Kullback-Leibler（KL）差异广泛用于贝叶斯神经网络（BNNS）的变异推理。然而，KL差异具有无限性和不对称性等局限性。我们检查了更通用，有限和对称的詹森 - 香农（JS）差异。我们根据几何JS差异为BNN制定新的损失函数，并表明基于KL差异的常规损失函数是其特殊情况。我们以封闭形式的高斯先验评估拟议损失函数的差异部分。对于任何其他一般的先验，都可以使用蒙特卡洛近似值。我们提供了实施这两种情况的算法。我们证明所提出的损失函数提供了一个可以调整的附加参数，以控制正则化程度。我们得出了所提出的损失函数在高斯先验和后代的基于KL差异的损失函数更好的条件。我们证明了基于嘈杂的CIFAR数据集和有偏见的组织病理学数据集的最新基于KL差异的BNN的性能提高。

尽管基于卷积神经网络（CNN）的组织病理学图像的分类模型，但量化其不确定性是不可行的。此外，当数据偏置时，CNN可以遭受过度装备。我们展示贝叶斯-CNN可以通过自动规范并通过量化不确定性来克服这些限制。我们开发了一种新颖的技术，利用贝叶斯-CNN提供的不确定性，这显着提高了大部分测试数据的性能（约为77％的测试数据的准确性提高了约6％）。此外，我们通过非线性维度降低技术将数据投射到低尺寸空间来提供对不确定性的新颖解释。该维度降低能够通过可视化解释测试数据，并在低维特征空间中揭示数据的结构。我们表明，贝叶斯-CNN可以通过分别将假阴性和假阳性降低11％和7.7％的最先进的转移学习CNN（TL-CNN）来表现出远得更好。它具有仅为186万个参数的这种性能，而TL-CNN的参数仅为134.33亿。此外，我们通过引入随机自适应激活功能来修改贝叶斯-CNN。修改后的贝叶斯-CNN在所有性能指标上的贝叶斯-CNN略胜一筹，并显着降低了误报和误报的数量（两者减少了3％）。我们还表明，通过执行McNemar的统计显着性测试，这些结果具有统计学意义。这项工作显示了贝叶斯-CNN对现有技术的优势，解释并利用组织病理学图像的不确定性。它应该在各种医学图像分类中找到应用程序。

变异推理（VI）的核心原理是将计算复杂后概率密度计算的统计推断问题转换为可拖动的优化问题。该属性使VI比几种基于采样的技术更快。但是，传统的VI算法无法扩展到大型数据集，并且无法轻易推断出越野数据点，而无需重新运行优化过程。该领域的最新发展，例如随机，黑框和摊销VI，已帮助解决了这些问题。如今，生成的建模任务广泛利用摊销VI来实现其效率和可扩展性，因为它利用参数化函数来学习近似的后验密度参数。在本文中，我们回顾了各种VI技术的数学基础，以构成理解摊销VI的基础。此外，我们还概述了最近解决摊销VI问题的趋势，例如摊销差距，泛化问题，不一致的表示学习和后验崩溃。最后，我们分析了改善VI优化的替代差异度量。

在这项工作中，我们使用变分推论来量化无线电星系分类的深度学习模型预测的不确定性程度。我们表明，当标记无线电星系时，个体测试样本的模型后差水平与人类不确定性相关。我们探讨了各种不同重量前沿的模型性能和不确定性校准，并表明稀疏事先产生更良好的校准不确定性估计。使用单个重量的后部分布，我们表明我们可以通过从最低信噪比（SNR）中除去权重来修剪30％的完全连接的层权重，而无需显着损失性能。我们证明，可以使用基于Fisher信息的排名来实现更大程度的修剪，但我们注意到两种修剪方法都会影响Failaroff-Riley I型和II型无线电星系的不确定性校准。最后，我们表明，与此领域的其他工作相比，我们经历了冷的后效，因此后部必须缩小后加权以实现良好的预测性能。我们检查是否调整成本函数以适应模型拼盘可以弥补此效果，但发现它不会产生显着差异。我们还研究了原则数据增强的效果，并发现这改善了基线，而且还没有弥补观察到的效果。我们将其解释为寒冷的后效，因为我们的培训样本过于有效的策划导致可能性拼盘，并将其提高到未来无线电银行分类的潜在问题。

Variational inference uses optimization, rather than integration, to approximate the marginal likelihood, and thereby the posterior, in a Bayesian model. Thanks to advances in computational scalability made in the last decade, variational inference is now the preferred choice for many high-dimensional models and large datasets. This tutorial introduces variational inference from the parametric perspective that dominates these recent developments, in contrast to the mean-field perspective commonly found in other introductory texts.

We investigate a local reparameterizaton technique for greatly reducing the variance of stochastic gradients for variational Bayesian inference (SGVB) of a posterior over model parameters, while retaining parallelizability. This local reparameterization translates uncertainty about global parameters into local noise that is independent across datapoints in the minibatch. Such parameterizations can be trivially parallelized and have variance that is inversely proportional to the minibatch size, generally leading to much faster convergence. Additionally, we explore a connection with dropout: Gaussian dropout objectives correspond to SGVB with local reparameterization, a scale-invariant prior and proportionally fixed posterior variance. Our method allows inference of more flexibly parameterized posteriors; specifically, we propose variational dropout, a generalization of Gaussian dropout where the dropout rates are learned, often leading to better models. The method is demonstrated through several experiments.

Large multilayer neural networks trained with backpropagation have recently achieved state-ofthe-art results in a wide range of problems. However, using backprop for neural net learning still has some disadvantages, e.g., having to tune a large number of hyperparameters to the data, lack of calibrated probabilistic predictions, and a tendency to overfit the training data. In principle, the Bayesian approach to learning neural networks does not have these problems. However, existing Bayesian techniques lack scalability to large dataset and network sizes. In this work we present a novel scalable method for learning Bayesian neural networks, called probabilistic backpropagation (PBP). Similar to classical backpropagation, PBP works by computing a forward propagation of probabilities through the network and then doing a backward computation of gradients. A series of experiments on ten real-world datasets show that PBP is significantly faster than other techniques, while offering competitive predictive abilities. Our experiments also show that PBP provides accurate estimates of the posterior variance on the network weights.

现代深度学习方法构成了令人难以置信的强大工具，以解决无数的挑战问题。然而，由于深度学习方法作为黑匣子运作，因此与其预测相关的不确定性往往是挑战量化。贝叶斯统计数据提供了一种形式主义来理解和量化与深度神经网络预测相关的不确定性。本教程概述了相关文献和完整的工具集，用于设计，实施，列车，使用和评估贝叶斯神经网络，即使用贝叶斯方法培训的随机人工神经网络。

We introduce a new, efficient, principled and backpropagation-compatible algorithm for learning a probability distribution on the weights of a neural network, called Bayes by Backprop. It regularises the weights by minimising a compression cost, known as the variational free energy or the expected lower bound on the marginal likelihood. We show that this principled kind of regularisation yields comparable performance to dropout on MNIST classification. We then demonstrate how the learnt uncertainty in the weights can be used to improve generalisation in non-linear regression problems, and how this weight uncertainty can be used to drive the exploration-exploitation trade-off in reinforcement learning.

We introduce a new, efficient, principled and backpropagation-compatible algorithm for learning a probability distribution on the weights of a neural network, called Bayes by Backprop. It regularises the weights by minimising a compression cost, known as the variational free energy or the expected lower bound on the marginal likelihood. We show that this principled kind of regularisation yields comparable performance to dropout on MNIST classification. We then demonstrate how the learnt uncertainty in the weights can be used to improve generalisation in non-linear regression problems, and how this weight uncertainty can be used to drive the exploration-exploitation trade-off in reinforcement learning.

We develop an optimization algorithm suitable for Bayesian learning in complex models. Our approach relies on natural gradient updates within a general black-box framework for efficient training with limited model-specific derivations. It applies within the class of exponential-family variational posterior distributions, for which we extensively discuss the Gaussian case for which the updates have a rather simple form. Our Quasi Black-box Variational Inference (QBVI) framework is readily applicable to a wide class of Bayesian inference problems and is of simple implementation as the updates of the variational posterior do not involve gradients with respect to the model parameters, nor the prescription of the Fisher information matrix. We develop QBVI under different hypotheses for the posterior covariance matrix, discuss details about its robust and feasible implementation, and provide a number of real-world applications to demonstrate its effectiveness.

随机梯度马尔可夫链蒙特卡洛（SGMCMC）被认为是大型模型（例如贝叶斯神经网络）中贝叶斯推断的金标准。由于从业人员在这些模型中面临速度与准确性权衡，因此变异推理（VI）通常是可取的选择。不幸的是，VI对后部的分解和功能形式做出了有力的假设。在这项工作中，我们提出了一个新的非参数变分近似，该近似没有对后验功能形式进行假设，并允许从业者指定算法应尊重或断裂的确切依赖性。该方法依赖于在修改的能量函数上运行的新的langevin型算法，其中潜在变量的一部分是在马尔可夫链的早期迭代中平均的。这样，统计依赖性可以以受控的方式破裂，从而使链条混合更快。可以以“辍学”方式进一步修改该方案，从而导致更大的可扩展性。我们在CIFAR-10，SVHN和FMNIST上测试RESNET-20的计划。在所有情况下，与SG-MCMC和VI相比，我们都会发现收敛速度和/或最终精度的提高。

One of the core problems of modern statistics is to approximate difficult-to-compute probability densities. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation involving the posterior density. In this paper, we review variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI has been used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling. The idea behind VI is to first posit a family of densities and then to find the member of that family which is close to the target. Closeness is measured by Kullback-Leibler divergence. We review the ideas behind mean-field variational inference, discuss the special case of VI applied to exponential family models, present a full example with a Bayesian mixture of Gaussians, and derive a variant that uses stochastic optimization to scale up to massive data. We discuss modern research in VI and highlight important open problems. VI is powerful, but it is not yet well understood. Our hope in writing this paper is to catalyze statistical research on this class of algorithms.

Compared to point estimates calculated by standard neural networks, Bayesian neural networks (BNN) provide probability distributions over the output predictions and model parameters, i.e., the weights. Training the weight distribution of a BNN, however, is more involved due to the intractability of the underlying Bayesian inference problem and thus, requires efficient approximations. In this paper, we propose a novel approach for BNN learning via closed-form Bayesian inference. For this purpose, the calculation of the predictive distribution of the output and the update of the weight distribution are treated as Bayesian filtering and smoothing problems, where the weights are modeled as Gaussian random variables. This allows closed-form expressions for training the network's parameters in a sequential/online fashion without gradient descent. We demonstrate our method on several UCI datasets and compare it to the state of the art.

近似复杂的概率密度是现代统计中的核心问题。在本文中，我们介绍了变分推理（VI）的概念，这是一种机器学习中的流行方法，该方法使用优化技术来估计复杂的概率密度。此属性允许VI汇聚速度比经典方法更快，例如Markov Chain Monte Carlo采样。概念上，VI通过选择一个概率密度函数，然后找到最接近实际概率密度的家庭 - 通常使用Kullback-Leibler（KL）发散作为优化度量。我们介绍了缩窄的证据，以促进近似的概率密度，我们审查了平均场变分推理背后的想法。最后，我们讨论VI对变分式自动编码器（VAE）和VAE-生成的对抗网络（VAE-GAN）的应用。用本文，我们的目标是解释VI的概念，并通过这种方法协助协助。

贝叶斯神经网络具有潜在变量（BNN + LVS）通过明确建模模型不确定性（通过网络权重）和环境暂停（通过潜在输入噪声变量）来捕获预测的不确定性。在这项工作中，我们首先表明BNN + LV具有严重形式的非可识别性：可以在模型参数和潜在变量之间传输解释性，同时拟合数据。我们证明，在无限数据的极限中，网络权重和潜变量的后部模式从地面真理渐近地偏离。由于这种渐近偏差，传统的推理方法可以在实践中，产量参数概括不确定和不确定的不确定性。接下来，我们开发一种新推断过程，明确地减轻了训练期间不可识别性的影响，并产生高质量的预测以及不确定性估计。我们展示我们的推理方法在一系列合成和实际数据集中改善了基准方法。

隐式过程（IPS）代表一个灵活的框架，可用于描述各种模型，从贝叶斯神经网络，神经抽样器和数据生成器到许多其他模型。 IP还允许在功能空间上进行大致推断。公式的这种变化解决了参数空间的固有退化问题近似推断，即参数数量及其在大型模型中的强大依赖性。为此，文献中先前的作品试图采用IPS来设置先验并近似产生的后部。但是，这被证明是一项具有挑战性的任务。现有的方法可以调整先前的IP导致高斯预测分布，该分布未能捕获重要的数据模式。相比之下，通过使用另一个IP近似后验过程产生灵活预测分布的方法不能将先前的IP调整到观察到的数据中。我们在这里建议第一个可以实现这两个目标的方法。为此，我们依赖于先前IP的诱导点表示，就像在稀疏高斯过程中所做的那样。结果是一种可扩展的方法，用于与IP的近似推断，可以将先前的IP参数调整到数据中，并提供准确的非高斯预测分布。

We introduce ensembles of stochastic neural networks to approximate the Bayesian posterior, combining stochastic methods such as dropout with deep ensembles. The stochastic ensembles are formulated as families of distributions and trained to approximate the Bayesian posterior with variational inference. We implement stochastic ensembles based on Monte Carlo dropout, DropConnect and a novel non-parametric version of dropout and evaluate them on a toy problem and CIFAR image classification. For CIFAR, the stochastic ensembles are quantitatively compared to published Hamiltonian Monte Carlo results for a ResNet-20 architecture. We also test the quality of the posteriors directly against Hamiltonian Monte Carlo simulations in a simplified toy model. Our results show that in a number of settings, stochastic ensembles provide more accurate posterior estimates than regular deep ensembles.

Recent advances in coreset methods have shown that a selection of representative datapoints can replace massive volumes of data for Bayesian inference, preserving the relevant statistical information and significantly accelerating subsequent downstream tasks. Existing variational coreset constructions rely on either selecting subsets of the observed datapoints, or jointly performing approximate inference and optimizing pseudodata in the observed space akin to inducing points methods in Gaussian Processes. So far, both approaches are limited by complexities in evaluating their objectives for general purpose models, and require generating samples from a typically intractable posterior over the coreset throughout inference and testing. In this work, we present a black-box variational inference framework for coresets that overcomes these constraints and enables principled application of variational coresets to intractable models, such as Bayesian neural networks. We apply our techniques to supervised learning problems, and compare them with existing approaches in the literature for data summarization and inference.

从降压和嘈杂的测量值（例如MRI和低剂量计算机断层扫描（CT））中重建图像是数学上不良的反问题。我们提出了一种基于期望传播（EP）技术的易于使用的重建方法。我们将蒙特卡洛（MC）方法，马尔可夫链蒙特卡洛（MCMC）和乘数（ADMM）算法的交替方向方法纳入EP方法，以解决EP中遇到的棘手性问题。我们在复杂的贝叶斯模型上演示了图像重建的方法。我们的技术应用于伽马相机扫描中的图像。我们仅将EPMC，EP-MCMC，EP-ADMM方法与MCMC进行比较。指标是更好的图像重建，速度和参数估计。在真实和模拟数据中使用伽马相机成像进行的实验表明，我们提出的方法在计算上比MCMC昂贵，并且产生相对更好的图像重建。