The Panel Vector Autoregressive (PVAR) model is a popular tool for macroeconomic forecasting and structural analysis in multi-country applications since it allows for spillovers between countries in a very flexible fashion. However, this flexibility means that the number of parameters to be estimated can be enormous leading to over-parameterization concerns. Bayesian global-local shrinkage priors, such as the Horseshoe prior used in this paper, can overcome these concerns, but they require the use of Markov Chain Monte Carlo (MCMC) methods rendering them computationally infeasible in high dimensions. In this paper, we develop computationally efficient Bayesian methods for estimating PVARs using an integrated rotated Gaussian approximation (IRGA). This exploits the fact that whereas own country information is often important in PVARs, information on other countries is often unimportant. Using an IRGA, we split the the posterior into two parts: one involving own country coefficients, the other involving other country coefficients. Fast methods such as approximate message passing or variational Bayes can be used on the latter and, conditional on these, the former are estimated with precision using MCMC methods. In a forecasting exercise involving PVARs with up to $18$ variables for each of $38$ countries, we demonstrate that our methods produce good forecasts quickly.

Fuzzy Cognitive Maps (FCMs) are considered a soft computing technique combining elements of fuzzy logic and recurrent neural networks. They found multiple application in such domains as modeling of system behavior, prediction of time series, decision making and process control. Less attention, however, has been turned towards using them in pattern classification. In this work we propose an FCM based classifier with a fully connected map structure. In contrast to methods that expect reaching a steady system state during reasoning, we chose to execute a few FCM iterations (steps) before collecting output labels. Weights were learned with a gradient algorithm and logloss or cross-entropy were used as the cost function. Our primary goal was to verify, whether such design would result in a descent general purpose classifier, with performance comparable to off the shelf classical methods. As the preliminary results were promising, we investigated the hypothesis that the performance of $d$-step classifier can be attributed to a fact that in previous $d-1$ steps it transforms the feature space by grouping observations belonging to a given class, so that they became more compact and separable. To verify this hypothesis we calculated three clustering scores for the transformed feature space. We also evaluated performance of pipelines built from FCM-based data transformer followed by a classification algorithm. The standard statistical analyzes confirmed both the performance of FCM based classifier and its capability to improve data. The supporting prototype software was implemented in Python using TensorFlow library.

Deep generative modelling is a class of techniques that train deep neural networks to model the distribution of training samples. Research has fragmented into various interconnected approaches, each of which making trade-offs including run-time, diversity, and architectural restrictions. In particular, this compendium covers energy-based models, variational autoencoders, generative adversarial networks, autoregressive models, normalizing flows, in addition to numerous hybrid approaches. These techniques are drawn under a single cohesive framework, comparing and contrasting to explain the premises behind each, while reviewing current state-of-the-art advances and implementations.

Unobserved confounding is one of the main challenges when estimating causal effects. We propose a novel causal reduction method that replaces an arbitrary number of possibly high-dimensional latent confounders with a single latent confounder that lives in the same space as the treatment variable without changing the observational and interventional distributions entailed by the causal model. After the reduction, we parameterize the reduced causal model using a flexible class of transformations, so-called normalizing flows. We propose a learning algorithm to estimate the parameterized reduced model jointly from observational and interventional data. This allows us to estimate the causal effect in a principled way from combined data. We perform a series of experiments on data simulated using nonlinear causal mechanisms and find that we can often substantially reduce the number of interventional samples when adding observational training samples without sacrificing accuracy. Thus, adding observational data may help to more accurately estimate causal effects even in the presence of unobserved confounders.

I trained a classifier on images of animals and gave it an image of myself, it's 98% confident I'm a dog. This is an exploration of a possible Bayesian fix. I trained a multi-class classifier on images of cats, dogs and wild animals and passed an image of myself, it's 98% confident I'm a dog. The problem isn't that I passed an inappropriate image because models in the real world are passed all sorts of garbage. It's that the model is overconfident about an image far away from the training data.

Deep learning has been the engine powering many successes of data science. However, the deep neural network (DNN), as the basic model of deep learning, is often excessively over-parameterized, causing many difficulties in training, prediction and interpretation. We propose a frequentist-like method for learning sparse DNNs and justify its consistency under the Bayesian framework: the proposed method could learn a sparse DNN with at most $O(n/\log(n))$ connections and nice theoretical guarantees such as posterior consistency, variable selection consistency and asymptotically optimal generalization bounds. In particular, we establish posterior consistency for the sparse DNN with a mixture Gaussian prior, show that the structure of the sparse DNN can be consistently determined using a Laplace approximation-based marginal posterior inclusion probability approach, and use Bayesian evidence to elicit sparse DNNs learned by an optimization method such as stochastic gradient descent in multiple runs with different initializations. The proposed method is computationally more efficient than standard Bayesian methods for large-scale sparse DNNs. The numerical results indicate that the proposed method can perform very well for large-scale network compression and high-dimensional nonlinear variable selection, both advancing interpretable machine learning.

Artificial intelligence systems, which are designed with a capability to learn from the data presented to them, are used throughout society. These systems are used to screen loan applicants, make sentencing recommendations for criminal defendants, scan social media posts for disallowed content and more. Because these systems don't assign meaning to their complex learned correlation network, they can learn associations that don't equate to causality, resulting in non-optimal and indefensible decisions being made. In addition to making decisions that are sub-optimal, these systems may create legal liability for their designers and operators by learning correlations that violate anti-discrimination and other laws regarding what factors can be used in different types of decision making. This paper presents the use of a machine learning expert system, which is developed with meaning-assigned nodes (facts) and correlations (rules). Multiple potential implementations are considered and evaluated under different conditions, including different network error and augmentation levels and different training levels. The performance of these systems is compared to random and fully connected networks.

Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. This paper introduces the sparse random feature method that learns parsimonious random feature models utilizing techniques from compressive sensing. We provide uniform bounds on the approximation error for functions in a reproducing kernel Hilbert space depending on the number of samples and the distribution of features. The error bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay.

"The key distinguishing property of a Bayesian approach is marginalization instead of optimization, where we represent solutions given by all settings of parameters weighted by their posterior probabilities, rather than bet everything on a single setting of parameters." The time is ripe to dig into marginalization vs optimization, and broaden our general understanding of the Bayesian approach.

The arguments / discussions between the Bayesian vs frequentist approaches in statistics are long running. I am interested in how these approaches impact machine learning. Often, books on machine learning combine the two approaches, or in some cases, take only one approach. This does not help from a learning standpoint. So, in this two-part blog we first discuss the differences between the Frequentist and Bayesian approaches.