Bayesian evidence helps compare models but can overfit.
problem Comparing hypotheses consistent with observations.
method Marginal likelihood, Occam's razor, PAC-Bayes bounds.
result Marginal likelihood can negatively correlate with generalization.
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48 results for “marginal likelihood”
New method estimates marginal likelihood for deep learning models using training data alone.
problem Estimation difficulties in marginal likelihood for model selection in deep learning.
method Scalable marginal likelihood estimation based on Laplace's method and Gauss-Newton approximations.
result Estimate outperforms cross-validation and manual tuning on various datasets.
We study the parameter estimation problem in mixture models with observational nonidentifiability: the full model (also containing hidden variables) is identifiable, but the marginal (observed) model is not. Hence global maxima of the marginal likelihood are (infinitely) degenerate and predictions of the marginal likel…
Stochastic Gradient Annealed Importance Sampling for Efficient Online Marginal Likelihood Estimationstat.ML
We consider estimating the marginal likelihood in settings with independent and identically distributed (i.i.d.) data. We propose estimating the predictive distributions in a sequential factorization of the marginal likelihood in such settings by using stochastic gradient Markov Chain Monte Carlo techniques. This appro…
Bayesian approach learns invariances from data alone, but last layer approximation is not always sufficient.
problem Learning invariances in neural networks using only training data.
method Bayesian marginal likelihood for last layer, custom optimisation routine, new lower bound.
result Partial success on standard benchmarks and medical imaging dataset, failure on CIFAR10.
Due to the intractable partition function, the exact likelihood function for a Markov random field (MRF), in many situations, can only be approximated. Major approximation approaches include pseudolikelihood and Laplace approximation. In this paper, we propose a novel way of approximating the likelihood function throug…
SUMO provides unbiased log marginal likelihood estimation for latent variable models.
problem Biased estimates of log marginal likelihood in latent variable models.
method Randomized truncation of infinite series for unbiased estimation.
result Models trained with SUMO give better test-set likelihoods than standard methods.
In this short note we provide an unbiased multilevel Monte Carlo estimator of the log marginal likelihood and discuss its application to variational Bayes.
Proposes a method to optimize neural network initialization using marginal likelihood maximization.
problem Optimizing hyperparameters for neural network initialization.
method Leverages the connection between neural networks and Gaussian processes to infer optimal hyperparameters.
result Marginal likelihood maximization provides near-optimal prediction performance on MNIST classification tasks.
Improved Gaussian process regression with tighter log marginal likelihood bounds.
problem Improving predictive performance in Gaussian process regression models.
method Lower bound on log marginal likelihood using conjugate gradients.
result Improved predictive performance compared to other conjugate gradient based approaches.
We present and implement two algorithms for analytic asymptotic evaluation of the marginal likelihood of data given a Bayesian network with hidden nodes. As shown by previous work, this evaluation is particularly hard for latent Bayesian network models, namely networks that include hidden variables, where asymptotic ap…
In Bayesian statistics, the marginal likelihood, also known as the evidence, is used to evaluate model fit as it quantifies the joint probability of the data under the prior. In contrast, non-Bayesian models are typically compared using cross-validation on held-out data, either through -fold partitioning or leave-$p…
The paper discusses the impact of prior densities on Bayesian model selection.
problem The sensitivity of marginal likelihood to prior choice in Bayesian model selection.
method Analyzes the role of prior densities in model selection, discusses improper priors, and proposes solutions.
result Marginal likelihood can be sensitive to prior choice, but improper priors can still be used with caution.
We consider the combined use of resampling and partial rejection control in sequential Monte Carlo methods, also known as particle filters. While the variance reducing properties of rejection control are known, there has not been (to the best of our knowledge) any work on unbiased estimation of the marginal likelihood …
Improves hyperparameter learning in GP models with non-conjugate likelihoods.
problem Hyperparameter learning entangled with approximate inference in GP models.
method Hybrid training procedure combining VI for inference and EP-like marginal likelihood approximation for hyperparameter learning.
result Empirically demonstrates the effectiveness of the proposed training procedure across various data sets.
The paper derives a formula for factorizing categorical data to improve Bayes classifiers.
problem Improving the accuracy of Bayes classifiers by effectively factoring multidimensional data.
method Derives an explicit formula for calculating the marginal likelihood of a factorized categorical dataset.
result The derived formula can be used to select the best factorization for constructing a Bayes classifier.
Improves Gaussian process regression without bias.
problem Bias in Gaussian process regression estimates.
method Adaptive computation selection to minimize bias.
result Guaranteed small bias in log marginal likelihood estimates.
Warm starts improve Gaussian process regression by up to 16x.
problem Optimizing hyperparameters for Gaussian processes.
method Iterative Gaussian processes with warm start optimization.
result Warm starts achieve the same results as conventional methods but significantly speed up computations.
It has been argued that in supervised classification tasks, in practice it may be more sensible to perform model selection with respect to some more focused model selection score, like the supervised (conditional) marginal likelihood, than with respect to the standard marginal likelihood criterion. However, for most Ba…
New algorithm improves latent variable model estimation.
problem Estimating parameters in latent variable models.
method Jarzynski-adjusted Langevin algorithm (JALA) for SMC methods.
result JALA-EM provides maximum marginal likelihood estimate.
This work gives an in-depth derivation of the trainable evidence lower bound obtained from the marginal joint log-Likelihood with the goal of training a Multi-Modal Variational Autoencoder (MVAE).
Generalising well in supervised learning tasks relies on correctly extrapolating the training data to a large region of the input space. One way to achieve this is to constrain the predictions to be invariant to transformations on the input that are known to be irrelevant (e.g. translation). Commonly, this is done thro…
Improving the Accuracy of Marginal Approximations in Likelihood-Free Inference via Localisationstat.ME
Improved likelihood-free inference by localizing and refining low-dimensional approximations.
problem Poor performance of common likelihood-free methods in high-dimensional models.
method Localisation followed by refinement of low-dimensional summaries.
result Improved accuracy in marginal posteriors through localized and refined approximations.
Deep Gaussian processes provide a flexible approach to probabilistic modelling of data using either supervised or unsupervised learning. For tractable inference approximations to the marginal likelihood of the model must be made. The original approach to approximate inference in these models used variational compressio…
We consider two connected aspects of maximum likelihood estimation of the parameter for high-dimensional discrete graphical models: the existence of the maximum likelihood estimate (mle) and its computation. When the data is sparse, there are many zeros in the contingency table and the maximum likelihood estimate of th…
Identifies interpretable generative model for multivariate data.
problem Black-box architectures of deep generative models are often unidentified and difficult to interpret.
method Introduces Deep Discrete Encoder (DDE) Copula, a hierarchical binary latent variable model inside a copula framework.
result Establishes conditions for identification of DDE copula parameters and proves posterior consistency.
We consider discrete graphical models Markov with respect to a graph and propose two distributed marginal methods to estimate the maximum likelihood estimate of the canonical parameter of the model. Both methods are based on a relaxation of the marginal likelihood obtained by considering the density of the variable…
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…
Optimizes AIS hyperparameters for efficient marginal likelihood estimation.
problem Limited computation budget affects AIS performance.
method Flexible intermediary distributions defined by residual density, parameter sharing, and fix linear schedule.
result Optimized-Path AIS reduces sampling iterations and improves performance.
We consider the problem of discriminative factor analysis for data that are in general non-Gaussian. A Bayesian model based on the ranks of the data is proposed. We first introduce a new {\em max-margin} version of the rank-likelihood. A discriminative factor model is then developed, integrating the max-margin rank-lik…
Graphical models with bi-directed edges (<->) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood estimation in the case of continuous variables with a Gaussian joint distributio…
Develops methods for constructing likelihoods and priors for Bayesian networks.
problem Learning parameters and structure of Bayesian networks from limited data.
method Introduces assumptions for constructing likelihoods and priors from small assessments.
result Allows construction of likelihoods and priors for a wide range of network structures.
Supervised topic models utilize document's side information for discovering predictive low dimensional representations of documents. Existing models apply the likelihood-based estimation. In this paper, we present a general framework of max-margin supervised topic models for both continuous and categorical response var…
ITF improves DSR but inflates curvature, while marginal likelihood reduces it, affecting QoIs.
problem Curvature mismatch between teacher forcing and marginal likelihood in chaotic dynamical systems.
method Comparing objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs.
result Curvature inflation by ITF and reduction by marginal likelihood affect dynamical quantities of interest.
We exhibit a strong link between frequentist PAC-Bayesian risk bounds and the Bayesian marginal likelihood. That is, for the negative log-likelihood loss function, we show that the minimization of PAC-Bayesian generalization risk bounds maximizes the Bayesian marginal likelihood. This provides an alternative explanatio…
AIS uses a suboptimal extended target distribution, which this paper improves using SGM.
problem Improving the efficiency of Annealed Importance Sampling for marginal likelihood estimation.
method Leveraging score-based generative modeling to approximate the optimal extended target distribution.
result Demonstrated novel, differentiable AIS procedures on synthetic and real-world data.
DAIS improves AIS for differentiable marginal likelihood estimation.
problem Differentiable marginal likelihood estimation for complex models.
method Proposes Differentiable Annealed Importance Sampling (DAIS) to make AIS differentiable.
result DAIS achieves convergence and consistency in Bayesian linear regression.
A new method estimates marginal likelihood using normalizing flows.
problem Estimating marginal likelihood in Bayesian model selection.
method Learned harmonic mean estimator using normalizing flows.
result Normalizing flows avoid the exploding variance problem.
Given a set of possible models (e.g., Bayesian network structures) and a data sample, in the unsupervised model selection problem the task is to choose the most accurate model with respect to the domain joint probability distribution. In contrast to this, in supervised model selection it is a priori known that the chos…
We define a generalized likelihood function based on uncertainty measures and show that maximizing such a likelihood function for different measures induces different types of classifiers. In the probabilistic framework, we obtain classifiers that optimize the cross-entropy function. In the possibilistic framework, we …
A fast method estimates group-adaptive elastic net penalties using co-data.
problem Computational inefficiency in estimating group-adaptive elastic net penalties.
method Derive low-dimensional representation of Taylor approximation for marginal likelihood and its derivative for group-adaptive ridge penalties; approximate elastic net marginal likelihood by ridge; transform ridge penalties to elastic net penalties.
result Significantly decreases computation time and outperforms other methods.
Given a set of possible models (e.g., Bayesian network structures) and a data sample, in the unsupervised model selection problem the task is to choose the most accurate model with respect to the domain joint probability distribution. In contrast to this, in supervised model selection it is a priori known that the chos…
Computing the marginal likelihood (ML) of a model requires marginalizing out all of the parameters and latent variables, a difficult high-dimensional summation or integration problem. To make matters worse, it is often hard to measure the accuracy of one's ML estimates. We present bidirectional Monte Carlo, a technique…
Proposes learning invariances in neural networks using a weight-space approach.
problem Learning invariances from data in neural networks remains an open problem.
method Minimizes a lower bound on the marginal likelihood in weight space.
result Results in higher performing models with naturally learned invariances.
Method estimates noise variance in Gaussian process regression.
problem Estimating noise variance in Gaussian process regression models.
method Reduces hyperparameter space, uses marginal likelihood function, derives bounds and asymptotes.
result Computational advantages and robustness compared to traditional methods.
Bayesian approach sparsifies neural networks efficiently.
problem Efficiently pruning neural networks to save resources.
method Sparsifiability via the Marginal likelihood (SpaM) framework.
result Prunes neural networks effectively without significant loss in performance.
Knot-based, sparse Gaussian processes have enjoyed considerable success as scalable approximations to full Gaussian processes. Problems can occur, however, when knot selection is done by optimizing the marginal likelihood. For example, the marginal likelihood surface is highly multimodal, which can cause suboptimal kno…
We present novel understandings of the Gamma-Poisson (GaP) model, a probabilistic matrix factorization model for count data. We show that GaP can be rewritten free of the score/activation matrix. This gives us new insights about the estimation of the topic/dictionary matrix by maximum marginal likelihood estimation. In…