757 research outputs found
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
Conditionally conjugate mean-field variational Bayes for logistic models
Variational Bayes (VB) is a common strategy for approximate Bayesian
inference, but simple methods are only available for specific classes of models
including, in particular, representations having conditionally conjugate
constructions within an exponential family. Models with logit components are an
apparently notable exception to this class, due to the absence of conjugacy
between the logistic likelihood and the Gaussian priors for the coefficients in
the linear predictor. To facilitate approximate inference within this widely
used class of models, Jaakkola and Jordan (2000) proposed a simple variational
approach which relies on a family of tangent quadratic lower bounds of logistic
log-likelihoods, thus restoring conjugacy between these approximate bounds and
the Gaussian priors. This strategy is still implemented successfully, but less
attempts have been made to formally understand the reasons underlying its
excellent performance. To cover this key gap, we provide a formal connection
between the above bound and a recent P\'olya-gamma data augmentation for
logistic regression. Such a result places the computational methods associated
with the aforementioned bounds within the framework of variational inference
for conditionally conjugate exponential family models, thereby allowing recent
advances for this class to be inherited also by the methods relying on Jaakkola
and Jordan (2000)
Bayesian dynamic financial networks with time-varying predictors
We propose a Bayesian nonparametric model including time-varying predictors
in dynamic network inference. The model is applied to infer the dependence
structure among financial markets during the global financial crisis,
estimating effects of verbal and material cooperation efforts. We interestingly
learn contagion effects, with increasing influence of verbal relations during
the financial crisis and opposite results during the United States housing
bubble
Nonparametric Bayes dynamic modeling of relational data
Symmetric binary matrices representing relations among entities are commonly
collected in many areas. Our focus is on dynamically evolving binary relational
matrices, with interest being in inference on the relationship structure and
prediction. We propose a nonparametric Bayesian dynamic model, which reduces
dimensionality in characterizing the binary matrix through a lower-dimensional
latent space representation, with the latent coordinates evolving in continuous
time via Gaussian processes. By using a logistic mapping function from the
probability matrix space to the latent relational space, we obtain a flexible
and computational tractable formulation. Employing P\`olya-Gamma data
augmentation, an efficient Gibbs sampler is developed for posterior
computation, with the dimension of the latent space automatically inferred. We
provide some theoretical results on flexibility of the model, and illustrate
performance via simulation experiments. We also consider an application to
co-movements in world financial markets
Locally Adaptive Dynamic Networks
Our focus is on realistically modeling and forecasting dynamic networks of
face-to-face contacts among individuals. Important aspects of such data that
lead to problems with current methods include the tendency of the contacts to
move between periods of slow and rapid changes, and the dynamic heterogeneity
in the actors' connectivity behaviors. Motivated by this application, we
develop a novel method for Locally Adaptive DYnamic (LADY) network inference.
The proposed model relies on a dynamic latent space representation in which
each actor's position evolves in time via stochastic differential equations.
Using a state space representation for these stochastic processes and
P\'olya-gamma data augmentation, we develop an efficient MCMC algorithm for
posterior inference along with tractable procedures for online updating and
forecasting of future networks. We evaluate performance in simulation studies,
and consider an application to face-to-face contacts among individuals in a
primary school
The Compass for Statistical Researchers
We have hiked many miles alongside several professors as we traversed our
statistical path -- a regime switching trail which changed direction following
a class on the foundations of our discipline. As we play the game of research
in that limbo between student and academic, one thing among Prof. Bernardi's
teachings has never been more clear: to draw a route in the research map you
not only need to know your destination, but you must also understand where you
are and how you arrived there
A nested expectation-maximization algorithm for latent class models with covariates
We develop a nested EM routine for latent class models with covariates which
allows maximization of the full-model log-likelihood and, differently from
current methods, guarantees monotone log-likelihood sequences along with
improved convergence rates
Bayesian inference on group differences in multivariate categorical data
Multivariate categorical data are common in many fields. We are motivated by
election polls studies assessing evidence of changes in voters opinions with
their candidates preferences in the 2016 United States Presidential primaries
or caucuses. Similar goals arise routinely in several applications, but current
literature lacks a general methodology which combines flexibility, efficiency,
and tractability in testing for group differences in multivariate categorical
data at different---potentially complex---scales. We address this goal by
leveraging a Bayesian representation which factorizes the joint probability
mass function for the group variable and the multivariate categorical data as
the product of the marginal probabilities for the groups, and the conditional
probability mass function of the multivariate categorical data, given the group
membership. To enhance flexibility, we define the conditional probability mass
function of the multivariate categorical data via a group-dependent mixture of
tensor factorizations, thus facilitating dimensionality reduction and borrowing
of information, while providing tractable procedures for computation, and
accurate tests assessing global and local group differences. We compare our
methods with popular competitors, and discuss improved performance in
simulations and in American election polls studies
The gravity fields of Jupiter and Saturn as determined by Juno and Cassini
This Doctoral Thesis reports on the analysis procedure for the data acquired by Juno and Cassini in the last year with the aim of determining, respectively, Jupiter and Saturn’s gravity fields. The radio science investigation exploits the Doppler shift of a microwave signal to precisely determine the Earth-Spacecraft radial velocity and to estimate the gravity field harmonic coefficients, along with other dynamical parameters. The basic concepts of Doppler data and orbit determination are reported. A discussion on the esti- mated gravity fields is presented, as well as the implications for the interior structure of the Solar System’s gas giants
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