34,512 research outputs found
Non-parametric Bayesian modeling of complex networks
Modeling structure in complex networks using Bayesian non-parametrics makes
it possible to specify flexible model structures and infer the adequate model
complexity from the observed data. This paper provides a gentle introduction to
non-parametric Bayesian modeling of complex networks: Using an infinite mixture
model as running example we go through the steps of deriving the model as an
infinite limit of a finite parametric model, inferring the model parameters by
Markov chain Monte Carlo, and checking the model's fit and predictive
performance. We explain how advanced non-parametric models for complex networks
can be derived and point out relevant literature
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
Bayesian Linear Regression
The paper is concerned with Bayesian analysis under prior-data conflict, i.e. the situation when observed data are rather unexpected under the prior (and the sample size is not large enough to eliminate the influence of the prior). Two approaches for Bayesian linear regression modeling based on conjugate priors are considered in detail, namely the standard approach also described in Fahrmeir, Kneib & Lang (2007) and an alternative adoption of the general construction procedure for exponential family sampling models. We recognize that - in contrast to some standard i.i.d. models like the scaled normal model and the Beta-Binomial / Dirichlet-Multinomial model, where prior-data conflict is completely ignored - the models may show some reaction to prior-data conflict, however in a rather unspecific way. Finally we briefly sketch the extension to a corresponding imprecise probability model, where, by considering sets of prior distributions instead of a single prior, prior-data conflict can be handled in a very appealing and intuitive way
Optimization Monte Carlo: Efficient and Embarrassingly Parallel Likelihood-Free Inference
We describe an embarrassingly parallel, anytime Monte Carlo method for
likelihood-free models. The algorithm starts with the view that the
stochasticity of the pseudo-samples generated by the simulator can be
controlled externally by a vector of random numbers u, in such a way that the
outcome, knowing u, is deterministic. For each instantiation of u we run an
optimization procedure to minimize the distance between summary statistics of
the simulator and the data. After reweighing these samples using the prior and
the Jacobian (accounting for the change of volume in transforming from the
space of summary statistics to the space of parameters) we show that this
weighted ensemble represents a Monte Carlo estimate of the posterior
distribution. The procedure can be run embarrassingly parallel (each node
handling one sample) and anytime (by allocating resources to the worst
performing sample). The procedure is validated on six experiments.Comment: NIPS 2015 camera read
Patterns of Scalable Bayesian Inference
Datasets are growing not just in size but in complexity, creating a demand
for rich models and quantification of uncertainty. Bayesian methods are an
excellent fit for this demand, but scaling Bayesian inference is a challenge.
In response to this challenge, there has been considerable recent work based on
varying assumptions about model structure, underlying computational resources,
and the importance of asymptotic correctness. As a result, there is a zoo of
ideas with few clear overarching principles.
In this paper, we seek to identify unifying principles, patterns, and
intuitions for scaling Bayesian inference. We review existing work on utilizing
modern computing resources with both MCMC and variational approximation
techniques. From this taxonomy of ideas, we characterize the general principles
that have proven successful for designing scalable inference procedures and
comment on the path forward
Non-linear regression models for Approximate Bayesian Computation
Approximate Bayesian inference on the basis of summary statistics is
well-suited to complex problems for which the likelihood is either
mathematically or computationally intractable. However the methods that use
rejection suffer from the curse of dimensionality when the number of summary
statistics is increased. Here we propose a machine-learning approach to the
estimation of the posterior density by introducing two innovations. The new
method fits a nonlinear conditional heteroscedastic regression of the parameter
on the summary statistics, and then adaptively improves estimation using
importance sampling. The new algorithm is compared to the state-of-the-art
approximate Bayesian methods, and achieves considerable reduction of the
computational burden in two examples of inference in statistical genetics and
in a queueing model.Comment: 4 figures; version 3 minor changes; to appear in Statistics and
Computin
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