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.

Unlike theoretical distributed learning (DL), DL over wireless edge networks faces the inherent dynamics/uncertainty of wireless connections and edge nodes, making DL less efficient or even inapplicable under the highly dynamic wireless edge networks (e.g., using mmW interfaces). This article addresses these problems by leveraging recent advances in coded computing and the deep dueling neural network architecture. By introducing coded structures/redundancy, a distributed learning task can be completed without waiting for straggling nodes. Unlike conventional coded computing that only optimizes the code structure, coded distributed learning over the wireless edge also requires to optimize the selection/scheduling of wireless edge nodes with heterogeneous connections, computing capability, and straggling effects. However, even neglecting the aforementioned dynamics/uncertainty, the resulting joint optimization of coding and scheduling to minimize the distributed learning time turns out to be NP-hard. To tackle this and to account for the dynamics and uncertainty of wireless connections and edge nodes, we reformulate the problem as a Markov Decision Process and then design a novel deep reinforcement learning algorithm that employs the deep dueling neural network architecture to find the jointly optimal coding scheme and the best set of edge nodes for different learning tasks without explicit information about the wireless environment and edge nodes' straggling parameters. Simulations show that the proposed framework reduces the average learning delay in wireless edge computing up to 66% compared with other DL approaches. The jointly optimal framework in this article is also applicable to any distributed learning scheme with heterogeneous and uncertain computing nodes.

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.

Inverse reinforcement learning (IRL) is the task of finding a reward function that generates a desired optimal policy for a given Markov Decision Process (MDP). This paper develops an information-theoretic lower bound for the sample complexity of the finite state, finite action IRL problem. A geometric construction of $\beta$-strict separable IRL problems using spherical codes is considered. Properties of the ensemble size as well as the Kullback-Leibler divergence between the generated trajectories are derived. The resulting ensemble is then used along with Fano's inequality to derive a sample complexity lower bound of $O(n \log n)$, where $n$ is the number of states in the MDP.

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.

People battle over which language is better for Statistical modeling, R or python or Julia. But if you know what you are doing, then any language or software is just a tool. Fluency in any language is less important than Concepts themselves. Here is Naive Bayes Learning explained clearly and implemented on Tableau from scratch with Data used-Predicting Churn for Bank Customers. Naive Bayes is a probabilistic model that assigns the probability of an event by calculating the individual probability of the variables.

This work proposes a semi-parametric approach to estimate Covid-19 (SARS-CoV-2) evolution in Spain. Considering the sequences of 14 days cumulative incidence of all Spanish regions, it combines modern Deep Learning (DL) techniques for analyzing sequences with the usual Bayesian Poisson-Gamma model for counts. DL model provides a suitable description of observed sequences but no reliable uncertainty quantification around it can be obtained. To overcome this we use the prediction from DL as an expert elicitation of the expected number of counts along with their uncertainty and thus obtaining the posterior predictive distribution of counts in an orthodox Bayesian analysis using the well known Poisson-Gamma model. The overall resulting model allows us to either predict the future evolution of the sequences on all regions, as well as, estimating the consequences of eventual scenarios.

"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.

Deep neural networks have become an extremely popular learning technique, with significant deployment in a wide variety of practical domains such as computer vision [1, 2, 3], biosignal processing [4, 5], image captioning [6] and speech recognition [7]. With the significant increase in dataset scale and subsequent increase in model complexity of multilayered neural network architectures, it has become imperative to learn networks that can offer performance guarantees, yield good generalization, and provide sparse representations. Vapnik's seminal work in computational learning theory [8] highlighted that a small VC dimension and good generalization go hand in hand, however, minimizing the VC dimension as a function of the weights of the class of networks has remained elusive. The representational redundancy in deep neural networks is well recognized. Some works on model complexity have recently appeared in literature [9, 10]. In many cases, the number of parameters exceeds the amount of training data resulting in severe overfitting [11]. Sontag [12] derived that the VC dimension of neural network with |W | weights is O(|W |log(|W |)), where |W | is the cardinality of total number of weights. Hence, it is essential to reduce the redundancy in weights and neurons and enforce sparsity in the structure to bring down the VC dimension [13, 14]. A number of methods have been proposed in the neural network domain to reduce model complexity [15, 16, 17], but these largely focus on pruning trained networks by removing synapses or neurons through heuristics or by applying sparsity inducing norms (e.g., L