40,101 research outputs found
Approximate Bayesian Computation with composite score functions
Both Approximate Bayesian Computation (ABC) and composite likelihood methods
are useful for Bayesian and frequentist inference, respectively, when the
likelihood function is intractable. We propose to use composite likelihood
score functions as summary statistics in ABC in order to obtain accurate
approximations to the posterior distribution. This is motivated by the use of
the score function of the full likelihood, and extended to general unbiased
estimating functions in complex models. Moreover, we show that if the composite
score is suitably standardised, the resulting ABC procedure is invariant to
reparameterisations and automatically adjusts the curvature of the composite
likelihood, and of the corresponding posterior distribution. The method is
illustrated through examples with simulated data, and an application to
modelling of spatial extreme rainfall data is discussed.Comment: Statistics and Computing (final version
Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses
We investigate the relationship between the structure of a discrete graphical
model and the support of the inverse of a generalized covariance matrix. We
show that for certain graph structures, the support of the inverse covariance
matrix of indicator variables on the vertices of a graph reflects the
conditional independence structure of the graph. Our work extends results that
have previously been established only in the context of multivariate Gaussian
graphical models, thereby addressing an open question about the significance of
the inverse covariance matrix of a non-Gaussian distribution. The proof
exploits a combination of ideas from the geometry of exponential families,
junction tree theory and convex analysis. These population-level results have
various consequences for graph selection methods, both known and novel,
including a novel method for structure estimation for missing or corrupted
observations. We provide nonasymptotic guarantees for such methods and
illustrate the sharpness of these predictions via simulations.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1162 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Total positivity in exponential families with application to binary variables
We study exponential families of distributions that are multivariate totally
positive of order 2 (MTP2), show that these are convex exponential families,
and derive conditions for existence of the MLE. Quadratic exponential familes
of MTP2 distributions contain attractive Gaussian graphical models and
ferromagnetic Ising models as special examples. We show that these are defined
by intersecting the space of canonical parameters with a polyhedral cone whose
faces correspond to conditional independence relations. Hence MTP2 serves as an
implicit regularizer for quadratic exponential families and leads to sparsity
in the estimated graphical model. We prove that the maximum likelihood
estimator (MLE) in an MTP2 binary exponential family exists if and only if both
of the sign patterns and are represented in the sample for
every pair of variables; in particular, this implies that the MLE may exist
with observations, in stark contrast to unrestricted binary exponential
families where observations are required. Finally, we provide a novel and
globally convergent algorithm for computing the MLE for MTP2 Ising models
similar to iterative proportional scaling and apply it to the analysis of data
from two psychological disorders
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