Standard variational lower bounds used to train latent variable models produce biased estimates of most quantities of interest. We introduce an unbiased estimator of the log marginal likelihood and its gradients for latent variable models based on randomized truncation of infinite series. If parameterized by an encoder-decoder architecture, the parameters of the encoder can be optimized to minimize its variance of this estimator. We show that models trained using our estimator give better test-set likelihoods than a standard importance-sampling based approach for the same average computational cost. This estimator also allows use of latent variable models for tasks where unbiased estimators, rather than marginal likelihood lower bounds, are preferred, such as minimizing reverse KL divergences and estimating score functions.
University of California, Berkeley（加州大学伯克利分校）；University of California, Davis（加州大学戴维斯分校）
We apply the technique of implicit differentiation to boost performance, reduce numerical error, and remove required user-tuning in the Marginal Unbiased Score Expansion (MUSE) algorithm for hierarchical Bayesian inference. We demonstrate these improvements on three representative inference problems: 1) an extended Neal's funnel 2) Bayesian neural networks, and 3) probabilistic principal component analysis. On our particular test cases, MUSE with implicit differentiation is faster than Hamiltonian Monte Carlo by factors of 155, 397, and 5, respectively, or factors of 65, 278, and 1 without imp…
School of Engineering, University of Warwick, UK（华威大学工程学院，英国）；Warwick Center for Predictive Modelling, University of Warwick, UK（华威大学预测建模中心，英国）；Department of Statistics, University of Warwick, UK（华威大学统计系，英国）；Division of Translational Medicine, Warwick Medical School, UK（华威医学院转化医学部，英国）；Dept of Clinical Biochemistry, Division of Pathology, University Hospital Coventry and Warwickshire, Coventry, UK（考文垂和华威郡大学医院病理学部临床生物化学系，考文垂，英国）
We introduce a Bayesian framework for inference with a supervised version of the Gaussian process latent variable model. The framework overcomes the high correlations between latent variables and hyperparameters by using an unbiased pseudo estimate for the marginal likelihood that approximately integrates over the latent variables. This is used to construct a Markov Chain to explore the posterior of the hyperparameters. We demonstrate the procedure on simulated and real examples, showing its ability to capture uncertainty and multimodality of the hyperparameters and improved uncertainty quanti…
Bayesian inference for Continuous-Time Markov Chains (CTMCs) on countably infinite spaces is notoriously difficult because evaluating the likelihood exactly is intractable. One way to address this challenge is to first build a non-negative and unbiased estimate of the likelihood -- involving the matrix exponential of finite truncations of the true rate matrix -- and then to use the estimates in a pseudo-marginal inference method. In this work, we show that we can dramatically increase the efficiency of this approach by avoiding the computation of exact matrix exponentials. In particular, we de…
We consider continuous time Markovian processes where populations of individual agents interact stochastically according to kinetic rules. Despite the increasing prominence of such models in fields ranging from biology to smart cities, Bayesian inference for such systems remains challenging, as these are continuous time, discrete state systems with potentially infinite state-space. Here we propose a novel efficient algorithm for joint state / parameter posterior sampling in population Markov Jump processes. We introduce a class of pseudo-marginal sampling algorithms based on a random truncatio…
Posterior distributions often feature intractable normalizing constants, called marginal likelihoods or evidence, that are useful for model comparison via Bayes factors. This has motivated a number of methods for estimating ratios of normalizing constants in statistics. In computational physics the logarithm of these ratios correspond to free energy differences. Combining unbiased Markov chain Monte Carlo estimators with path sampling, also called thermodynamic integration, we propose new unbiased estimators of the logarithm of ratios of normalizing constants. As a by-product, we propose unbia…
In applications of Gaussian processes where quantification of uncertainty is of primary interest, it is necessary to accurately characterize the posterior distribution over covariance parameters. This paper proposes an adaptation of the Stochastic Gradient Langevin Dynamics algorithm to draw samples from the posterior distribution over covariance parameters with negligible bias and without the need to compute the marginal likelihood. In Gaussian process regression, this has the enormous advantage that stochastic gradients can be computed by solving linear systems only. A novel unbiased linear …
Manning College of Information and Computer Sciences, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校曼宁信息与计算机科学学院）
Variational inference for state space models (SSMs) is known to be hard in general. Recent works focus on deriving variational objectives for SSMs from unbiased sequential Monte Carlo estimators. We reveal that the marginal particle filter is obtained from sequential Monte Carlo by applying Rao-Blackwellization operations, which sacrifices the trajectory information for reduced variance and differentiability. We propose the variational marginal particle filter (VMPF), which is a differentiable and reparameterizable variational filtering objective for SSMs based on an unbiased estimator. We fin…
Kernel methods have revolutionized the fields of pattern recognition and machine learning. Their success, however, critically depends on the choice of kernel parameters. Using Gaussian process (GP) classification as a working example, this paper focuses on Bayesian inference of covariance (kernel) parameters using Markov chain Monte Carlo (MCMC) methods. The motivation is that, compared to standard optimization of kernel parameters, they have been systematically demonstrated to be superior in quantifying uncertainty in predictions. Recently, the Pseudo-Marginal MCMC approach has been proposed …
In this short note we provide an unbiased multilevel Monte Carlo estimator of the log marginal likelihood and discuss its application to variational Bayes.
We consider lithological tomography in which the posterior distribution of (hydro)geological parameters of interest is inferred from geophysical data by treating the intermediate geophysical properties as latent variables. In such a latent variable model, one needs to estimate the intractable likelihood of the (hydro)geological parameters given the geophysical data. The pseudo-marginal method is an adaptation of the Metropolis-Hastings algorithm in which an unbiased approximation of this likelihood is obtained by Monte Carlo averaging over samples from, in this setting, the noisy petrophysical…
Our article deals with Bayesian inference for a general state space model with the simulated likelihood computed by the particle filter. We show empirically that the partially or fully adapted particle filters can be much more efficient than the standard particle, especially when the signal to noise ratio is high. This is especially important because using the particle filter within MCMC sampling is O(T^2), where T is the sample size. We also show that an adaptive independent proposal for the unknown parameters based on a mixture of normals can be much more efficient than the usual optimal ran…
Survey data are often collected under multistage sampling designs where units are binned to clusters that are sampled in a first stage. The unit-indexed population variables of interest are typically dependent within cluster. We propose a Fully Bayesian method that constructs an exact likelihood for the observed sample to incorporate unit-level marginal sampling weights for performing unbiased inference for population parameters while simultaneously accounting for the dependence induced by sampling clusters of units to produce correct uncertainty quantification. Our approach parameterizes clus…
The variational auto-encoder (VAE) is a deep latent variable model that has two neural networks in an autoencoder-like architecture; one of them parameterizes the model's likelihood. Fitting its parameters via maximum likelihood (ML) is challenging since the computation of the marginal likelihood involves an intractable integral over the latent space; thus the VAE is trained instead by maximizing a variational lower bound. Here, we develop a ML training scheme for VAEs by introducing unbiased estimators of the log-likelihood gradient. We obtain the estimators by augmenting the latent space wit…
This paper is concerned with Bayesian inference when the likelihood is analytically intractable but can be unbiasedly estimated. We propose an annealed importance sampling procedure for estimating expectations with respect to the posterior. The proposed algorithm is useful in cases where finding a good proposal density is challenging, and when estimates of the marginal likelihood are required. The effect of likelihood estimation is investigated, and the results provide guidelines on how to set up the precision of the likelihood estimation in order to optimally implement the procedure. The meth…
An econometric or statistical model may undergo a marginal gain if we admit a new variable to the model, and a marginal loss if we remove an existing variable from the model. Assuming equality of opportunity among all candidate variables, we derive a valuation framework by the expected marginal gain and marginal loss in all potential modeling scenarios. However, marginal gain and loss are not symmetric; thus, we introduce three unbiased solutions. When used in variable selection, our new approaches significantly outperform several popular methods used in practice. The results also explore some…
We show that unconverged stochastic gradient descent can be interpreted as a procedure that samples from a nonparametric variational approximate posterior distribution. This distribution is implicitly defined as the transformation of an initial distribution by a sequence of optimization updates. By tracking the change in entropy over this sequence of transformations during optimization, we form a scalable, unbiased estimate of the variational lower bound on the log marginal likelihood. We can use this bound to optimize hyperparameters instead of using cross-validation. This Bayesian interpreta…
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 …
Learning from N-Tuple Data with M Positive Instances: Unbiased Risk Estimation and Theoretical Guarantees
Engineering Research Center of the Ministry of Education for Intelligent Control System and Intelligent Equipment, Yanshan University（教育部智能控制系统与智能设备工程研究中心，燕山大学）
