We present a deep transformation model for probabilistic regression. Deep learning is known for outstandingly accurate predictions on complex data but in regression tasks, it is predominantly used to just predict a single number. This ignores the non-deterministic character of most tasks. Especially if crucial decisions are based on the predictions, like in medical applications, it is essential to quantify the prediction uncertainty. The presented deep learning transformation model estimates the whole conditional probability distribution, which is the most thorough way to capture uncertainty about the outcome. We combine ideas from a statistical transformation model (most likely transformation) with recent transformation models from deep learning (normalizing flows) to predict complex outcome distributions. The core of the method is a parameterized transformation function which can be trained with the usual maximum likelihood framework using gradient descent. The method can be combined with existing deep learning architectures. For small machine learning benchmark datasets, we report state of the art performance for most dataset and partly even outperform it. Our method works for complex input data, which we demonstrate by employing a CNN architecture on image data.

It has recently been established that the numerical solution of ordinary differential equations can be posed as a nonlinear Bayesian inference problem, which can be approximately solved via Gaussian filtering and smoothing, whenever a Gauss--Markov prior is used. In this paper the class of $\nu$ times differentiable linear time invariant Gauss--Markov priors is considered. A taxonomy of Gaussian estimators is established, with the maximum a posteriori estimate at the top of the hierarchy, which can be computed with the iterated extended Kalman smoother. The remaining three classes are termed explicit, semi-implicit, and implicit, which are in similarity with the classical notions corresponding to conditions on the vector field, under which the filter update produces a local maximum a posteriori estimate. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness $\nu+1$. Consequently, using methods from scattered data approximation and nonlinear analysis in Sobolev spaces, it is shown that the maximum a posteriori estimate converges to the true solution at a polynomial rate in the fill-distance (maximum step size) subject to mild conditions on the vector field. The methodology developed provides a novel and more natural approach to study the convergence of these estimators than classical methods of convergence analysis. The methods and theoretical results are demonstrated in numerical examples.

Standard variational lower bounds used to train latent variable models produce biased estimates of most quantities of interest. We introduce an unbiased estimator of the log marginal likelihood and its gradients for latent variable models based on randomized truncation of infinite series. If parameterized by an encoder-decoder architecture, the parameters of the encoder can be optimized to minimize its variance of this estimator. We show that models trained using our estimator give better test-set likelihoods than a standard importance-sampling based approach for the same average computational cost. This estimator also allows use of latent variable models for tasks where unbiased estimators, rather than marginal likelihood lower bounds, are preferred, such as minimizing reverse KL divergences and estimating score functions.

Researchers have long been interested in the role that norms can play in governing agent actions in multi-agent systems. Much work has been done on formalising normative concepts from human society and adapting them for the government of open software systems, and on the simulation of normative processes in human and artificial societies. However, there has been comparatively little work on applying normative MAS mechanisms to understanding the norms in human society. This work investigates this issue in the context of international politics. Using the GDELT dataset, containing machine-encoded records of international events extracted from news reports, we extracted bilateral sequences of inter-country events and applied a Bayesian norm mining mechanism to identify norms that best explained the observed behaviour. A statistical evaluation showed that the normative model fitted the data significantly better than a probabilistic discrete event model.

Assume we have potential "causes" $z\in Z$, which produce "events" $w$ with known probabilities $\beta(w|z)$. We observe $w_1,w_2,...,w_n$, what can we say about the distribution of the causes? A Bayesian estimate will assume a prior on distributions on $Z$ (we assume a Dirichlet prior) and calculate a posterior. An average over that posterior then gives a distribution on $Z$, which estimates how much each cause $z$ contributed to our observations. This is the setting of Latent Dirichlet Allocation, which can be applied e.g. to topics "producing" words in a document. In this setting usually the number of observed words is large, but the number of potential topics is small. We are here interested in applications with many potential "causes" (e.g. locations on the globe), but only a few observations. We show that the exact Bayesian estimate can be computed in linear time (and constant space) in $|Z|$ for a given upper bound on $n$ with a surprisingly simple formula. We generalize this algorithm to the case of sparse probabilities $\beta(w|z)$, in which we only need to assume that the tree width of an "interaction graph" on the observations is limited. On the other hand we also show that without such limitation the problem is NP-hard.

We revisit the long-standing problem of \emph{learning from small sample}. In recent years major efforts have been invested into the generation of new samples from a small set of training data points. Some use classical transformations, others synthesize new examples. Our approach belongs to the second one. We propose a new model based on conditional Generative Latent Optimization (cGLO). Our model learns to synthesize completely new samples for every class just by interpolating between samples in the latent space. The proposed method samples the learned latent space using spherical interpolations (\emph{slerp}) and generates a new sample using the trained generator. Our empirical results show that the new sampled set is diverse enough, leading to improvement in image classification in comparison to the state of the art, when trained on small samples of CIFAR-100 and CUB-200.

We consider inference problems over probabilistic graphical models with aggregate data. In particular, we propose a new efficient belief propagation type algorithm over tree-structured graphs with polynomial computational complexity as well as a global convergence guarantee. This is in contrast to previous methods that either exhibit prohibitive complexity as the population grows or do not guarantee convergence. Our method is based on optimal transport, or more specifically, multi-marginal optimal transport theory. In particular, the inference problem with aggregate observations we consider in this paper can be seen as a structured multi-marginal optimal transport problem, where the cost function decomposes according to the underlying graph. Consequently, the celebrated Sinkhorn algorithm for multi-marginal optimal transport can be leveraged, together with the standard belief propagation algorithm to establish an efficient inference scheme. We demonstrate the performance of our algorithm on applications such as inferring population flow from aggregate observations.

We introduce manifold-modeling flows (MFMFs), a new class of generative models that simultaneously learn the data manifold as well as a tractable probability density on that manifold. Combining aspects of normalizing flows, GANs, autoencoders, and energy-based models, they have the potential to represent data sets with a manifold structure more faithfully and provide handles on dimensionality reduction, denoising, and out-of-distribution detection. We argue why such models should not be trained by maximum likelihood alone and present a new training algorithm that separates manifold and density updates. With two pedagogical examples we demonstrate how manifold-modeling flows let us learn the data manifold and allow for better inference than standard flows in the ambient data space.

The artificial segmentation of an investment management process into a workflow with silos of offline human operators can restrict silos from collectively and adaptively pursuing a unified optimal investment goal. To meet the investor's objectives, an online algorithm can provide an explicit incremental approach that makes sequential updates as data arrives at the process level. This is in stark contrast to offline (or batch) processes that are focused on making component level decisions prior to process level integration. Here we present and report results for an integrated, and online framework for algorithmic portfolio management. This article provides a workflow that can in-turn be embedded into a process level learning framework. The workflow can be enhanced to refine signal generation and asset-class evolution and definitions. Our results confirm that we can use our framework in conjunction with resampling methods to outperform naive market capitalisation benchmarks while making clear the extent of back-test over-fitting. We consider such an online update framework to be a crucial step towards developing intelligent portfolio selection algorithms that integrate financial theory, investor views, and data analysis with process-level learning.

In linear regression models, a fusion of the coefficients is used to identify the predictors having similar relationships with the response. This is called variable fusion. This paper presents a novel variable fusion method in terms of Bayesian linear regression models. We focus on hierarchical Bayesian models based on a spike-and-slab prior approach. A spike-and-slab prior is designed to perform variable fusion. To obtain estimates of parameters, we develop a Gibbs sampler for the parameters. Simulation studies and a real data analysis show that our proposed method has better performances than previous methods.