583 research outputs found
Efficient computational strategies for doubly intractable problems with applications to Bayesian social networks
Powerful ideas recently appeared in the literature are adjusted and combined
to design improved samplers for Bayesian exponential random graph models.
Different forms of adaptive Metropolis-Hastings proposals (vertical, horizontal
and rectangular) are tested and combined with the Delayed rejection (DR)
strategy with the aim of reducing the variance of the resulting Markov chain
Monte Carlo estimators for a given computational time. In the examples treated
in this paper the best combination, namely horizontal adaptation with delayed
rejection, leads to a variance reduction that varies between 92% and 144%
relative to the adaptive direction sampling approximate exchange algorithm of
Caimo and Friel (2011). These results correspond to an increased performance
which varies from 10% to 94% if we take simulation time into account. The
highest improvements are obtained when highly correlated posterior
distributions are considered.Comment: 23 pages, 8 figures. Accepted to appear in Statistics and Computin
Bayesian model selection for exponential random graph models via adjusted pseudolikelihoods
Models with intractable likelihood functions arise in areas including network
analysis and spatial statistics, especially those involving Gibbs random
fields. Posterior parameter es timation in these settings is termed a
doubly-intractable problem because both the likelihood function and the
posterior distribution are intractable. The comparison of Bayesian models is
often based on the statistical evidence, the integral of the un-normalised
posterior distribution over the model parameters which is rarely available in
closed form. For doubly-intractable models, estimating the evidence adds
another layer of difficulty. Consequently, the selection of the model that best
describes an observed network among a collection of exponential random graph
models for network analysis is a daunting task. Pseudolikelihoods offer a
tractable approximation to the likelihood but should be treated with caution
because they can lead to an unreasonable inference. This paper specifies a
method to adjust pseudolikelihoods in order to obtain a reasonable, yet
tractable, approximation to the likelihood. This allows implementation of
widely used computational methods for evidence estimation and pursuit of
Bayesian model selection of exponential random graph models for the analysis of
social networks. Empirical comparisons to existing methods show that our
procedure yields similar evidence estimates, but at a lower computational cost.Comment: Supplementary material attached. To view attachments, please download
and extract the gzzipped source file listed under "Other formats
Noisy Hamiltonian Monte Carlo for doubly-intractable distributions
Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the
statistician's toolbox as an alternative sampling method in settings when
standard Metropolis-Hastings is inefficient. HMC generates a Markov chain on an
augmented state space with transitions based on a deterministic differential
flow derived from Hamiltonian mechanics. In practice, the evolution of
Hamiltonian systems cannot be solved analytically, requiring numerical
integration schemes. Under numerical integration, the resulting approximate
solution no longer preserves the measure of the target distribution, therefore
an accept-reject step is used to correct the bias. For doubly-intractable
distributions -- such as posterior distributions based on Gibbs random fields
-- HMC suffers from some computational difficulties: computation of gradients
in the differential flow and computation of the accept-reject proposals poses
difficulty. In this paper, we study the behaviour of HMC when these quantities
are replaced by Monte Carlo estimates
Bayesian exponential random graph modelling of interhospital patient referral networks
Using original data that we have collected on referral relations between 110 hospitals serving a large regional community, we show how recently derived Bayesian exponential random graph models may be adopted to illuminate core empirical issues in research on relational coordination among health care organisations. We show how a rigorous Bayesian computation approach supports a fully probabilistic analytical framework that alleviates well-known problems in the estimation of model parameters of exponential random graph models. We also show how the main structural features of interhospital patient referral networks that prior studies have described, can be reproduced with accuracy by specifying the system of local dependencies that produce – but at the same time are induced by – decentralised collaborative arrangements between hospitals
Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels
Monte Carlo algorithms often aim to draw from a distribution by
simulating a Markov chain with transition kernel such that is
invariant under . However, there are many situations for which it is
impractical or impossible to draw from the transition kernel . For instance,
this is the case with massive datasets, where is it prohibitively expensive to
calculate the likelihood and is also the case for intractable likelihood models
arising from, for example, Gibbs random fields, such as those found in spatial
statistics and network analysis. A natural approach in these cases is to
replace by an approximation . Using theory from the stability of
Markov chains we explore a variety of situations where it is possible to
quantify how 'close' the chain given by the transition kernel is to
the chain given by . We apply these results to several examples from spatial
statistics and network analysis.Comment: This version: results extended to non-uniformly ergodic Markov chain
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