MCMC for doubly-intractable distributions

Markov Chain Monte Carlo (MCMC) algorithms are routinely used to draw samples from distributions with intractable normalization constants. However, standard MCMC algorithms do not apply to doubly-intractable distributions in which there are additional parameter-dependent normalization terms; for example, the posterior over parameters of an undirected graphical model. An ingenious auxiliary-variable scheme (Møller et al., 2004) offers a solution: exact sampling (Propp and Wilson, 1996) is used to sample from a Metropolis–Hastings proposal for which the acceptance probability is tractable. Unfortunately the acceptance probability of these expensive updates can be low. This paper provides a generalization of Møller et al. (2004) and a new MCMC algorithm, which obtains better acceptance probabilities for the same amount of exact sampling, and removes the need to estimate model parameters before sampling begins

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

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

Markov Chain Monte Carlo Based on Deterministic Transformations

In this article we propose a novel MCMC method based on deterministic transformations T: X x D --> X where X is the state-space and D is some set which may or may not be a subset of X. We refer to our new methodology as Transformation-based Markov chain Monte Carlo (TMCMC). One of the remarkable advantages of our proposal is that even if the underlying target distribution is very high-dimensional, deterministic transformation of a one-dimensional random variable is sufficient to generate an appropriate Markov chain that is guaranteed to converge to the high-dimensional target distribution. Apart from clearly leading to massive computational savings, this idea of deterministically transforming a single random variable very generally leads to excellent acceptance rates, even though all the random variables associated with the high-dimensional target distribution are updated in a single block. Since it is well-known that joint updating of many random variables using Metropolis-Hastings (MH) algorithm generally leads to poor acceptance rates, TMCMC, in this regard, seems to provide a significant advance. We validate our proposal theoretically, establishing the convergence properties. Furthermore, we show that TMCMC can be very effectively adopted for simulating from doubly intractable distributions. TMCMC is compared with MH using the well-known Challenger data, demonstrating the effectiveness of of the former in the case of highly correlated variables. Moreover, we apply our methodology to a challenging posterior simulation problem associated with the geostatistical model of Diggle et al. (1998), updating 160 unknown parameters jointly, using a deterministic transformation of a one-dimensional random variable. Remarkable computational savings as well as good convergence properties and acceptance rates are the results.Comment: 28 pages, 3 figures; Longer abstract inside articl

Delayed acceptance ABC-SMC

Approximate Bayesian computation (ABC) is now an established technique for statistical inference used in cases where the likelihood function is computationally expensive or not available. It relies on the use of a~model that is specified in the form of a~simulator, and approximates the likelihood at a~parameter value $\theta$ by simulating auxiliary data sets $x$ and evaluating the distance of $x$ from the true data $y$. However, ABC is not computationally feasible in cases where using the simulator for each $\theta$ is very expensive. This paper investigates this situation in cases where a~cheap, but approximate, simulator is available. The approach is to employ delayed acceptance Markov chain Monte Carlo (MCMC) within an ABC sequential Monte Carlo (SMC) sampler in order to, in a~first stage of the kernel, use the cheap simulator to rule out parts of the parameter space that are not worth exploring, so that the ``true'' simulator is only run (in the second stage of the kernel) where there is a~reasonable chance of accepting proposed values of $\theta$. We show that this approach can be used quite automatically, with few tuning parameters. Applications to stochastic differential equation models and latent doubly intractable distributions are presented

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