Latent Dirichlet allocation (LDA) is useful in document analysis, image processing, and many information systems; however, its generalization performance has been left unknown because it is a singular learning machine to which regular statistical theory can not be applied. Stochastic matrix factorization (SMF) is a restricted matrix factorization in which matrix factors are stochastic; the column of the matrix is in a simplex. SMF is being applied to image recognition and text mining. We can understand SMF as a statistical model by which a stochastic matrix of given data is represented by a product of two stochastic matrices, whose generalization performance has also been left unknown because of non-regularity. In this paper, by using an algebraic and geometric method, we show the analytic equivalence of LDA and SMF, both of which have the same real log canonical threshold (RLCT), resulting in that they asymptotically have the same Bayesian generalization error and the same log marginal likelihood. Moreover, we derive the upper bound of the RLCT and prove that it is smaller than the dimension of the parameter divided by two, hence the Bayesian generalization errors of them are smaller than those of regular statistical models.
Stein variational gradient descent (SVGD) is a recently proposed particle-based Bayesian inference method, which has attracted a lot of interest due to its remarkable approximation ability and particle efficiency compared to traditional variational inference and Markov Chain Monte Carlo methods. However, we observed that particles of SVGD tend to collapse to modes of the target distribution, and this particle degeneracy phenomenon becomes more severe with higher dimensions. Our theoretical analysis finds out that there exists a negative correlation between the dimensionality and the repulsive force of SVGD which should be blamed for this phenomenon. We propose Message Passing SVGD (MP-SVGD) to solve this problem. By leveraging the conditional independence structure of probabilistic graphical models (PGMs), MP-SVGD converts the original high-dimensional global inference problem into a set of local ones over the Markov blanket with lower dimensions. Experimental results show its advantages of preventing vanishing repulsive force in high-dimensional space over SVGD, and its particle efficiency and approximation flexibility over other inference methods on graphical models.
While off-policy temporal difference (TD) methods have widely been used in reinforcement learning due to their efficiency and simple implementation, their Bayesian counterparts have not been utilized as frequently. One reason is that the non-linear max operation in the Bellman optimality equation makes it difficult to define conjugate distributions over the value functions. In this paper, we introduce a novel Bayesian approach to off-policy TD methods, called as ADFQ, which updates beliefs on state-action values, Q, through an online Bayesian inference method known as Assumed Density Filtering. We formulate an efficient closed-form solution for the value update by approximately estimating analytic parameters of the posterior of the Q-beliefs. Uncertainty measures in the beliefs not only are used in exploration but also provide a natural regularization for the value update considering all next available actions. ADFQ converges to Q-learning as the uncertainty measures of the Q-beliefs decrease and improves common drawbacks of other Bayesian RL algorithms such as computational complexity. We extend ADFQ with a neural network. Our empirical results demonstrate that ADFQ outperforms comparable algorithms on various Atari 2600 games, with drastic improvements in highly stochastic domains or domains with a large action space.
We introduce a novel uncertainty estimation for classification tasks for Bayesian convolutional neural networks with variational inference. By normalizing the output of a Softplus function in the final layer, we estimate aleatoric and epistemic uncertainty in a coherent manner. The intractable posterior probability distributions over weights are inferred by Bayes by Backprop. Firstly, we demonstrate how this reliable variational inference method can serve as a fundamental construct for various network architectures. On multiple datasets in supervised learning settings (MNIST, CIFAR-10, CIFAR-100), this variational inference method achieves performances equivalent to frequentist inference in identical architectures, while the two desiderata, a measure for uncertainty and regularization are incorporated naturally. Secondly, we examine how our proposed measure for aleatoric and epistemic uncertainties is derived and validate it on the aforementioned datasets.
The use of ensembles of neural networks (NNs) for the quantification of predictive uncertainty is widespread. However, the current justification is intuitive rather than analytical. This work proposes one minor modification to the normal ensembling methodology, which we prove allows the ensemble to perform Bayesian inference, hence converging to the corresponding Gaussian Process as both the total number of NNs, and the size of each, tend to infinity. This working paper provides early-stage results in a reinforcement learning setting, analysing the practicality of the technique for an ensemble of small, finite number. Using the uncertainty estimates produced by anchored ensembles to govern the exploration-exploitation process results in steadier, more stable learning.
Multi-task neural network architectures provide a mechanism that jointly integrates information from distinct sources. It is ideal in the context of MR-only radiotherapy planning as it can jointly regress a synthetic CT (synCT) scan and segment organs-at-risk (OAR) from MRI. We propose a probabilistic multi-task network that estimates: 1) intrinsic uncertainty through a heteroscedastic noise model for spatially-adaptive task loss weighting and 2) parameter uncertainty through approximate Bayesian inference. This allows sampling of multiple segmentations and synCTs that share their network representation. We test our model on prostate cancer scans and show that it produces more accurate and consistent synCTs with a better estimation in the variance of the errors, state of the art results in OAR segmentation and a methodology for quality assurance in radiotherapy treatment planning.
Effective learning of asymmetric and local features in images and other data observed on multi-dimensional grids is a challenging objective critical for a wide range of image processing applications involving biomedical and natural images. It requires methods that are sensitive to local details while fast enough to handle massive numbers of images of ever increasing sizes. We introduce a probabilistic model-based framework that achieves these objectives by incorporating adaptivity into discrete wavelet transforms (DWT) through Bayesian hierarchical modeling, thereby allowing wavelet bases to adapt to the geometric structure of the data while maintaining the high computational scalability of wavelet methods---linear in the sample size (e.g., the resolution of an image). We derive a recursive representation of the Bayesian posterior model which leads to an exact message passing algorithm to complete learning and inference. While our framework is applicable to a range of problems including multi-dimensional signal processing, compression, and structural learning, we illustrate its work and evaluate its performance in the context of image reconstruction using real images from the ImageNet database, two widely used benchmark datasets, and a dataset from retinal optical coherence tomography and compare its performance to state-of-the-art methods based on basis transforms and deep learning.