arXiv subjects

Robert Kohn

Publications and source records attributed to Robert Kohn.

At least 73 records · Page 4Linked to original sources

arXiv2018

Speeding Up MCMC by Efficient Data Subsampling

We propose Subsampling MCMC, a Markov Chain Monte Carlo (MCMC) framework where the likelihood function for $n$ observations is estimated from a random subset of $m$ observations. We introduce a highly efficient unbiased estimator of the log-likelihood based on control variates, such that the computing cost is much smaller than that of the full log-likelihood in standard MCMC. The likelihood estimate is bias-corrected and used in two dependent pseudo-marginal algorithms to sample from a perturbed posterior, for which we derive the asymptotic error with respect to $n$ and $m$, respectively. We propose a practical estimator of the error and show that the error is negligible even for a very small $m$ in our applications. We demonstrate that Subsampling MCMC is substantially more efficient than standard MCMC in terms of sampling efficiency for a given computational budget, and that it outperforms other subsampling methods for MCMC proposed in the literature.

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arXiv2017

Efficient Bayesian estimation for flexible panel models for multivariate outcomes: Impact of life events on mental health and excessive alcohol consumption

The problem we consider considers estimating a multivariate longitudinal panel data model whose outcomes can be a combination of discrete and continuous variables. This problem is challenging because the likelihood is usually analytically intractable. Our article makes both a methodological contribution and also a substantive contribution to the application. The methodological contribution is to introduce into the panel data literature a particle Metropolis within Gibbs method to carry out Bayesian inference, using a Hamiltonian Monte Carlo (Neal 2011} proposal for sampling the vector of unknown parameters. We note that in panel data models the Our second contribution is to apply our method to carry out a serious analysis of the impact of serious life events on mental health and excessive alcohol consumption. The dependence between these two outcomes may be more pronounced when consumption of alcohol is excessive and mental health poor, which in turn has implications for how life events impact the joint distribution of the outcomes.

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arXiv2017

Mixed Marginal Copula Modeling

This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density of the observations with respect to a mixed measure. The treatment is quite general, although we focus focus on mixtures of Gaussian and Archimedean copulas. The inference is Bayesian with the estimation carried out by Markov chain Monte Carlo. We illustrate the methodology and algorithms by applying them to estimate a multivariate income dynamics model.

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arXiv2017

Scalable MCMC for Large Data Problems using Data Subsampling and the Difference Estimator

We propose a generic Markov Chain Monte Carlo (MCMC) algorithm to speed up computations for datasets with many observations. A key feature of our approach is the use of the highly efficient difference estimator from the survey sampling literature to estimate the log-likelihood accurately using only a small fraction of the data. Our algorithm improves on the $O(n)$ complexity of regular MCMC by operating over local data clusters instead of the full sample when computing the likelihood. The likelihood estimate is used in a Pseudo-marginal framework to sample from a perturbed posterior which is within $O(m^{-1/2})$ of the true posterior, where $m$ is the subsample size. The method is applied to a logistic regression model to predict firm bankruptcy for a large data set. We document a significant speed up in comparison to the standard MCMC on the full dataset.

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arXiv2017

Efficient Bayesian inference for multivariate factor stochastic volatility models with leverage

This paper discusses the efficient Bayesian estimation of a multivariate factor stochastic volatility (Factor MSV) model with leverage. We propose a novel approach to construct the sampling schemes that converges to the posterior distribution of the latent volatilities and the parameters of interest of the Factor MSV model based on recent advances in Particle Markov chain Monte Carlo (PMCMC). As opposed to the approach of Chib et al. (2006} and Omori et al. (2007}, our approach does not require approximating the joint distribution of outcome and volatility innovations by a mixture of bivariate normal distributions. To sample the free elements of the loading matrix we employ the interweaving method used in Kastner et al. (2017} in the Particle Metropolis within Gibbs (PMwG) step. The proposed method is illustrated empirically using a simulated dataset and a sample of daily US stock returns.

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arXiv2017

Fast Inference for Intractable Likelihood Problems using Variational Bayes

Variational Bayes (VB) is a popular estimation method for Bayesian inference. However, most existing VB algorithms are restricted to cases where the likelihood is tractable, which precludes their use in many important situations. Tran et al. (2017) extend the scope of application of VB to cases where the likelihood is intractable but can be estimated unbiasedly, and name the method Variational Bayes with Intractable Likelihood (VBIL). This paper presents a version of VBIL, named Variational Bayes with Intractable Log-Likelihood (VBILL), that is useful for cases as Big Data and Big Panel Data models, where unbiased estimators of the gradient of the log-likelihood are available. We demonstrate that such estimators can be easily obtained in many Big Data applications. The proposed method is exact in the sense that, apart from an extra Monte Carlo error which can be controlled, it is able to produce estimators as if the true likelihood, or full-data likelihood, is used. In particular, we develop a computationally efficient approach, based on data subsampling and the MapReduce programming technique, for analyzing massive datasets which cannot fit into the memory of a single desktop PC. We illustrate the method using several simulated datasets and a big real dataset based on the arrival time status of U. S. airlines.

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arXiv2017

Speeding Up MCMC by Delayed Acceptance and Data Subsampling

The complexity of the Metropolis-Hastings (MH) algorithm arises from the requirement of a likelihood evaluation for the full data set in each iteration. Payne and Mallick (2015) propose to speed up the algorithm by a delayed acceptance approach where the acceptance decision proceeds in two stages. In the first stage, an estimate of the likelihood based on a random subsample determines if it is likely that the draw will be accepted and, if so, the second stage uses the full data likelihood to decide upon final acceptance. Evaluating the full data likelihood is thus avoided for draws that are unlikely to be accepted. We propose a more precise likelihood estimator which incorporates auxiliary information about the full data likelihood while only operating on a sparse set of the data. We prove that the resulting delayed acceptance MH is more efficient compared to that of Payne and Mallick (2015). The caveat of this approach is that the full data set needs to be evaluated in the second stage. We therefore propose to substitute this evaluation by an estimate and construct a state-dependent approximation thereof to use in the first stage. This results in an algorithm that (i) can use a smaller subsample m by leveraging on recent advances in Pseudo-Marginal MH (PMMH) and (ii) is provably within $O(m^{-2})$ of the true posterior.

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arXiv2016

Variational Bayes with Intractable Likelihood

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 inference in many statistical applications, including state space models and ABC. The method is generic in the sense that it can be applied to almost all statistical models without requiring too much model-based derivation, which is a drawback of many existing VB algorithms. We also show how the proposed method can be used to obtain highly accurate VB approximations of marginal posterior distributions.

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arXiv2016

Importance sampling squared for Bayesian inference in latent variable models

We consider Bayesian inference by importance sampling when the likelihood is analytically intractable but can be unbiasedly estimated. We refer to this procedure as importance sampling squared (IS2), as we can often estimate the likelihood itself by importance sampling. We provide a formal justification for importance sampling when working with an estimate of the likelihood and study its convergence properties. We analyze the effect of estimating the likelihood on the resulting inference and provide guidelines on how to set up the precision of the likelihood estimate in order to obtain an optimal tradeoff? between computational cost and accuracy for posterior inference on the model parameters. We illustrate the procedure in empirical applications for a generalized multinomial logit model and a stochastic volatility model. The results show that the IS2 method can lead to fast and accurate posterior inference under the optimal implementation.

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arXiv2015

Exact ABC using Importance Sampling

Approximate Bayesian Computation (ABC) is a powerful method for carrying out Bayesian inference when the likelihood is computationally intractable. However, a drawback of ABC is that it is an approximate method that induces a systematic error because it is necessary to set a tolerance level to make the computation tractable. The issue of how to optimally set this tolerance level has been the subject of extensive research. This paper proposes an ABC algorithm based on importance sampling that estimates expectations with respect to the "exact" posterior distribution given the observed summary statistics. This overcomes the need to select the tolerance level. By "exact" we mean that there is no systematic error and the Monte Carlo error can be made arbitrarily small by increasing the number of importance samples. We provide a formal justification for the method and study its convergence properties. The method is illustrated in two applications and the empirical results suggest that the proposed ABC based estimators consistently converge to the true values as the number of importance samples increases. Our proposed approach can be applied more generally to any importance sampling problem where an unbiased estimate of the likelihood is required.

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arXiv2015

Bayesian inference for latent factor GARCH models

Latent factor GARCH models are difficult to estimate using Bayesian methods because standard Markov chain Monte Carlo samplers produce slowly mixing and inefficient draws from the posterior distributions of the model parameters. This paper describes how to apply the particle Gibbs algorithm to estimate factor GARCH models efficiently. The method has two advantages over previous approaches. First, it generalises in a straightfoward way to models with multiple factors and to various members of the GARCH family. Second, it scales up well as the dimension of the o, bservation vector increases.

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arXiv2015

Markov Interacting Importance Samplers

We introduce a new Markov chain Monte Carlo (MCMC) sampler called the Markov Interacting Importance Sampler (MIIS). The MIIS sampler uses conditional importance sampling (IS) approximations to jointly sample the current state of the Markov Chain and estimate conditional expectations, possibly by incorporating a full range of variance reduction techniques. We compute Rao-Blackwellized estimates based on the conditional expectations to construct control variates for estimating expectations under the target distribution. The control variates are particularly efficient when there are substantial correlations between the variables in the target distribution, a challenging setting for MCMC. An important motivating application of MIIS occurs when the exact Gibbs sampler is not available because it is infeasible to directly simulate from the conditional distributions. In this case the MIIS method can be more efficient than a Metropolis-within-Gibbs approach. We also introduce the MIIS random walk algorithm, designed to accelerate convergence and improve upon the computational efficiency of standard random walk samplers. Simulated and empirical illustrations for Bayesian analysis show that the method significantly reduces the variance of Monte Carlo estimates compared to standard MCMC approaches, at equivalent implementation and computational effort.

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arXiv2014

Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator

When an unbiased estimator of the likelihood is used within a Metropolis--Hastings chain, it is necessary to trade off the number of Monte Carlo samples used to construct this estimator against the asymptotic variances of averages computed under this chain. Many Monte Carlo samples will typically result in Metropolis--Hastings averages with lower asymptotic variances than the corresponding Metropolis--Hastings averages using fewer samples. However, the computing time required to construct the likelihood estimator increases with the number of Monte Carlo samples. Under the assumption that the distribution of the additive noise introduced by the log-likelihood estimator is Gaussian with variance inversely proportional to the number of Monte Carlo samples and independent of the parameter value at which it is evaluated, we provide guidelines on the number of samples to select. We demonstrate our results by considering a stochastic volatility model applied to stock index returns.

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arXiv2014

An extended space approach for particle Markov chain Monte Carlo methods

In this paper we consider fully Bayesian inference in general state space models. Existing particle Markov chain Monte Carlo (MCMC) algorithms use an augmented model that takes into account all the variable sampled in a sequential Monte Carlo algorithm. This paper describes an approach that also uses sequential Monte Carlo to construct an approximation to the state space, but generates extra states using MCMC runs at each time point. We construct an augmented model for our extended space with the marginal distribution of the sampled states matching the posterior distribution of the state vector. We show how our method may be combined with particle independent Metropolis-Hastings or particle Gibbs steps to obtain a smoothing algorithm. All the Metropolis acceptance probabilities are identical to those obtained in existing approaches, so there is no extra cost in term of Metropolis-Hastings rejections when using our approach. The number of MCMC iterates at each time point is chosen by the used and our augmented model collapses back to the model in Olsson and Ryden (2011) when the number of MCMC iterations reduces. We show empirically that our approach works well on applied examples and can outperform existing methods.

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arXiv2013

Particle Efficient Importance Sampling

The efficient importance sampling (EIS) method is a general principle for the numerical evaluation of high-dimensional integrals that uses the sequential structure of target integrands to build variance minimising importance samplers. Despite a number of successful applications in high dimensions, it is well known that importance sampling strategies are subject to an exponential growth in variance as the dimension of the integration increases. We solve this problem by recognising that the EIS framework has an offline sequential Monte Carlo interpretation. The particle EIS method is based on non-standard resampling weights that take into account the look-ahead construction of the importance sampler. We apply the method for a range of univariate and bivariate stochastic volatility specifications. We also develop a new application of the EIS approach to state space models with Student's t state innovations. Our results show that the particle EIS method strongly outperforms both the standard EIS method and particle filters for likelihood evaluation in high dimensions. Moreover, the ratio between the variances of the particle EIS and particle filter methods remains stable as the time series dimension increases. We illustrate the efficiency of the method for Bayesian inference using the particle marginal Metropolis-Hastings and importance sampling squared algorithms.

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arXiv2013

Adaptive Metropolis-Hastings Sampling using Reversible Dependent Mixture Proposals

This article develops a general-purpose adaptive sampler that approximates the target density by a mixture of multivariate t densities. The adaptive sampler is based on reversible proposal distributions each of which has the mixture of multivariate t densities as its invariant density. The reversible proposals consist of a combination of independent and correlated steps that allow the sampler to traverse the parameter space efficiently as well as allowing the sampler to keep moving and locally exploring the parameter space. We employ a two-chain approach, in which a trial chain is used to adapt the proposal densities used in the main chain. Convergence of the main chain and a strong law of large numbers are proved under reasonable conditions, and without imposing a Diminishing Adaptation condition. The mixtures of multivariate t densities are fitted by an efficient Variational Approximation algorithm in which the number of components is determined automatically. The performance of the sampler is evaluated using simulated and real examples. Our autocorrelated framework is quite general and can handle mixtures other than multivariate t.

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arXiv2013

Efficient variational inference for generalized linear mixed models with large datasets

The article develops a hybrid Variational Bayes algorithm that combines the mean-field and fixed-form Variational Bayes methods. The new estimation algorithm can be used to approximate any posterior without relying on conjugate priors. We propose a divide and recombine strategy for the analysis of large datasets, which partitions a large dataset into smaller pieces and then combines the variational distributions that have been learnt in parallel on each separate piece using the hybrid Variational Bayes algorithm. The proposed method is applied to fitting generalized linear mixed models. The computational efficiency of the parallel and hybrid Variational Bayes algorithm is demonstrated on several simulated and real datasets.

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arXiv2013

On the existence of moments for high dimensional importance sampling

Theoretical results for importance sampling rely on the existence of certain moments of the importance weights, which are the ratios between the proposal and target densities. In particular, a finite variance ensures square root convergence and asymptotic normality of the importance sampling estimate, and can be important for the reliability of the method in practice. We derive conditions for the existence of any required moments of the weights for Gaussian proposals and show that these conditions are almost necessary and sufficient for a wide range of models with latent Gaussian components. Important examples are time series and panel data models with measurement densities which belong to the exponential family. We introduce practical and simple methods for checking and imposing the conditions for the existence of the desired moments. We develop a two component mixture proposal that allows us to flexibly adapt a given proposal density into a robust importance density. These methods are illustrated on a wide range of models including generalized linear mixed models, non-Gaussian nonlinear state space models and panel data models with autoregressive random effects.

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