Bayesian inference for models that have an intractable partition function is known as a doubly intractable problem, where standard Monte Carlo methods are not applicable. The past decade has seen the development of auxiliary variable Monte Carlo techniques (Møller et al., 2006; Murray et al., 2006) for tackling this problem; these approaches being members of the more general class of pseudo-marginal, or exact-approximate, Monte Carlo algorithms (Andrieu and Roberts, 2009), which make use of unbiased estimates of intractable posteriors. Everitt et al. (2017) investigated the use of exact-approximate importance sampling (IS) and sequential Monte Carlo (SMC) in doubly intractable problems, but focussed only on SMC algorithms that used data-point tempering. This paper describes SMC samplers that may use alternative sequences of distributions, and describes ways in which likelihood estimates may be improved adaptively as the algorithm progresses, building on ideas from Moores et al. (2015). This approach is compared with a number of alternative algorithms for doubly intractable problems, including approximate Bayesian computation (ABC), which we show is closely related to the method of Møller et al. (2006).
Markov chain Monte Carlo (MCMC) algorithms are widely used to sample from complicated distributions, especially to sample from the posterior distribution in Bayesian inference. However, MCMC is not directly applicable when facing the doubly intractable problem. In this paper, we discussed and compared two existing solutions -- Pseudo-marginal Monte Carlo and Exchange Algorithm. This paper also proposes a novel algorithm: Multi-armed Bandit MCMC (MABMC), which chooses between two (or more) randomized acceptance ratios in each step. MABMC could be applied directly to incorporate Pseudo-marginal …
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 (Moeller 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. Unfo…
Stanford University（斯坦福大学）；University of Edinburgh（爱丁堡大学）
Computing partition functions, the normalizing constants of probability distributions, is often hard. Variants of importance sampling give unbiased estimates of a normalizer Z, however, unbiased estimates of the reciprocal 1/Z are harder to obtain. Unbiased estimates of 1/Z allow Markov chain Monte Carlo sampling of "doubly-intractable" distributions, such as the parameter posterior for Markov Random Fields or Exponential Random Graphs. We demonstrate how to construct unbiased estimates for 1/Z given access to black-box importance sampling estimators for Z. We adapt recent work on random serie…
Bayesian inference in the presence of an intractable likelihood function is computationally challenging. When following a Markov chain Monte Carlo (MCMC) approach to approximate the posterior distribution in this context, one typically either uses MCMC schemes which target the joint posterior of the parameters and some auxiliary latent variables, or pseudo-marginal Metropolis--Hastings (MH) schemes. The latter mimic a MH algorithm targeting the marginal posterior of the parameters by approximating unbiasedly the intractable likelihood. However, in scenarios where the parameters and auxiliary v…
Gaussian processes (GPs) are frequently used in machine learning and statistics to construct powerful models. However, when employing GPs in practice, important considerations must be made, regarding the high computational burden, approximation of the posterior, choice of the covariance function and inference of its hyperparmeters. To address these issues, Hensman et al. (2015) combine variationally sparse GPs with Markov chain Monte Carlo (MCMC) to derive a scalable, flexible and general framework for GP models. Nevertheless, the resulting approach requires intractable likelihood evaluations …
We consider online computation of expectations of additive state functionals under general path probability measures proportional to products of unnormalised transition densities. These transition densities are assumed to be intractable but possible to estimate, with or without bias. Using pseudo-marginalisation techniques we are able to extend the particle-based, rapid incremental smoother (PaRIS) algorithm proposed in [J.Olsson and J.Westerborn. Efficient particle-based online smoothing in general hidden Markov models: The PaRIS algorithm. Bernoulli, 23(3):1951--1996, 2017] to this setting. …
Intractable posterior distributions of parameters with intractable normalizing constants depending upon the parameters are known as doubly intractable posterior distributions. The terminology itself indicates that obtaining Bayesian inference from such posteriors is doubly difficult compared to traditional intractable posteriors where the normalizing constants are tractable and admit traditional Markov Chain Monte Carlo (MCMC) solutions. As can be anticipated, a plethora of MCMC-based methods have originated in the literature to deal with doubly intractable distributions. Yet, it remains very …
Samsung AI Center Moscow（三星人工智能中心莫斯科）；National Research University Higher School of Economics（国立研究型高等经济大学）
We extend the existing framework of semi-implicit variational inference (SIVI) and introduce doubly semi-implicit variational inference (DSIVI), a way to perform variational inference and learning when both the approximate posterior and the prior distribution are semi-implicit. In other words, DSIVI performs inference in models where the prior and the posterior can be expressed as an intractable infinite mixture of some analytic density with a highly flexible implicit mixing distribution. We provide a sandwich bound on the evidence lower bound (ELBO) objective that can be made arbitrarily tigh…
For many stochastic models of interest in systems biology, such as those describing biochemical reaction networks, exact quantification of parameter uncertainty through statistical inference is intractable. Likelihood-free computational inference techniques enable parameter inference when the likelihood function for the model is intractable but the generation of many sample paths is feasible through stochastic simulation of the forward problem. The most common likelihood-free method in systems biology is approximate Bayesian computation that accepts parameters that result in low discrepancy be…
Parameter inference for stochastic differential equation mixed effects models (SDEMEMs) is a challenging problem. Analytical solutions for these models are rarely available, which means that the likelihood is also intractable. In this case, exact inference is possible using the pseudo-marginal method, where the intractable likelihood is replaced by its nonnegative unbiased estimate. A useful application of this idea is particle MCMC, which uses a particle filter estimate of the likelihood. While the exact posterior is targeted by these methods, a naive implementation for SDEMEMs can be highly …
Recent advances in Markov chain Monte Carlo (MCMC) extend the scope of Bayesian inference to models for which the likelihood function is intractable. Although these developments allow us to estimate model parameters, other basic problems such as estimating the marginal likelihood, a fundamental tool in Bayesian model selection, remain challenging. This is an important scientific limitation because testing psychological hypotheses with hierarchical models has proven difficult with current model selection methods. We propose an efficient method for estimating the marginal likelihood for models w…
Variational Bayes (VB) is rapidly becoming a popular tool for Bayesian inference in statistical modeling. However, the existing VB algorithms are restricted to cases where the likelihood is tractable, which precludes the use of VB in many interesting situations such as in state space models and in approximate Bayesian computation (ABC), where application of VB methods was previously impossible. This paper extends the scope of application of VB to cases where the likelihood is intractable, but can be estimated unbiasedly. The proposed VB method therefore makes it possible to carry out Bayesian …
A large number of statistical models are "doubly-intractable": the likelihood normalising term, which is a function of the model parameters, is intractable, as well as the marginal likelihood (model evidence). This means that standard inference techniques to sample from the posterior, such as Markov chain Monte Carlo (MCMC), cannot be used. Examples include, but are not confined to, massive Gaussian Markov random fields, autologistic models and Exponential random graph models. A number of approximate schemes based on MCMC techniques, Approximate Bayesian computation (ABC) or analytic approxima…
We consider the efficient use of an approximation within Markov chain Monte Carlo (MCMC), with subsequent importance sampling (IS) correction of the Markov chain inexact output, leading to asymptotically exact inference. We detail convergence and central limit theorems for the resulting MCMC-IS estimators. We also consider the case where the approximate Markov chain is pseudo-marginal, requiring unbiased estimators for its approximate marginal target. Convergence results with asymptotic variance formulae are shown for this case, and for the case where the IS weights based on unbiased estimator…
We study the existence of algorithms generating almost surely nonnegative unbiased estimators. We show that given a nonconstant real-valued function $f$ and a sequence of unbiased estimators of $λ\in\mathbb{R}$, there is no algorithm yielding almost surely nonnegative unbiased estimators of $f(λ)\in\mathbb{R}^+$. The study is motivated by pseudo-marginal Monte Carlo algorithms that rely on such nonnegative unbiased estimators. These methods allow "exact inference" in intractable models, in the sense that integrals with respect to a target distribution can be estimated without any systematic er…
Performing numerical integration when the integrand itself cannot be evaluated point-wise is a challenging task that arises in statistical analysis, notably in Bayesian inference for models with intractable likelihood functions. Markov chain Monte Carlo (MCMC) algorithms have been proposed for this setting, such as the pseudo-marginal method for latent variable models and the exchange algorithm for a class of undirected graphical models. As with any MCMC algorithm, the resulting estimators are justified asymptotically in the limit of the number of iterations, but exhibit a bias for any fixed n…
Bayesian inference of Gibbs random fields (GRFs) is often referred to as a doubly intractable problem, since the likelihood function is intractable. The exploration of the posterior distribution of such models is typically carried out with a sophisticated Markov chain Monte Carlo (MCMC) method, the exchange algorithm (Murray et al., 2006), which requires simulations from the likelihood function at each iteration. The purpose of this paper is to consider an approach to dramatically reduce this computational overhead. To this end we introduce a novel class of algorithms which use realizations of…
Bayesian inference in state-space models is challenging due to high-dimensional state trajectories. A viable approach is particle Markov chain Monte Carlo, combining MCMC and sequential Monte Carlo to form "exact approximations" to otherwise intractable MCMC methods. The performance of the approximation is limited to that of the exact method. We focus on particle Gibbs and particle Gibbs with ancestor sampling, improving their performance beyond that of the underlying Gibbs sampler (which they approximate) by marginalizing out one or more parameters. This is possible when the parameter prior i…
Google Brain（谷歌大脑）；New York University（纽约大学）；University of Oxford（牛津大学）；DeepMind（深度思维）
Deep latent variable models have become a popular model choice due to the scalable learning algorithms introduced by (Kingma & Welling, 2013; Rezende et al., 2014). These approaches maximize a variational lower bound on the intractable log likelihood of the observed data. Burda et al. (2015) introduced a multi-sample variational bound, IWAE, that is at least as tight as the standard variational lower bound and becomes increasingly tight as the number of samples increases. Counterintuitively, the typical inference network gradient estimator for the IWAE bound performs poorly as the number of sa…
In this comment we consider whether QMC methods can be further embedded within SMC schemes in settings in which the transition density of the latent process is intractable and pseudo-marginal methods are deployed.
Stochastic approximation methods play a central role in maximum likelihood estimation problems involving intractable likelihood functions, such as marginal likelihoods arising in problems with missing or incomplete data, and in parametric empirical Bayesian estimation. Combined with Markov chain Monte Carlo algorithms, these stochastic optimisation methods have been successfully applied to a wide range of problems in science and industry. However, this strategy scales poorly to large problems because of methodological and theoretical difficulties related to using high-dimensional Markov chain …
We investigate nonlinear state-space models without a closed-form transition density, and propose reformulating such models over their latent noise variables rather than their latent state variables. In doing so the tractable noise density emerges in place of the intractable transition density. For importance sampling methods such as the auxiliary particle filter, this enables importance weights to be computed where they could not be otherwise. As case studies we take two multivariate marine biogeochemical models and perform state and parameter estimation using the particle marginal Metropolis…
Doubly intractable distributions arise in many settings, for example in Markov models for point processes and exponential random graph models for networks. Bayesian inference for these models is challenging because they involve intractable normalising "constants" that are actually functions of the parameters of interest. Although several clever computational methods have been developed for these models, each method suffers from computational issues that makes it computationally burdensome or even infeasible for many problems. We propose a novel algorithm that provides computational gains over …
In the following article we consider approximate Bayesian parameter inference for observation driven time series models. Such statistical models appear in a wide variety of applications, including econometrics and applied mathematics. This article considers the scenario where the likelihood function cannot be evaluated point-wise; in such cases, one cannot perform exact statistical inference, including parameter estimation, which often requires advanced computational algorithms, such as Markov chain Monte Carlo (MCMC). We introduce a new approximation based upon approximate Bayesian computatio…
Pseudo-marginal Markov chain Monte Carlo methods for sampling from intractable distributions have gained recent interest and have been theoretically studied in considerable depth. Their main appeal is that they are exact, in the sense that they target marginally the correct invariant distribution. However, the pseudo-marginal Markov chain can exhibit poor mixing and slow convergence towards its target. As an alternative, a subtly different Markov chain can be simulated, where better mixing is possible but the exactness property is sacrificed. This is the noisy algorithm, initially conceptualis…
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 9…
Markov chain Monte Carlo is a class of algorithms for drawing Markovian samples from high-dimensional target densities to approximate the numerical integration associated with computing statistical expectation, especially in Bayesian statistics. However, many Markov chain Monte Carlo algorithms do not seem to share the same theoretical support and each algorithm is proven in a different way. This incurs many terminologies and ancillary concepts, which makes Markov chain Monte Carlo literature seems to be scattered and intimidating to researchers from many other fields, including new researcher…
Markov chain Monte Carlo (MCMC) is an established approach for uncertainty quantification and propagation in scientific applications. A key challenge in applying MCMC to scientific domains is computation: the target density of interest is often a function of expensive computations, such as a high-fidelity physical simulation, an intractable integral, or a slowly-converging iterative algorithm. Thus, using an MCMC algorithms with an expensive target density becomes impractical, as these expensive computations need to be evaluated at each iteration of the algorithm. In practice, these computatio…
Bayesian Likelihood-Free Inference (LFI) approaches allow to obtain posterior distributions for stochastic models with intractable likelihood, by relying on model simulations. In Approximate Bayesian Computation (ABC), a popular LFI method, summary statistics are used to reduce data dimensionality. ABC algorithms adaptively tailor simulations to the observation in order to sample from an approximate posterior, whose form depends on the chosen statistics. In this work, we introduce a new way to learn ABC statistics: we first generate parameter-simulation pairs from the model independently on th…
In this paper, a multivariate count distribution with Conway-Maxwell (COM)-Poisson marginals is proposed. To do this, we develop a modification of the Sarmanov method for constructing multivariate distributions. Our multivariate COM-Poisson (MultCOMP) model has desirable features such as (i) it admits a flexible covariance matrix allowing for both negative and positive non-diagonal entries; (ii) it overcomes the limitation of the existing bivariate COM-Poisson distributions in the literature that do not have COM-Poisson marginals; (iii) it allows for the analysis of multivariate counts and is …
Performing inference over simulators is generally intractable as their runtime means we cannot compute a marginal likelihood. We develop a likelihood-free inference method to infer parameters for a cardiac simulator, which replicates electrical flow through the heart to the body surface. We improve the fit of a state-of-the-art simulator to an electrocardiogram (ECG) recorded from a real patient.
The pseudo-marginal algorithm is a popular variant of the Metropolis--Hastings scheme which allows us to sample asymptotically from a target probability density $π$, when we are only able to estimate an unnormalized version of $π$ pointwise unbiasedly. It has found numerous applications in Bayesian statistics as there are many scenarios where the likelihood function is intractable but can be estimated unbiasedly using Monte Carlo samples. Using many samples will typically result in averages computed under this chain with lower asymptotic variances than the corresponding averages that use fewer…
University of Liège（列日大学）；University of Vienna（维也纳大学）
Posterior inference with an intractable likelihood is becoming an increasingly common task in scientific domains which rely on sophisticated computer simulations. Typically, these forward models do not admit tractable densities forcing practitioners to make use of approximations. This work introduces a novel approach to address the intractability of the likelihood and the marginal model. We achieve this by learning a flexible amortized estimator which approximates the likelihood-to-evidence ratio. We demonstrate that the learned ratio estimator can be embedded in MCMC samplers to approximate l…
This article addresses the problem of efficient Bayesian inference in dynamic systems using particle methods and makes a number of contributions. First, we develop a correlated pseudo-marginal (CPM) approach for Bayesian inference in state space (SS) models that is based on filtering the disturbances, rather than the states. This approach is useful when the state transition density is intractable or inefficient to compute, and also when the dimension of the disturbance is lower than the dimension of the state. Second, we propose a block pseudo-marginal (BPM) method that uses as the estimate of…
ETH Zurich（苏黎世联邦理工学院）；Swiss Data Science Center（瑞士数据科学中心）
Annealed Importance Sampling (AIS) is a popular algorithm used to estimates the intractable marginal likelihood of deep generative models. Although AIS is guaranteed to provide unbiased estimate for any set of hyperparameters, the common implementations rely on simple heuristics such as the geometric average bridging distributions between initial and the target distribution which affect the estimation performance when the computation budget is limited. In order to reduce the number of sampling iterations, we present a parameteric AIS process with flexible intermediary distributions defined by …
Inference for doubly intractable distributions is challenging because the intractable normalizing functions of these models include parameters of interest. Previous auxiliary variable MCMC algorithms are infeasible for multi-dimensional models with large data sets because they depend on expensive auxiliary variable simulation at each iteration. We develop a fast Bayesian indirect algorithm by replacing an expensive auxiliary variable simulation from a probability model with a computationally cheap simulation from a surrogate model. We learn the relationship between the surrogate model paramete…
In this expository paper we abstract and describe a simple MCMC scheme for sampling from intractable target densities. The approach has been introduced in Gonçalves et al. (2017a) in the specific context of jump-diffusions, and is based on the Barker's algorithm paired with a simple Bernoulli factory type scheme, the so called 2-coin algorithm. In many settings it is an alternative to standard Metropolis-Hastings pseudo-marginal method for simulating from intractable target densities. Although Barker's is well-known to be slightly less efficient than Metropolis-Hastings, the key advantage of o…
In the present article we derive an explicit expression for the trun- cated mean and variance for the multivariate normal distribution with ar- bitrary rectangular double truncation. We use the moment generating ap- proach of Tallis (1961) and extend it to general μ, Σ and all combinations of truncation. As part of the solution we also give a formula for the bivari- ate marginal density of truncated multinormal variates. We also prove an invariance property of some elements of the inverse covariance after trunca- tion. Computer algorithms for computing the truncated mean, variance and the biva…
This article surveys computational methods for posterior inference with intractable likelihoods, that is where the likelihood function is unavailable in closed form, or where evaluation of the likelihood is infeasible. We review recent developments in pseudo-marginal methods, approximate Bayesian computation (ABC), the exchange algorithm, thermodynamic integration, and composite likelihood, paying particular attention to advancements in scalability for large datasets. We also mention R and MATLAB source code for implementations of these algorithms, where they are available.
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 anot…
For multivariate nonparametric regression, doubly penalized ANOVA modeling (DPAM) has recently been proposed, using hierarchical total variations (HTVs) and empirical norms as penalties on the component functions such as main effects and multi-way interactions in a functional ANOVA decomposition of the underlying regression function. The two penalties play complementary roles: the HTV penalty promotes sparsity in the selection of basis functions within each component function, whereas the empirical-norm penalty promotes sparsity in the selection of component functions. We adopt backfitting or …
In this work we consider Bayesian inference problems with intractable likelihood functions. We present a method to compute an approximate of the posterior with a limited number of model simulations. The method features an inverse Gaussian Process regression (IGPR), i.e., one from the output of a simulation model to the input of it. Within the method, we provide an adaptive algorithm with a tempering procedure to construct the approximations of the marginal posterior distributions. With examples we demonstrate that IGPR has a competitive performance compared to some commonly used algorithms, es…
We present a data augmentation scheme to perform Markov chain Monte Carlo inference for models where data generation involves a rejection sampling algorithm. Our idea, which seems to be missing in the literature, is a simple scheme to instantiate the rejected proposals preceding each data point. The resulting joint probability over observed and rejected variables can be much simpler than the marginal distribution over the observed variables, which often involves intractable integrals. We consider three problems, the first being the modeling of flow-cytometry measurements subject to truncation.…
