In Bayesian regression, we often use a Gaussian observation model, where we control the level of aleatoric uncertainty with a noise variance parameter. By contrast, for Bayesian classification we use a categorical distribution with no mechanism to represent our beliefs about aleatoric uncertainty. Our work shows that explicitly accounting for aleatoric uncertainty significantly improves the performance of Bayesian neural networks. We note that many standard benchmarks, such as CIFAR-10, have essentially no aleatoric uncertainty. Moreover, we show that data augmentation in approximate inference softens the likelihood, leading to underconfidence and misrepresenting our beliefs about aleatoric uncertainty. Accordingly, we find that a cold posterior, tempered by a power greater than one, often more honestly reflects our beliefs about aleatoric uncertainty than no tempering --- providing an explicit link between data augmentation and cold posteriors. We further show that we can match or exceed the performance of posterior tempering by using a Dirichlet observation model, where we explicitly control the level of aleatoric uncertainty, without any need for tempering.

In Bayesian regression, we often use a Gaussian observation model, where we control the level of aleatoric uncertainty with a noise variance parameter. By contrast, for Bayesian classification we use a categorical distribution with no mechanism to represent our beliefs about aleatoric uncertainty. Our work shows that explicitly accounting for aleatoric uncertainty significantly improves the performance of Bayesian neural networks. We note that many standard benchmarks, such as CIFAR-10, have essentially no aleatoric uncertainty. Moreover, we show that data augmentation in approximate inference softens the likelihood, leading to underconfidence and misrepresenting our beliefs about aleatoric uncertainty.

In Bayesian regression, we often use a Gaussian observation model, where we control the level of aleatoric uncertainty with a noise variance parameter. By contrast, for Bayesian classification we use a categorical distribution with no mechanism to represent our beliefs about aleatoric uncertainty. Our work shows that explicitly accounting for aleatoric uncertainty significantly improves the performance of Bayesian neural networks. We note that many standard benchmarks, such as CIFAR-10, have essentially no aleatoric uncertainty. Moreover, we show that data augmentation in approximate inference softens the likelihood, leading to underconfidence and misrepresenting our beliefs about aleatoric uncertainty.

Triplet Markov chains are general generative models for sequential data which take into account three kinds of random variables: (noisy) observations, their associated discrete labels and latent variables which aim at strengthening the distribution of the observations and their associated labels. However, in practice, we do not have at our disposal all the labels associated to the observations to estimate the parameters of such models. In this paper, we propose a general framework based on a variational Bayesian inference to train parameterized triplet Markov chain models in a semi-supervised context. The generality of our approach enables us to derive semi-supervised algorithms for a variety of generative models for sequential Bayesian classification.

The current definition of a Bayesian credible set cannot, in general, achieve an arbitrarily preassigned credible level. This drawback is particularly acute for classification problems, where there are only a finite number of achievable credible levels. As a result, there is as of today no general way to construct an exact credible set for classification. In this paper, we introduce a generalized credible set that can achieve any preassigned credible level. The key insight is a simple connection between the Bayesian highest posterior density credible set and the Neyman--Pearson lemma, which, as far as we know, hasn't been noticed before. Using this connection, we introduce a randomized decision rule to fill the gaps among the discrete credible levels. Accompanying this methodology, we also develop the Steering Wheel Plot to represent the credible set, which is useful in visualizing the uncertainty in classification. By developing the exact credible set for discrete parameters, we make the theory of Bayesian inference more complete.

The full Bayesian method for applying neural networks to a pre(cid:173) diction problem is to set up the prior/hyperprior structure for the net and then perform the necessary integrals. However, these inte(cid:173) grals are not tractable analytically, and Markov Chain Monte Carlo (MCMC) methods are slow, especially if the parameter space is high-dimensional. Using Gaussian processes we can approximate the weight space integral analytically, so that only a small number of hyperparameters need be integrated over by MCMC methods. We have applied this idea to classification problems, obtaining ex(cid:173) cellent results on the real-world problems investigated so far .

In Bayesian classification, it is important to establish a probabilistic model for each class for likelihood estimation. Most of the previous methods modeled the probability distribution in the whole sample space. However, real-world problems are usually too complex to model in the whole sample space; some fundamental assumptions are required to simplify the global model, for example, the class conditional independence assumption for naive Bayesian classification. In this paper, with the insight that the distribution in a local sample space should be simpler than that in the whole sample space, a local probabilistic model established for a local region is expected much simpler and can relax the fundamental assumptions that may not be true in the whole sample space. Based on these advantages we propose establishing local probabilistic models for Bayesian classification. In addition, a Bayesian classifier adopting a local probabilistic model can even be viewed as a generalized local classification model; by tuning the size of the local region and the corresponding local model assumption, a fitting model can be established for a particular classification problem. The experimental results on several real-world datasets demonstrate the effectiveness of local probabilistic models for Bayesian classification.

Paul Graham popularized the term "Bayesian Classification" (or more accurately "Naïve Bayesian Classification") after his "A Plan for Spam" article was published (http://www.paulgraham.com/spam.html). In fact, text classifiers based on naïve Bayesian and other techniques have been around for many years. Companies such as Autonomy and Interwoven incorporate machine-learning techniques to automatically classify documents of all kinds; one such machine-learning technique is naïve Bayesian text classification. Naïve Bayesian text classifiers are fast, accurate, simple, and easy to implement. In this article, I present a complete naïve Bayesian text classifier written in 100 lines of commented, nonobfuscated Perl.

The full Bayesian method for applying neural networks to a prediction problem is to set up the prior/hyperprior structure for the net and then perform the necessary integrals. However, these integrals are not tractable analytically, and Markov Chain Monte Carlo (MCMC) methods are slow, especially if the parameter space is high-dimensional. Using Gaussian processes we can approximate the weight space integral analytically, so that only a small number of hyperparameters need be integrated over by MCMC methods. We have applied this idea to classification problems, obtaining excellent results on the real-world problems investigated so far. 1 INTRODUCTION To make predictions based on a set of training data, fundamentally we need to combine our prior beliefs about possible predictive functions with the data at hand. In the Bayesian approach to neural networks a prior on the weights in the net induces a prior distribution over functions.