients of the intensity function are assigned
independent gamma priors, leading to a closed form posterior distribution. On
the theoretical side, we prove that as the posterior
asymptotically concentrates around the "true", data-generating intensity
function at an optimal rate for -H\"older regular intensity functions (). In the second approach we employ a gamma Markov chain prior on the
coefficients of the intensity function. The posterior distribution is no longer
available in closed form, but inference can be performed using a
straightforward version of the Gibbs sampler. Both approaches scale well with
sample size, but the second is much less sensitive to the choice of .
Practical performance of our methods is first demonstrated via synthetic data
examples. We compare our second method with other existing approaches on the UK
coal mining disasters data. Furthermore, we apply it to the US mass shootings
data and Donald Trump's Twitter data.Comment: 45 pages, 22 figure
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