The Laplace approximation (LA) of the Bayesian posterior is a Gaussian distribution centered at the maximum a posteriori estimate. Its appeal in Bayesian deep learning stems from the ability to quantify uncertainty post-hoc (i.e., after standard network parameter optimization), the ease of sampling from the approximate posterior, and the analytic form of model evidence. However, an important computational bottleneck of LA is the necessary step of calculating and inverting the Hessian matrix of the log posterior. The Hessian may be approximated in a variety of ways, with quality varying with a number of factors including the network, dataset, and inference task. In this paper, we propose an alternative framework that sidesteps Hessian calculation and inversion. The Hessian-free Laplace (HFL) approximation uses curvature of both the log posterior and network prediction to estimate its variance. Only two point estimates are needed: the standard maximum a posteriori parameter and the optimal parameter under a loss regularized by the network prediction. We show that, under standard assumptions of LA in Bayesian deep learning, HFL targets the same variance as LA, and can be efficiently amortized in a pre-trained network. Experiments demonstrate comparable performance to that of exact and approximate Hessians, with excellent coverage for in-between uncertainty.

Tabular data is common yet typically incomplete, small in volume, and access-restricted due to privacy concerns. Synthetic data generation offers potential solutions. Many metrics exist for evaluating the quality of synthetic tabular data; however, we lack an objective, coherent interpretation of the many metrics. To address this issue, we propose an evaluation framework with a single, mathematical objective that posits that the synthetic data should be drawn from the same distribution as the observed data. Through various structural decomposition of the objective, this framework allows us to reason for the first time the completeness of any set of metrics, as well as unifies existing metrics, including those that stem from fidelity considerations, downstream application, and model-based approaches. Moreover, the framework motivates model-free baselines and a new spectrum of metrics. We evaluate structurally informed synthesizers and synthesizers powered by deep learning. Using our structured framework, we show that synthetic data generators that explicitly represent tabular structure outperform other methods, especially on smaller datasets.

Modern classification tasks usually involve many class labels and can be informed by a broad range of features. Many of these tasks are tackled by constructing a set of classifiers, which are then applied at test time and then pieced together in a fixed procedure determined in advance or at training time. We present an active classification process at the test time, where each classifier in a large ensemble is viewed as a potential observation that might inform our classification process. Observations are then selected dynamically based on previous observations, using a value-theoretic computation that balances an estimate of the expected classification gain from each observation as well as its computational cost. The expected classification gain is computed using a probabilistic model that uses the outcome from previous observations. This active classification process is applied at test time for each individual test instance, resulting in an efficient instance-specific decision path. We demonstrate the benefit of the active scheme on various real-world datasets, and show that it can achieve comparable or even higher classification accuracy at a fraction of the computational costs of traditional methods.

In this paper, we derive a method to refine a Bayes network diagnostic model by exploiting constraints implied by expert decisions on test ordering. At each step, the expert executes an evidence gathering test, which suggests the test's relative diagnostic value. We demonstrate that consistency with an expert's test selection leads to non-convex constraints on the model parameters. We incorporate these constraints by augmenting the network with nodes that represent the constraint likelihoods. Gibbs sampling, stochastic hill climbing and greedy search algorithms are proposed to find a MAP estimate that takes into account test ordering constraints and any data available. We demonstrate our approach on diagnostic sessions from a manufacturing scenario.

This work considers the problem of learning the structure of multivariate linear tree models, which include a variety of directed tree graphical models with continuous, discrete, and mixed latent variables such as linear-Gaussian models, hidden Markov models, Gaussian mixture models, and Markov evolutionary trees. The setting is one where we only have samples from certain observed variables in the tree, and our goal is to estimate the tree structure (i.e., the graph of how the underlying hidden variables are connected to each other and to the observed variables). We propose the Spectral Recursive Grouping algorithm, an efficient and simple bottom-up procedure for recovering the tree structure from independent samples of the observed variables. Our finite sample size bounds for exact recovery of the tree structure reveal certain natural dependencies on underlying statistical and structural properties of the underlying joint distribution. Furthermore, our sample complexity guarantees have no explicit dependence on the dimensionality of the observed variables, making the algorithm applicable to many high-dimensional settings. At the heart of our algorithm is a spectral quartet test for determining the relative topology of a quartet of variables from second-order statistics.

Bayesian filtering of stochastic stimuli has received a great deal of attention recently. It has been applied to describe the way in which biological systems dynamically represent and make decisions about the environment. There have been no exact results for the error in the biologically plausible setting of inference on point process, however. We present an exact analysis of the evolution of the meansquared error in a state estimation task using Gaussian-tuned point processes as sensors. This allows us to study the dynamics of the error of an optimal Bayesian decoder, providing insights into the limits obtainable in this task. This is done for Markovian and a class of non-Markovian Gaussian processes. We find that there is an optimal tuning width for which the error is minimized. This leads to a characterization of the optimal encoding for the setting as a function of the statistics of the stimulus, providing a mathematically sound primer for an ecological theory of sensory processing.

A sizeable literature has focused on the problem of estimating a low-dimensional feature space for a neuron's stimulus sensitivity. However, comparatively little work has addressed the problem of estimating the nonlinear function from feature space to spike rate. Here, we use a Gaussian process (GP) prior over the infinitedimensional space of nonlinear functions to obtain Bayesian estimates of the "nonlinearity" in the linear-nonlinear-Poisson (LNP) encoding model. This approach offers increased flexibility, robustness, and computational tractability compared to traditional methods (e.g., parametric forms, histograms, cubic splines). We then develop a framework for optimal experimental design under the GP-Poisson model using uncertainty sampling.

Nonparametric Bayesian methods are developed for analysis of multi-channel spike-train data, with the feature learning and spike sorting performed jointly. The feature learning and sorting are performed simultaneously across all channels. Dictionary learning is implemented via the beta-Bernoulli process, with spike sorting performed via the dynamic hierarchical Dirichlet process (dHDP), with these two models coupled. The dHDP is augmented to eliminate refractoryperiod violations, it allows the "appearance" and "disappearance" of neurons over time, and it models smooth variation in the spike statistics.

Biased labelers are a systemic problem in crowdsourcing, and a comprehensive toolbox for handling their responses is still being developed. A typical crowdsourcing application can be divided into three steps: data collection, data curation, and learning. At present these steps are often treated separately. We present Bayesian Bias Mitigation for Crowdsourcing (BBMC), a Bayesian model to unify all three. Most data curation methods account for the effects of labeler bias by modeling all labels as coming from a single latent truth.

We derive a plausible learning rule for feedforward, feedback and lateral connections in a recurrent network of spiking neurons. Operating in the context of a generative model for distributions of spike sequences, the learning mechanism is derived from variational inference principles. The synaptic plasticity rules found are interesting in that they are strongly reminiscent of experimental Spike Time Dependent Plasticity, and in that they differ for excitatory and inhibitory neurons. A simulation confirms the method's applicability to learning both stationary and temporal spike patterns.